TPTP Problem File: ITP210^3.p
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%------------------------------------------------------------------------------
% File : ITP210^3 : TPTP v9.0.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00353_010205
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0024_Assertions_00353_010205 [Des22]
% Status : Theorem
% Rating : 0.62 v9.0.0, 0.70 v8.2.0, 0.69 v8.1.0
% Syntax : Number of formulae : 11329 (5578 unt;1603 typ; 0 def)
% Number of atoms : 25969 (11559 equ; 0 cnn)
% Maximal formula atoms : 48 ( 2 avg)
% Number of connectives : 96556 (2581 ~; 330 |;1588 &;82679 @)
% ( 0 <=>;9378 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 124 ( 123 usr)
% Number of type conns : 8510 (8510 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1483 (1480 usr; 132 con; 0-11 aty)
% Number of variables : 25098 (3455 ^;21378 !; 265 ?;25098 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 14:36:05.698
%------------------------------------------------------------------------------
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set_Pr4944855402158566229_o_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
set_Pr2101469702781467981_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
set_Pr8834758594704517033_o_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
produc6271795597528267376eger_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Rat__Orat_M_Eo_J_J,type,
set_Sum_sum_rat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_J,type,
set_Sum_sum_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Rat__Orat_J_J,type,
set_Sum_sum_o_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_J,type,
set_Sum_sum_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Int__Oint_J_J,type,
set_Sum_sum_o_int: $tType ).
thf(ty_n_t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
heap_e7401611519738050253t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
product_prod_num_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
product_prod_nat_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
set_Product_prod_o_o: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Rat__Orat_J_J,type,
set_option_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
set_option_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
set_list_char: $tType ).
thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
set_filter_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Set__Oset_It__Nat__Onat_J_J,type,
filter_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
set_Sum_sum_o_o: $tType ).
thf(ty_n_t__Predicate__Opred_It__Product____Type__Ounit_J,type,
pred_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
list_Code_integer: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
set_Code_integer: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
set_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
option_num: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
set_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Code____Numeral__Onatural,type,
code_natural: $tType ).
thf(ty_n_t__Code____Numeral__Ointeger,type,
code_integer: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Assertions__Oassn,type,
assn: $tType ).
thf(ty_n_t__String__Oliteral,type,
literal: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (1480)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
archim2889992004027027881ng_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
archimedean_frac_rat: rat > rat ).
thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
thf(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: ( produc3658429121746597890et_nat > $o ) > assn ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > produc3658429121746597890et_nat > $o ).
thf(sy_c_Assertions_Oin__range,type,
in_range: produc3658429121746597890et_nat > $o ).
thf(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: produc3658429121746597890et_nat > $o ).
thf(sy_c_Assertions_Oproper,type,
proper: ( produc3658429121746597890et_nat > $o ) > $o ).
thf(sy_c_Assertions_OrelH,type,
relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).
thf(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).
thf(sy_c_Assertions_Owand__assn,type,
wand_assn: assn > assn > assn ).
thf(sy_c_Assertions_Owand__raw,type,
wand_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set_Pr5094982260447487303t_unit ).
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set_Product_prod_o_o ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
bNF_Cardinal_czero_o: set_Product_prod_o_o ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001_Eo,type,
bNF_Ca3172636760949973492d_of_o: set_o > set_Product_prod_o_o ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Product____Type__Ounit,type,
bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
bNF_Ca8331644756375544342r_on_o: set_o > set_Product_prod_o_o > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Nat__Onat,type,
bNF_Ca1281551314933786834on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_Eo_J_J,type,
bNF_id2059982186938784458at_o_o: ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_Eo_J,type,
bNF_id4470576313039291732at_o_o: ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
bNF_id_bnf_o_o_o: ( $o > $o > $o ) > $o > $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_Eo_M_062_It__Int__Oint_M_Eo_J_J,type,
bNF_id_bnf_o_int_o: ( $o > int > $o ) > $o > int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_Eo_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_id_bnf_o_nat_o: ( $o > nat > $o ) > $o > nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_I_Eo_M_Eo_J,type,
bNF_id_bnf_o_o: ( $o > $o ) > $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Int__Oint_M_062_I_Eo_M_Eo_J_J,type,
bNF_id_bnf_int_o_o: ( int > $o > $o ) > int > $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
bNF_id_bnf_int_int_o: ( int > int > $o ) > int > int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_id_bnf_int_nat_o: ( int > nat > $o ) > int > nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Int__Oint_M_Eo_J,type,
bNF_id_bnf_int_o: ( int > $o ) > int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Nat__Onat_M_062_I_Eo_M_Eo_J_J,type,
bNF_id_bnf_nat_o_o: ( nat > $o > $o ) > nat > $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Nat__Onat_M_062_It__Int__Oint_M_Eo_J_J,type,
bNF_id_bnf_nat_int_o: ( nat > int > $o ) > nat > int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_id_bnf_nat_nat_o: ( nat > nat > $o ) > nat > nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_id_bnf_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
bNF_id146787821168362314_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bNF_id5045731369644952736_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_062_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
bNF_id5557877041440844106_nat_o: ( produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
bNF_id1044457264471098170_nat_o: ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Set__Oset_It__Nat__Onat_J_M_062_I_Eo_M_Eo_J_J,type,
bNF_id1382368747416007662at_o_o: ( set_nat > $o > $o ) > set_nat > $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Set__Oset_It__Nat__Onat_J_M_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
bNF_id293821497076346774_nat_o: ( set_nat > set_nat > $o ) > set_nat > set_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bNF_id_bnf_set_nat_o: ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J_J,type,
bNF_id745042670895148362_int_o: ( set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
bNF_id9176111459293603464_int_o: ( set_Pr958786334691620121nt_int > $o ) > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001_Eo,type,
bNF_id_bnf_o: $o > $o ).
thf(sy_c_BNF__Composition_Oid__bnf_001t__Int__Oint,type,
bNF_id_bnf_int: int > int ).
thf(sy_c_BNF__Composition_Oid__bnf_001t__List__Olist_It__Nat__Onat_J,type,
bNF_id_bnf_list_nat: list_nat > list_nat ).
thf(sy_c_BNF__Composition_Oid__bnf_001t__Nat__Onat,type,
bNF_id_bnf_nat: nat > nat ).
thf(sy_c_BNF__Composition_Oid__bnf_001t__Rat__Orat,type,
bNF_id_bnf_rat: rat > rat ).
thf(sy_c_BNF__Def_Oeq__onp_001t__List__Olist_It__String__Ochar_J,type,
bNF_eq_onp_list_char: ( list_char > $o ) > list_char > list_char > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J,type,
bNF_re5598663031114558885t_char: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o ) > ( $o > $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
bNF_re8691941649098027561iteral: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > literal > literal ) > $o ) > ( $o > $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J,type,
bNF_re7347849368244578525t_char: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > list_char > list_char ) > $o ) > ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
bNF_re5719858693005739489iteral: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > literal > literal ) > $o ) > ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J,type,
bNF_re7972843584066424825t_char: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > list_char > list_char ) > $o ) > ( $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
bNF_re136309367175754621iteral: ( $o > $o > $o ) > ( ( $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > literal > literal ) > $o ) > ( $o > $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J,type,
bNF_re796896441190010845t_char: ( $o > $o > $o ) > ( ( $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > list_char > list_char ) > $o ) > ( $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J,type,
bNF_re2365650225897248225iteral: ( $o > $o > $o ) > ( ( $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > literal > literal ) > $o ) > ( $o > $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
bNF_re4583855572895197261t_char: ( $o > $o > $o ) > ( ( $o > $o > list_char > list_char ) > ( $o > $o > list_char > list_char ) > $o ) > ( $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
bNF_re5778109699181701841iteral: ( $o > $o > $o ) > ( ( $o > $o > list_char > list_char ) > ( $o > $o > literal > literal ) > $o ) > ( $o > $o > $o > list_char > list_char ) > ( $o > $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
bNF_re441219624009962717t_char: ( $o > $o > $o ) > ( ( $o > list_char > list_char ) > ( $o > list_char > list_char ) > $o ) > ( $o > $o > list_char > list_char ) > ( $o > $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
bNF_re1595102025518187489iteral: ( $o > $o > $o ) > ( ( $o > list_char > list_char ) > ( $o > literal > literal ) > $o ) > ( $o > $o > list_char > list_char ) > ( $o > $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
bNF_re6157522605288126113t_char: ( $o > $o > $o ) > ( ( list_char > list_char ) > ( list_char > list_char ) > $o ) > ( $o > list_char > list_char ) > ( $o > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001_062_It__String__Oliteral_Mt__String__Oliteral_J,type,
bNF_re6148748351543924773iteral: ( $o > $o > $o ) > ( ( list_char > list_char ) > ( literal > literal ) > $o ) > ( $o > list_char > list_char ) > ( $o > literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint,type,
bNF_re3804157879324367682nt_int: ( int > code_integer > $o ) > ( int > int > $o ) > ( int > int ) > ( code_integer > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re982302072995117890nteger: ( int > int > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( int > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
bNF_re5828396963709814525t_char: ( list_Code_integer > list_Code_integer > $o ) > ( list_char > list_char > $o ) > ( list_Code_integer > list_char ) > ( list_Code_integer > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
bNF_re7775871201573978401char_o: ( list_char > list_char > $o ) > ( ( list_char > $o ) > ( list_char > $o ) > $o ) > ( list_char > list_char > $o ) > ( list_char > list_char > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
bNF_re7399511318543894245t_char: ( list_char > list_char > $o ) > ( ( list_char > list_char ) > ( list_char > list_char ) > $o ) > ( list_char > list_char > list_char ) > ( list_char > list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001_Eo_001_Eo,type,
bNF_re6668474094832959905ar_o_o: ( list_char > list_char > $o ) > ( $o > $o > $o ) > ( list_char > $o ) > ( list_char > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
bNF_re901197953848981245nteger: ( list_char > list_char > $o ) > ( list_Code_integer > list_Code_integer > $o ) > ( list_char > list_Code_integer ) > ( list_char > list_Code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
bNF_re1407854171455731685t_char: ( list_char > list_char > $o ) > ( list_char > list_char > $o ) > ( list_char > list_char ) > ( list_char > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re76688982720252355at_nat: ( list_char > list_char > $o ) > ( nat > nat > $o ) > ( list_char > nat ) > ( list_char > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_001_062_It__String__Oliteral_M_Eo_J,type,
bNF_re7831395832754058691eral_o: ( list_char > literal > $o ) > ( ( list_char > $o ) > ( literal > $o ) > $o ) > ( list_char > list_char > $o ) > ( literal > literal > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001_Eo_001_Eo,type,
bNF_re8730477052009823892al_o_o: ( list_char > literal > $o ) > ( $o > $o > $o ) > ( list_char > $o ) > ( literal > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
bNF_re4991257583289732696t_char: ( list_char > literal > $o ) > ( list_char > list_char > $o ) > ( list_char > list_char ) > ( literal > list_char ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
bNF_re5453467372598631581iteral: ( list_char > literal > $o ) > ( list_char > literal > $o ) > ( list_char > list_char ) > ( literal > literal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
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bij_be5333170631980326235at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_be5432664580149595207at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Fun_Ocomp_001t__String__Ochar_001t__String__Ochar_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Oid_001_Eo,type,
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thf(sy_c_Fun_Oid_001t__Int__Oint,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
id_nat: nat > nat ).
thf(sy_c_Fun_Oid_001t__Num__Onum,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J_J,type,
map_fu1104370778922912482iteral: ( $o > $o ) > ( ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > $o > $o > literal > literal ) > ( $o > $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J_J,type,
map_fu7309004089335677210iteral: ( $o > $o ) > ( ( $o > $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > $o > literal > literal ) > ( $o > $o > $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J_J_J,type,
map_fu1929762555066093878iteral: ( $o > $o ) > ( ( $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > literal > literal ) > ( $o > $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > $o > literal > literal ).
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map_fu5333924603067766298iteral: ( $o > $o ) > ( ( $o > $o > $o > list_char > list_char ) > $o > $o > $o > literal > literal ) > ( $o > $o > $o > $o > list_char > list_char ) > $o > $o > $o > $o > literal > literal ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_J_001_062_I_Eo_M_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J_J,type,
map_fu1888220742600445322iteral: ( $o > $o ) > ( ( $o > $o > list_char > list_char ) > $o > $o > literal > literal ) > ( $o > $o > $o > list_char > list_char ) > $o > $o > $o > literal > literal ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_I_Eo_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_I_Eo_M_062_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
map_fu5700776415251466522iteral: ( $o > $o ) > ( ( $o > list_char > list_char ) > $o > literal > literal ) > ( $o > $o > list_char > list_char ) > $o > $o > literal > literal ).
thf(sy_c_Fun_Omap__fun_001_Eo_001_Eo_001_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_001_062_It__String__Oliteral_Mt__String__Oliteral_J,type,
map_fu4627020735845646302iteral: ( $o > $o ) > ( ( list_char > list_char ) > literal > literal ) > ( $o > list_char > list_char ) > $o > literal > literal ).
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map_fu3424225382358772769nt_rat: ( int > int ) > ( product_prod_int_int > rat ) > ( int > product_prod_int_int ) > int > rat ).
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map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).
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map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
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map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
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map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
thf(sy_c_Fun_Omap__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
map_fu6891787308814931657nteger: ( num > num ) > ( ( num > int ) > num > code_integer ) > ( num > num > int ) > num > num > code_integer ).
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map_fu8638147718074629079nteger: ( num > num ) > ( int > code_integer ) > ( num > int ) > num > code_integer ).
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map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
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map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_001_062_It__String__Oliteral_M_Eo_J,type,
map_fu2661873803724901240eral_o: ( literal > list_char ) > ( ( list_char > $o ) > literal > $o ) > ( list_char > list_char > $o ) > literal > literal > $o ).
thf(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001_Eo_001_Eo,type,
map_fu6698795530259886495ar_o_o: ( literal > list_char ) > ( $o > $o ) > ( list_char > $o ) > literal > $o ).
thf(sy_c_Fun_Omap__fun_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
map_fu7398138089906338514iteral: ( literal > list_char ) > ( list_char > literal ) > ( list_char > list_char ) > literal > literal ).
thf(sy_c_Fun__Def_Omax__strict,type,
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thf(sy_c_Fun__Def_Omax__weak,type,
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thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set_Pr4329608150637261639at_nat ).
thf(sy_c_Fun__Def_Omin__weak,type,
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thf(sy_c_Fun__Def_Opair__leq,type,
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thf(sy_c_Fun__Def_Opair__less,type,
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thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
gcd_Gcd_int: set_int > int ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_GCD_OGcd__class_OLcm_001t__Int__Oint,type,
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thf(sy_c_GCD_OGcd__class_OLcm_001t__Nat__Onat,type,
gcd_Lcm_nat: set_nat > nat ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_GCD_Obezw__rel,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
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thf(sy_c_GCD_Ogcd__class_Olcm_001t__Int__Oint,type,
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thf(sy_c_GCD_Ogcd__class_Olcm_001t__Nat__Onat,type,
gcd_lcm_nat: nat > nat > nat ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ocomm__monoid_001t__Assertions__Oassn,type,
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thf(sy_c_Groups_Ocomm__monoid_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ocomm__monoid_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ocomm__monoid_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ocomm__monoid_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
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sup_su1630790145277462993_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
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sup_su8003631835303541148at_o_o: ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
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sup_su4182031696650224058_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).
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sup_sup_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
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thf(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
sup_sup_assn: assn > assn > assn ).
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thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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sup_sup_set_int: set_int > set_int > set_int ).
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sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
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lattic7826324295020591184_F_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Lifting_OQuotient_001t__List__Olist_It__String__Ochar_J_001t__String__Oliteral,type,
quotie6109894551476798619iteral: ( list_char > list_char > $o ) > ( list_char > literal ) > ( literal > list_char ) > ( list_char > literal > $o ) > $o ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__String__Ochar,type,
append_char: list_char > list_char > list_char ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat2: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
fold_int_int: ( int > int > int ) > list_int > int > int ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__String__Ochar,type,
cons_char: char > list_char > list_char ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__String__Ochar,type,
nil_char: list_char ).
thf(sy_c_List_Olist_Olist__all2_001t__String__Ochar_001t__String__Ochar,type,
list_all2_char_char: ( char > char > $o ) > list_char > list_char > $o ).
thf(sy_c_List_Olist_Olist__all_001t__String__Ochar,type,
list_all_char: ( char > $o ) > list_char > $o ).
thf(sy_c_List_Olist_Omap_001t__Code____Numeral__Ointeger_001t__String__Ochar,type,
map_Co843364623960961780r_char: ( code_integer > char ) > list_Code_integer > list_char ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
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map_ch5843194146513361140nteger: ( char > code_integer ) > list_char > list_Code_integer ).
thf(sy_c_List_Olist_Omap_001t__String__Ochar_001t__String__Ochar,type,
map_char_char: ( char > char ) > list_char > list_char ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__String__Ochar,type,
set_char2: list_char > set_char ).
thf(sy_c_List_Oord_Olexordp_001t__String__Ochar,type,
lexordp_char: ( char > char > $o ) > list_char > list_char > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__String__Ochar,type,
lexordp_eq_char: ( char > char > $o ) > list_char > list_char > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Misc_Obijective_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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pairse6848479906795794847e_term: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ).
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rel_re3331361160397388719_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o ).
thf(sy_c_Misc_Orel__restrict_001_Eo,type,
rel_restrict_o: set_Product_prod_o_o > set_o > set_Product_prod_o_o ).
thf(sy_c_Misc_Orel__restrict_001t__Int__Oint,type,
rel_restrict_int: set_Pr958786334691620121nt_int > set_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Misc_Orel__restrict_001t__Nat__Onat,type,
rel_restrict_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).
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rel_re2197483123211154633at_nat: set_Pr4759807026041935047at_nat > set_Pr8551490117392284871at_nat > set_Pr4759807026041935047at_nat ).
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rel_re2767227651283840073at_nat: set_Pr5564308138774400199at_nat > set_Pr4329608150637261639at_nat > set_Pr5564308138774400199at_nat ).
thf(sy_c_Misc_Orel__restrict_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
uncurry_int_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
uncurry_int_int_int: ( int > int > int ) > product_prod_int_int > int ).
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thf(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Multiset_Oms__strict,type,
ms_strict: set_Pr8551490117392284871at_nat ).
thf(sy_c_Multiset_Oms__weak,type,
ms_weak: set_Pr8551490117392284871at_nat ).
thf(sy_c_Multiset_Opw__leq,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
semiri681578069525770553at_rat: nat > rat ).
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Ochar_J,type,
size_size_list_char: list_char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Oliteral,type,
size_size_literal: literal > nat ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > list_nat ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list_nat > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set_nat > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > sum_sum_nat_nat ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: sum_sum_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
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neg_nu3811975205180677377ec_int: int > int ).
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neg_nu5851722552734809277nc_int: int > int ).
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neg_numeral_sub_rat: num > num > rat ).
thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
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numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
numeral_numeral_rat: num > rat ).
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ring_1_iszero_rat: rat > $o ).
thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
some_num: num > option_num ).
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thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
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type_d4410041424927559462al_nat: ( code_natural > nat ) > ( nat > code_natural ) > set_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Int__Oint_001t__Int__Oint,type,
type_d7247357190169752966nt_int: ( int > int ) > ( int > int ) > set_int > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
type_d8752208705193531015at_nat: ( int > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > int ) > set_se7855581050983116737at_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
type_d5972188401318659438st_nat: ( list_nat > list_nat ) > ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Nat__Onat,type,
type_d6250493948777748686at_nat: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Rat__Orat_001t__Rat__Orat,type,
type_d5298809244756387038at_rat: ( rat > rat ) > ( rat > rat ) > set_rat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Rat__Orat_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
type_d8554052265237484179nt_int: ( rat > set_Pr958786334691620121nt_int ) > ( set_Pr958786334691620121nt_int > rat ) > set_se6260736226359567993nt_int > $o ).
thf(sy_c_Typedef_Otype__definition_001t__String__Oliteral_001t__List__Olist_It__String__Ochar_J,type,
type_d4752411451802217481t_char: ( literal > list_char ) > ( list_char > literal ) > set_list_char > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
accp_P8928870874622223812nt_int: ( produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ) > produc7773217078559923341nt_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
member6576561426505652726_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Filter__Ofilter_It__Nat__Onat_J,type,
member_filter_nat: filter_nat > set_filter_nat > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
member8781333585448626064_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member6689249552917799696at_nat: produc4166570645942440679at_nat > set_Pr8551490117392284871at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member4095101504841534314at_nat: produc732395585841259969at_nat > set_Pr457366540195662369at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member5325734588016868240nt_int: produc412284730452459111nt_int > set_Pr8057934254988874055nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_c_member_001t__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_v_x,type,
x: assn ).
thf(sy_v_y,type,
y: assn ).
% Relevant facts (9688)
thf(fact_0_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_1_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_2_Abs__assn__eqI_I2_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
( ! [H: produc3658429121746597890et_nat] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_3_Abs__assn__eqI_I1_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
( ! [H: produc3658429121746597890et_nat] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_4_wand__assn__def,axiom,
( wand_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% wand_assn_def
thf(fact_5_Abs__assn__inverse,axiom,
! [Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ Y ) )
= Y ) ) ).
% Abs_assn_inverse
thf(fact_6_inf__assn__def,axiom,
( inf_inf_assn
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ P2 @ H2 )
& ( rep_assn @ Q @ H2 ) ) ) ) ) ).
% inf_assn_def
thf(fact_7_sup__assn__def,axiom,
( sup_sup_assn
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ P2 @ H2 )
| ( rep_assn @ Q @ H2 ) ) ) ) ) ).
% sup_assn_def
thf(fact_8_bot__assn__def,axiom,
( bot_bot_assn
= ( abs_assn
@ ^ [Uu: produc3658429121746597890et_nat] : $false ) ) ).
% bot_assn_def
thf(fact_9_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: produc3658429121746597890et_nat > $o] :
( ( X
= ( abs_assn @ Y2 ) )
=> ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_10_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
=> ( P @ ( abs_assn @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_assn_induct
thf(fact_11_Abs__assn__inject,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( ( abs_assn @ X )
= ( abs_assn @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_assn_inject
thf(fact_12_Rep__assn,axiom,
! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).
% Rep_assn
thf(fact_13_Rep__assn__cases,axiom,
! [Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ~ ! [X2: assn] :
( Y
!= ( rep_assn @ X2 ) ) ) ).
% Rep_assn_cases
thf(fact_14_Rep__assn__induct,axiom,
! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ! [X2: assn] : ( P @ ( rep_assn @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_15_times__assn__def,axiom,
( times_times_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% times_assn_def
thf(fact_16_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu: produc3658429121746597890et_nat] : $false ) ).
% bool_assn_proper(2)
thf(fact_17_bool__assn__proper_I3_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H2: produc3658429121746597890et_nat] :
( ( P @ H2 )
| ( Q2 @ H2 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_18_bool__assn__proper_I4_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H2: produc3658429121746597890et_nat] :
( ( P @ H2 )
& ( Q2 @ H2 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_19_wand__proper,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% wand_proper
thf(fact_20_times__assn__proper,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% times_assn_proper
thf(fact_21_assn__times__comm,axiom,
( times_times_assn
= ( ^ [P2: assn,Q: assn] : ( times_times_assn @ Q @ P2 ) ) ) ).
% assn_times_comm
thf(fact_22_assn__times__assoc,axiom,
! [P: assn,Q2: assn,R: assn] :
( ( times_times_assn @ ( times_times_assn @ P @ Q2 ) @ R )
= ( times_times_assn @ P @ ( times_times_assn @ Q2 @ R ) ) ) ).
% assn_times_assoc
thf(fact_23_mod__starE,axiom,
! [A: assn,B: assn,H3: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A @ B ) @ H3 )
=> ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
=> ! [H_2: produc3658429121746597890et_nat] :
~ ( rep_assn @ B @ H_2 ) ) ) ).
% mod_starE
thf(fact_24_mod__starD,axiom,
! [A2: assn,B2: assn,H3: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A2 @ B2 ) @ H3 )
=> ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
( ( rep_assn @ A2 @ H1 )
& ( rep_assn @ B2 @ H22 ) ) ) ).
% mod_starD
thf(fact_25_inf__sup__absorb,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_26_inf__sup__absorb,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_27_inf__sup__absorb,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( sup_sup_list_char_o @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_28_inf__sup__absorb,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_29_inf__sup__absorb,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_30_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_31_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_32_inf__sup__absorb,axiom,
! [X: int,Y: int] :
( ( inf_inf_int @ X @ ( sup_sup_int @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_33_sup__inf__absorb,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_34_sup__inf__absorb,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_35_sup__inf__absorb,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( sup_sup_list_char_o @ X @ ( inf_inf_list_char_o @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_36_sup__inf__absorb,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_37_sup__inf__absorb,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_38_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_39_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_40_sup__inf__absorb,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( inf_inf_int @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_41_sup__bot__left,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_42_sup__bot__left,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_43_sup__bot__left,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_44_sup__bot__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X )
= X ) ).
% sup_bot_left
thf(fact_45_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_46_sup__bot__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ X )
= X ) ).
% sup_bot_left
thf(fact_47_sup__bot__right,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ bot_bo8422036546324065075at_nat )
= X ) ).
% sup_bot_right
thf(fact_48_sup__bot__right,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
= X ) ).
% sup_bot_right
thf(fact_49_sup__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
= X ) ).
% sup_bot_right
thf(fact_50_sup__bot__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% sup_bot_right
thf(fact_51_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_52_sup__bot__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% sup_bot_right
thf(fact_53_bot__eq__sup__iff,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( bot_bo8422036546324065075at_nat
= ( sup_su3035147773818789531at_nat @ X @ Y ) )
= ( ( X = bot_bo8422036546324065075at_nat )
& ( Y = bot_bo8422036546324065075at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_54_bot__eq__sup__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat
= ( sup_su5525570899277871387at_nat @ X @ Y ) )
= ( ( X = bot_bo228742789529271731at_nat )
& ( Y = bot_bo228742789529271731at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_55_bot__eq__sup__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( sup_su6327502436637775413at_nat @ X @ Y ) )
= ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = bot_bo2099793752762293965at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_56_bot__eq__sup__iff,axiom,
! [X: set_o,Y: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X @ Y ) )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_57_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_58_bot__eq__sup__iff,axiom,
! [X: set_int,Y: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ X @ Y ) )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% bot_eq_sup_iff
thf(fact_59_sup__eq__bot__iff,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( ( sup_su3035147773818789531at_nat @ X @ Y )
= bot_bo8422036546324065075at_nat )
= ( ( X = bot_bo8422036546324065075at_nat )
& ( Y = bot_bo8422036546324065075at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_60_sup__eq__bot__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( ( sup_su5525570899277871387at_nat @ X @ Y )
= bot_bo228742789529271731at_nat )
= ( ( X = bot_bo228742789529271731at_nat )
& ( Y = bot_bo228742789529271731at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_61_sup__eq__bot__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( sup_su6327502436637775413at_nat @ X @ Y )
= bot_bo2099793752762293965at_nat )
= ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_62_sup__eq__bot__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= bot_bot_set_o )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_63_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_64_sup__eq__bot__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( sup_sup_set_int @ X @ Y )
= bot_bot_set_int )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% sup_eq_bot_iff
thf(fact_65_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ( sup_su3035147773818789531at_nat @ A @ B )
= bot_bo8422036546324065075at_nat )
= ( ( A = bot_bo8422036546324065075at_nat )
& ( B = bot_bo8422036546324065075at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_66_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ( sup_su5525570899277871387at_nat @ A @ B )
= bot_bo228742789529271731at_nat )
= ( ( A = bot_bo228742789529271731at_nat )
& ( B = bot_bo228742789529271731at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_67_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( sup_su6327502436637775413at_nat @ A @ B )
= bot_bo2099793752762293965at_nat )
= ( ( A = bot_bo2099793752762293965at_nat )
& ( B = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_68_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_69_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_70_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( sup_sup_set_int @ A @ B )
= bot_bot_set_int )
= ( ( A = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_71_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_72_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_73_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_74_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_75_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_76_sup__bot_Oleft__neutral,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_77_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( bot_bo8422036546324065075at_nat
= ( sup_su3035147773818789531at_nat @ A @ B ) )
= ( ( A = bot_bo8422036546324065075at_nat )
& ( B = bot_bo8422036546324065075at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_78_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat
= ( sup_su5525570899277871387at_nat @ A @ B ) )
= ( ( A = bot_bo228742789529271731at_nat )
& ( B = bot_bo228742789529271731at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_79_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( sup_su6327502436637775413at_nat @ A @ B ) )
= ( ( A = bot_bo2099793752762293965at_nat )
& ( B = bot_bo2099793752762293965at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_80_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B ) )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_81_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_82_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_int,B: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ A @ B ) )
= ( ( A = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_83_sup__bot_Oright__neutral,axiom,
! [A: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A @ bot_bo8422036546324065075at_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_84_sup__bot_Oright__neutral,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_85_sup__bot_Oright__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_86_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_87_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_88_sup__bot_Oright__neutral,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ bot_bot_set_int )
= A ) ).
% sup_bot.right_neutral
thf(fact_89_inf__bot__left,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ bot_bo8147686125503663512_int_o @ X )
= bot_bo8147686125503663512_int_o ) ).
% inf_bot_left
thf(fact_90_inf__bot__left,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ bot_bot_list_char_o @ X )
= bot_bot_list_char_o ) ).
% inf_bot_left
thf(fact_91_inf__bot__left,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
= bot_bo2099793752762293965at_nat ) ).
% inf_bot_left
thf(fact_92_inf__bot__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_93_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_94_inf__bot__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ X )
= bot_bot_set_int ) ).
% inf_bot_left
thf(fact_95_inf__bot__right,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ bot_bo8147686125503663512_int_o )
= bot_bo8147686125503663512_int_o ) ).
% inf_bot_right
thf(fact_96_inf__bot__right,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ bot_bot_list_char_o )
= bot_bot_list_char_o ) ).
% inf_bot_right
thf(fact_97_inf__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
= bot_bo2099793752762293965at_nat ) ).
% inf_bot_right
thf(fact_98_inf__bot__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_99_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_100_inf__bot__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ bot_bot_set_int )
= bot_bot_set_int ) ).
% inf_bot_right
thf(fact_101_inf__apply,axiom,
( inf_in3604695632404883862_int_o
= ( ^ [F: product_prod_int_int > $o,G: product_prod_int_int > $o,X3: product_prod_int_int] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_apply
thf(fact_102_inf__apply,axiom,
( inf_inf_list_char_o
= ( ^ [F: list_char > $o,G: list_char > $o,X3: list_char] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_apply
thf(fact_103_inf__right__idem,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y )
= ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_104_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_105_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_106_inf__right__idem,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y )
= ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_107_inf__right__idem,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Y )
= ( inf_inf_list_char_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_108_inf_Oright__idem,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B )
= ( inf_in2572325071724192079at_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_109_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_110_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_111_inf_Oright__idem,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ B )
= ( inf_in3604695632404883862_int_o @ A @ B ) ) ).
% inf.right_idem
thf(fact_112_inf_Oright__idem,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ B )
= ( inf_inf_list_char_o @ A @ B ) ) ).
% inf.right_idem
thf(fact_113_inf__left__idem,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
= ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_114_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_115_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_116_inf__left__idem,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
= ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_117_inf__left__idem,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ X @ Y ) )
= ( inf_inf_list_char_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_118_inf_Oleft__idem,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ A @ B ) )
= ( inf_in2572325071724192079at_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_119_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_120_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_121_inf_Oleft__idem,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ A @ B ) )
= ( inf_in3604695632404883862_int_o @ A @ B ) ) ).
% inf.left_idem
thf(fact_122_inf_Oleft__idem,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( inf_inf_list_char_o @ A @ ( inf_inf_list_char_o @ A @ B ) )
= ( inf_inf_list_char_o @ A @ B ) ) ).
% inf.left_idem
thf(fact_123_inf__idem,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_124_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_125_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_126_inf__idem,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_127_inf__idem,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_128_inf_Oidem,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_129_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_130_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_131_inf_Oidem,axiom,
! [A: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ A @ A )
= A ) ).
% inf.idem
thf(fact_132_inf_Oidem,axiom,
! [A: list_char > $o] :
( ( inf_inf_list_char_o @ A @ A )
= A ) ).
% inf.idem
thf(fact_133_mem__Collect__eq,axiom,
! [A: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( member2340774599025711042nt_int @ A @ ( collec5210948495886036740nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_143_Collect__cong,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ! [X2: set_Pr958786334691620121nt_int] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collec5210948495886036740nt_int @ P )
= ( collec5210948495886036740nt_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_144_Collect__cong,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_145_Collect__cong,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_146_Collect__cong,axiom,
! [P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_147_Collect__cong,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ! [X2: produc3658429121746597890et_nat > $o] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collec939566748876313656_nat_o @ P )
= ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_cong
thf(fact_148_sup_Oright__idem,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ B )
= ( sup_su3035147773818789531at_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_149_sup_Oright__idem,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ B )
= ( sup_su5525570899277871387at_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_150_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_151_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_152_sup_Oright__idem,axiom,
! [A: int,B: int] :
( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ B )
= ( sup_sup_int @ A @ B ) ) ).
% sup.right_idem
thf(fact_153_sup__left__idem,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
= ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_154_sup__left__idem,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
= ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_155_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_156_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_157_sup__left__idem,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_158_sup_Oleft__idem,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ A @ B ) )
= ( sup_su3035147773818789531at_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_159_sup_Oleft__idem,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) )
= ( sup_su5525570899277871387at_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_160_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_161_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_162_sup_Oleft__idem,axiom,
! [A: int,B: int] :
( ( sup_sup_int @ A @ ( sup_sup_int @ A @ B ) )
= ( sup_sup_int @ A @ B ) ) ).
% sup.left_idem
thf(fact_163_sup__idem,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_164_sup__idem,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_165_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_166_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_167_sup__idem,axiom,
! [X: int] :
( ( sup_sup_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_168_sup_Oidem,axiom,
! [A: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_169_sup_Oidem,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_170_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_171_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_172_sup_Oidem,axiom,
! [A: int] :
( ( sup_sup_int @ A @ A )
= A ) ).
% sup.idem
thf(fact_173_inf__fun__def,axiom,
( inf_in3604695632404883862_int_o
= ( ^ [F: product_prod_int_int > $o,G: product_prod_int_int > $o,X3: product_prod_int_int] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_174_inf__fun__def,axiom,
( inf_inf_list_char_o
= ( ^ [F: list_char > $o,G: list_char > $o,X3: list_char] : ( inf_inf_o @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_175_inf__left__commute,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_176_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_177_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_178_inf__left__commute,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( inf_in3604695632404883862_int_o @ Y @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_179_inf__left__commute,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( inf_inf_list_char_o @ Y @ ( inf_inf_list_char_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_180_inf_Oleft__commute,axiom,
! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ B @ ( inf_in2572325071724192079at_nat @ A @ C ) )
= ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_181_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_182_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_183_inf_Oleft__commute,axiom,
! [B: product_prod_int_int > $o,A: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ B @ ( inf_in3604695632404883862_int_o @ A @ C ) )
= ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_184_inf_Oleft__commute,axiom,
! [B: list_char > $o,A: list_char > $o,C: list_char > $o] :
( ( inf_inf_list_char_o @ B @ ( inf_inf_list_char_o @ A @ C ) )
= ( inf_inf_list_char_o @ A @ ( inf_inf_list_char_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_185_inf__commute,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_186_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_187_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_188_inf__commute,axiom,
( inf_in3604695632404883862_int_o
= ( ^ [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_189_inf__commute,axiom,
( inf_inf_list_char_o
= ( ^ [X3: list_char > $o,Y3: list_char > $o] : ( inf_inf_list_char_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_190_inf_Ocommute,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_191_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_192_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A3: nat,B3: nat] : ( inf_inf_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_193_inf_Ocommute,axiom,
( inf_in3604695632404883862_int_o
= ( ^ [A3: product_prod_int_int > $o,B3: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_194_inf_Ocommute,axiom,
( inf_inf_list_char_o
= ( ^ [A3: list_char > $o,B3: list_char > $o] : ( inf_inf_list_char_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_195_inf__assoc,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
= ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_196_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_197_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_198_inf__assoc,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
= ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_199_inf__assoc,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Z )
= ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_200_inf_Oassoc,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C )
= ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_201_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_202_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_203_inf_Oassoc,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C )
= ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ).
% inf.assoc
thf(fact_204_inf_Oassoc,axiom,
! [A: list_char > $o,B: list_char > $o,C: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ C )
= ( inf_inf_list_char_o @ A @ ( inf_inf_list_char_o @ B @ C ) ) ) ).
% inf.assoc
thf(fact_205_inf__sup__aci_I1_J,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_206_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_207_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_208_inf__sup__aci_I1_J,axiom,
( inf_in3604695632404883862_int_o
= ( ^ [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_209_inf__sup__aci_I1_J,axiom,
( inf_inf_list_char_o
= ( ^ [X3: list_char > $o,Y3: list_char > $o] : ( inf_inf_list_char_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_210_inf__sup__aci_I2_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
= ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_211_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_212_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_213_inf__sup__aci_I2_J,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
= ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_214_inf__sup__aci_I2_J,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Z )
= ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_215_inf__sup__aci_I3_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_216_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_217_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_218_inf__sup__aci_I3_J,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( inf_in3604695632404883862_int_o @ Y @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_219_inf__sup__aci_I3_J,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( inf_inf_list_char_o @ Y @ ( inf_inf_list_char_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_220_inf__sup__aci_I4_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
= ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_221_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_222_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_223_inf__sup__aci_I4_J,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
= ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_224_inf__sup__aci_I4_J,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ X @ Y ) )
= ( inf_inf_list_char_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_225_sup__left__commute,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
= ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_226_sup__left__commute,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
= ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_227_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_228_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_229_sup__left__commute,axiom,
! [X: int,Y: int,Z: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_230_sup_Oleft__commute,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ C ) )
= ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_231_sup_Oleft__commute,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ C ) )
= ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_232_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_233_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_234_sup_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( sup_sup_int @ B @ ( sup_sup_int @ A @ C ) )
= ( sup_sup_int @ A @ ( sup_sup_int @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_235_sup__commute,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_236_sup__commute,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_237_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_238_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_239_sup__commute,axiom,
( sup_sup_int
= ( ^ [X3: int,Y3: int] : ( sup_sup_int @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_240_sup_Ocommute,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_241_sup_Ocommute,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_242_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_243_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_244_sup_Ocommute,axiom,
( sup_sup_int
= ( ^ [A3: int,B3: int] : ( sup_sup_int @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_245_sup__assoc,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
= ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_246_sup__assoc,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
= ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_247_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_248_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_249_sup__assoc,axiom,
! [X: int,Y: int,Z: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_250_sup_Oassoc,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ C )
= ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_251_sup_Oassoc,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C )
= ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_252_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_253_sup_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_254_sup_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ C )
= ( sup_sup_int @ A @ ( sup_sup_int @ B @ C ) ) ) ).
% sup.assoc
thf(fact_255_inf__sup__aci_I5_J,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_256_inf__sup__aci_I5_J,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_257_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_258_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_259_inf__sup__aci_I5_J,axiom,
( sup_sup_int
= ( ^ [X3: int,Y3: int] : ( sup_sup_int @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_260_inf__sup__aci_I6_J,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
= ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_261_inf__sup__aci_I6_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
= ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_262_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_263_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_264_inf__sup__aci_I6_J,axiom,
! [X: int,Y: int,Z: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_265_inf__sup__aci_I7_J,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
= ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_266_inf__sup__aci_I7_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
= ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_267_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_268_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_269_inf__sup__aci_I7_J,axiom,
! [X: int,Y: int,Z: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_270_inf__sup__aci_I8_J,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
= ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_271_inf__sup__aci_I8_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
= ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_272_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_273_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_274_inf__sup__aci_I8_J,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_275_sup__inf__distrib2,axiom,
! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_276_sup__inf__distrib2,axiom,
! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z ) @ X )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_277_sup__inf__distrib2,axiom,
! [Y: list_char > $o,Z: list_char > $o,X: list_char > $o] :
( ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ Y @ Z ) @ X )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ Y @ X ) @ ( sup_sup_list_char_o @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_278_sup__inf__distrib2,axiom,
! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z ) @ X )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_279_sup__inf__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z ) @ X )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_280_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_281_sup__inf__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_282_sup__inf__distrib2,axiom,
! [Y: int,Z: int,X: int] :
( ( sup_sup_int @ ( inf_inf_int @ Y @ Z ) @ X )
= ( inf_inf_int @ ( sup_sup_int @ Y @ X ) @ ( sup_sup_int @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_283_sup__inf__distrib1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_284_sup__inf__distrib1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_285_sup__inf__distrib1,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( sup_sup_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ X @ Y ) @ ( sup_sup_list_char_o @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_286_sup__inf__distrib1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_287_sup__inf__distrib1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_288_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_289_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_290_sup__inf__distrib1,axiom,
! [X: int,Y: int,Z: int] :
( ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_291_inf__sup__distrib2,axiom,
! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_292_inf__sup__distrib2,axiom,
! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z ) @ X )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_293_inf__sup__distrib2,axiom,
! [Y: list_char > $o,Z: list_char > $o,X: list_char > $o] :
( ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ Y @ Z ) @ X )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ Y @ X ) @ ( inf_inf_list_char_o @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_294_inf__sup__distrib2,axiom,
! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_295_inf__sup__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_296_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_297_inf__sup__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_298_inf__sup__distrib2,axiom,
! [Y: int,Z: int,X: int] :
( ( inf_inf_int @ ( sup_sup_int @ Y @ Z ) @ X )
= ( sup_sup_int @ ( inf_inf_int @ Y @ X ) @ ( inf_inf_int @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_299_inf__sup__distrib1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_300_inf__sup__distrib1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_301_inf__sup__distrib1,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( sup_sup_list_char_o @ Y @ Z ) )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ ( inf_inf_list_char_o @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_302_inf__sup__distrib1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_303_inf__sup__distrib1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_304_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_305_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_306_inf__sup__distrib1,axiom,
! [X: int,Y: int,Z: int] :
( ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) )
= ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_307_distrib__imp2,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y2 @ Z2 ) )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ Y2 ) @ ( sup_su6327502436637775413at_nat @ X2 @ Z2 ) ) )
=> ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_308_distrib__imp2,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ X2 @ ( inf_in3604695632404883862_int_o @ Y2 @ Z2 ) )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X2 @ Y2 ) @ ( sup_su8463660629351352368_int_o @ X2 @ Z2 ) ) )
=> ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_309_distrib__imp2,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ! [X2: list_char > $o,Y2: list_char > $o,Z2: list_char > $o] :
( ( sup_sup_list_char_o @ X2 @ ( inf_inf_list_char_o @ Y2 @ Z2 ) )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ X2 @ Y2 ) @ ( sup_sup_list_char_o @ X2 @ Z2 ) ) )
=> ( ( inf_inf_list_char_o @ X @ ( sup_sup_list_char_o @ Y @ Z ) )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ ( inf_inf_list_char_o @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_310_distrib__imp2,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X2 @ ( inf_in1697001100524423349at_nat @ Y2 @ Z2 ) )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X2 @ Y2 ) @ ( sup_su3035147773818789531at_nat @ X2 @ Z2 ) ) )
=> ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_311_distrib__imp2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ Y2 @ Z2 ) )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y2 ) @ ( sup_su5525570899277871387at_nat @ X2 @ Z2 ) ) )
=> ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_312_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_313_distrib__imp2,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y2: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_314_distrib__imp2,axiom,
! [X: int,Y: int,Z: int] :
( ! [X2: int,Y2: int,Z2: int] :
( ( sup_sup_int @ X2 @ ( inf_inf_int @ Y2 @ Z2 ) )
= ( inf_inf_int @ ( sup_sup_int @ X2 @ Y2 ) @ ( sup_sup_int @ X2 @ Z2 ) ) )
=> ( ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) )
= ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_315_distrib__imp1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y2 @ Z2 ) )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Z2 ) ) )
=> ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_316_distrib__imp1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X2 @ ( sup_su8463660629351352368_int_o @ Y2 @ Z2 ) )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X2 @ Y2 ) @ ( inf_in3604695632404883862_int_o @ X2 @ Z2 ) ) )
=> ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_317_distrib__imp1,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ! [X2: list_char > $o,Y2: list_char > $o,Z2: list_char > $o] :
( ( inf_inf_list_char_o @ X2 @ ( sup_sup_list_char_o @ Y2 @ Z2 ) )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ X2 @ Y2 ) @ ( inf_inf_list_char_o @ X2 @ Z2 ) ) )
=> ( ( sup_sup_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ X @ Y ) @ ( sup_sup_list_char_o @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_318_distrib__imp1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ X2 @ ( sup_su3035147773818789531at_nat @ Y2 @ Z2 ) )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X2 @ Y2 ) @ ( inf_in1697001100524423349at_nat @ X2 @ Z2 ) ) )
=> ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_319_distrib__imp1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y2 @ Z2 ) )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y2 ) @ ( inf_in7913087082777306421at_nat @ X2 @ Z2 ) ) )
=> ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_320_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_321_distrib__imp1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y2: nat,Z2: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_322_distrib__imp1,axiom,
! [X: int,Y: int,Z: int] :
( ! [X2: int,Y2: int,Z2: int] :
( ( inf_inf_int @ X2 @ ( sup_sup_int @ Y2 @ Z2 ) )
= ( sup_sup_int @ ( inf_inf_int @ X2 @ Y2 ) @ ( inf_inf_int @ X2 @ Z2 ) ) )
=> ( ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_323_boolean__algebra_Oconj__zero__right,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ bot_bo8147686125503663512_int_o )
= bot_bo8147686125503663512_int_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_324_boolean__algebra_Oconj__zero__right,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ bot_bot_list_char_o )
= bot_bot_list_char_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_325_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
= bot_bo2099793752762293965at_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_326_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_327_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_328_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ bot_bot_set_int )
= bot_bot_set_int ) ).
% boolean_algebra.conj_zero_right
thf(fact_329_boolean__algebra_Oconj__zero__left,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ bot_bo8147686125503663512_int_o @ X )
= bot_bo8147686125503663512_int_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_330_boolean__algebra_Oconj__zero__left,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ bot_bot_list_char_o @ X )
= bot_bot_list_char_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_331_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
= bot_bo2099793752762293965at_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_332_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_333_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_334_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ X )
= bot_bot_set_int ) ).
% boolean_algebra.conj_zero_left
thf(fact_335_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_336_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z ) @ X )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_337_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: list_char > $o,Z: list_char > $o,X: list_char > $o] :
( ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ Y @ Z ) @ X )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ Y @ X ) @ ( sup_sup_list_char_o @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_338_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z ) @ X )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_339_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z ) @ X )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_340_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_341_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_342_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z ) @ X )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_343_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: list_char > $o,Z: list_char > $o,X: list_char > $o] :
( ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ Y @ Z ) @ X )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ Y @ X ) @ ( inf_inf_list_char_o @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_344_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_345_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_346_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_347_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_348_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_349_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( sup_sup_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ X @ Y ) @ ( sup_sup_list_char_o @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_350_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_351_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_352_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_353_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_354_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
= ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_355_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( sup_sup_list_char_o @ Y @ Z ) )
= ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ ( inf_inf_list_char_o @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_356_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_357_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_358_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_359_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ bot_bo8422036546324065075at_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_360_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_361_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_362_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_363_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_364_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_365_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_366_type__definition__assn,axiom,
type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).
% type_definition_assn
thf(fact_367_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( B2
= ( sup_su3035147773818789531at_nat @ K @ B ) )
=> ( ( sup_su3035147773818789531at_nat @ A @ B2 )
= ( sup_su3035147773818789531at_nat @ K @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_368_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( B2
= ( sup_su5525570899277871387at_nat @ K @ B ) )
=> ( ( sup_su5525570899277871387at_nat @ A @ B2 )
= ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_369_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_370_boolean__algebra__cancel_Osup2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_371_boolean__algebra__cancel_Osup2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( sup_sup_int @ K @ B ) )
=> ( ( sup_sup_int @ A @ B2 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_372_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Pr8551490117392284871at_nat,K: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( A2
= ( sup_su3035147773818789531at_nat @ K @ A ) )
=> ( ( sup_su3035147773818789531at_nat @ A2 @ B )
= ( sup_su3035147773818789531at_nat @ K @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_373_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( A2
= ( sup_su5525570899277871387at_nat @ K @ A ) )
=> ( ( sup_su5525570899277871387at_nat @ A2 @ B )
= ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_374_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_375_boolean__algebra__cancel_Osup1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A2 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_376_boolean__algebra__cancel_Osup1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( sup_sup_int @ K @ A ) )
=> ( ( sup_sup_int @ A2 @ B )
= ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_377_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( A2
= ( inf_in2572325071724192079at_nat @ K @ A ) )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ B )
= ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_378_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_379_boolean__algebra__cancel_Oinf1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A2 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_380_boolean__algebra__cancel_Oinf1,axiom,
! [A2: product_prod_int_int > $o,K: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( A2
= ( inf_in3604695632404883862_int_o @ K @ A ) )
=> ( ( inf_in3604695632404883862_int_o @ A2 @ B )
= ( inf_in3604695632404883862_int_o @ K @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_381_boolean__algebra__cancel_Oinf1,axiom,
! [A2: list_char > $o,K: list_char > $o,A: list_char > $o,B: list_char > $o] :
( ( A2
= ( inf_inf_list_char_o @ K @ A ) )
=> ( ( inf_inf_list_char_o @ A2 @ B )
= ( inf_inf_list_char_o @ K @ ( inf_inf_list_char_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_382_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( B2
= ( inf_in2572325071724192079at_nat @ K @ B ) )
=> ( ( inf_in2572325071724192079at_nat @ A @ B2 )
= ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_383_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_384_boolean__algebra__cancel_Oinf2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B2 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_385_boolean__algebra__cancel_Oinf2,axiom,
! [B2: product_prod_int_int > $o,K: product_prod_int_int > $o,B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( B2
= ( inf_in3604695632404883862_int_o @ K @ B ) )
=> ( ( inf_in3604695632404883862_int_o @ A @ B2 )
= ( inf_in3604695632404883862_int_o @ K @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_386_boolean__algebra__cancel_Oinf2,axiom,
! [B2: list_char > $o,K: list_char > $o,B: list_char > $o,A: list_char > $o] :
( ( B2
= ( inf_inf_list_char_o @ K @ B ) )
=> ( ( inf_inf_list_char_o @ A @ B2 )
= ( inf_inf_list_char_o @ K @ ( inf_inf_list_char_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_387_type__definition_ORep,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( member_o @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_388_type__definition_ORep,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( member6576561426505652726_nat_o @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_389_type__definition_ORep,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,X: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( member_list_char @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_390_type__definition_ORep,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,X: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( member2340774599025711042nt_int @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_391_type__definition_ORep,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,X: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( member2643936169264416010at_nat @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_392_type__definition_Ointro,axiom,
! [Rep: product_unit > $o,A2: set_o,Abs: $o > product_unit] :
( ! [X2: product_unit] : ( member_o @ ( Rep @ X2 ) @ A2 )
=> ( ! [X2: product_unit] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_393_type__definition_Ointro,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,Abs: ( produc3658429121746597890et_nat > $o ) > assn] :
( ! [X2: assn] : ( member6576561426505652726_nat_o @ ( Rep @ X2 ) @ A2 )
=> ( ! [X2: assn] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_394_type__definition_Ointro,axiom,
! [Rep: literal > list_char,A2: set_list_char,Abs: list_char > literal] :
( ! [X2: literal] : ( member_list_char @ ( Rep @ X2 ) @ A2 )
=> ( ! [X2: literal] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: list_char] :
( ( member_list_char @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_395_type__definition_Ointro,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,Abs: set_Pr958786334691620121nt_int > rat] :
( ! [X2: rat] : ( member2340774599025711042nt_int @ ( Rep @ X2 ) @ A2 )
=> ( ! [X2: rat] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_396_type__definition_Ointro,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat,Abs: set_Pr1261947904930325089at_nat > int] :
( ! [X2: int] : ( member2643936169264416010at_nat @ ( Rep @ X2 ) @ A2 )
=> ( ! [X2: int] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_397_type__definition_OAbs__cases,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ~ ! [Y2: $o] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member_o @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_398_type__definition_OAbs__cases,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ~ ! [Y2: produc3658429121746597890et_nat > $o] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member6576561426505652726_nat_o @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_399_type__definition_OAbs__cases,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,X: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ~ ! [Y2: list_char] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member_list_char @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_400_type__definition_OAbs__cases,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,X: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ~ ! [Y2: set_Pr958786334691620121nt_int] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member2340774599025711042nt_int @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_401_type__definition_OAbs__cases,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,X: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ~ ! [Y2: set_Pr1261947904930325089at_nat] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member2643936169264416010at_nat @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_402_type__definition_ORep__cases,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( member_o @ Y @ A2 )
=> ~ ! [X2: product_unit] :
( Y
= ( ~ ( Rep @ X2 ) ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_403_type__definition_ORep__cases,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( member6576561426505652726_nat_o @ Y @ A2 )
=> ~ ! [X2: assn] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_404_type__definition_ORep__cases,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,Y: list_char] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( member_list_char @ Y @ A2 )
=> ~ ! [X2: literal] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_405_type__definition_ORep__cases,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( member2340774599025711042nt_int @ Y @ A2 )
=> ~ ! [X2: rat] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_406_type__definition_ORep__cases,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( member2643936169264416010at_nat @ Y @ A2 )
=> ~ ! [X2: int] :
( Y
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_407_type__definition_OAbs__induct,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,P: product_unit > $o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_408_type__definition_OAbs__induct,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,P: assn > $o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_409_type__definition_OAbs__induct,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,P: literal > $o,X: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ! [Y2: list_char] :
( ( member_list_char @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_410_type__definition_OAbs__induct,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,P: rat > $o,X: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ! [Y2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_411_type__definition_OAbs__induct,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,P: int > $o,X: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ! [Y2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_412_type__definition_OAbs__inject,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,X: $o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( member_o @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_413_type__definition_OAbs__inject,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ( member6576561426505652726_nat_o @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_414_type__definition_OAbs__inject,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,X: list_char,Y: list_char] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( member_list_char @ X @ A2 )
=> ( ( member_list_char @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_415_type__definition_OAbs__inject,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( member2340774599025711042nt_int @ X @ A2 )
=> ( ( member2340774599025711042nt_int @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_416_type__definition_OAbs__inject,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( member2643936169264416010at_nat @ X @ A2 )
=> ( ( member2643936169264416010at_nat @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_417_type__definition_ORep__induct,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,Y: $o,P: $o > $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( member_o @ Y @ A2 )
=> ( ! [X2: product_unit] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_418_type__definition_ORep__induct,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( member6576561426505652726_nat_o @ Y @ A2 )
=> ( ! [X2: assn] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_419_type__definition_ORep__induct,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,Y: list_char,P: list_char > $o] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( member_list_char @ Y @ A2 )
=> ( ! [X2: literal] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_420_type__definition_ORep__induct,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( member2340774599025711042nt_int @ Y @ A2 )
=> ( ! [X2: rat] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_421_type__definition_ORep__induct,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( member2643936169264416010at_nat @ Y @ A2 )
=> ( ! [X2: int] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_422_type__definition_ORep__inject,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,X: product_unit,Y: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_423_type__definition_ORep__inject,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,X: assn,Y: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_424_type__definition_ORep__inject,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,X: literal,Y: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_425_type__definition_ORep__inject,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,X: rat,Y: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_426_type__definition_ORep__inject,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,X: int,Y: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_427_type__definition_OAbs__inverse,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( member_o @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_428_type__definition_OAbs__inverse,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( member6576561426505652726_nat_o @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_429_type__definition_OAbs__inverse,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,Y: list_char] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( member_list_char @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_430_type__definition_OAbs__inverse,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( member2340774599025711042nt_int @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_431_type__definition_OAbs__inverse,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( member2643936169264416010at_nat @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_432_type__definition_ORep__inverse,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_433_type__definition_ORep__inverse,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A2: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_434_type__definition_ORep__inverse,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,A2: set_list_char,X: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_435_type__definition_ORep__inverse,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,A2: set_se6260736226359567993nt_int,X: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_436_type__definition_ORep__inverse,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,A2: set_se7855581050983116737at_nat,X: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_437_type__definition__def,axiom,
( type_d6188575255521822967unit_o
= ( ^ [Rep2: product_unit > $o,Abs2: $o > product_unit,A4: set_o] :
( ! [X3: product_unit] : ( member_o @ ( Rep2 @ X3 ) @ A4 )
& ! [X3: product_unit] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: $o] :
( ( member_o @ Y3 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_438_type__definition__def,axiom,
( type_d3909072315231072503_nat_o
= ( ^ [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A4: set_Pr4532377907799695533_nat_o] :
( ! [X3: assn] : ( member6576561426505652726_nat_o @ ( Rep2 @ X3 ) @ A4 )
& ! [X3: assn] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_439_type__definition__def,axiom,
( type_d4752411451802217481t_char
= ( ^ [Rep2: literal > list_char,Abs2: list_char > literal,A4: set_list_char] :
( ! [X3: literal] : ( member_list_char @ ( Rep2 @ X3 ) @ A4 )
& ! [X3: literal] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: list_char] :
( ( member_list_char @ Y3 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_440_type__definition__def,axiom,
( type_d8554052265237484179nt_int
= ( ^ [Rep2: rat > set_Pr958786334691620121nt_int,Abs2: set_Pr958786334691620121nt_int > rat,A4: set_se6260736226359567993nt_int] :
( ! [X3: rat] : ( member2340774599025711042nt_int @ ( Rep2 @ X3 ) @ A4 )
& ! [X3: rat] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y3 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_441_type__definition__def,axiom,
( type_d8752208705193531015at_nat
= ( ^ [Rep2: int > set_Pr1261947904930325089at_nat,Abs2: set_Pr1261947904930325089at_nat > int,A4: set_se7855581050983116737at_nat] :
( ! [X3: int] : ( member2643936169264416010at_nat @ ( Rep2 @ X3 ) @ A4 )
& ! [X3: int] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y3 @ A4 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_442_mult_Oright__assoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_443_mult_Oright__assoc,axiom,
! [A: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
= ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_444_mult_Oright__assoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_445_mult_Oright__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_446_mult_Oright__assoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_447_mult_Oright__commute,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_448_mult_Oright__commute,axiom,
! [A: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
= ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_449_mult_Oright__commute,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_450_mult_Oright__commute,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ ( times_times_nat @ A @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_451_mult_Oright__commute,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ ( times_times_int @ A @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_452_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ B @ ( times_3573771949741848930nteger @ A @ C ) )
= ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_453_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: assn,A: assn,C: assn] :
( ( times_times_assn @ B @ ( times_times_assn @ A @ C ) )
= ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_454_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_455_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_456_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_457_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_3573771949741848930nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( times_3573771949741848930nteger @ B3 @ A3 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_458_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_assn
= ( ^ [A3: assn,B3: assn] : ( times_times_assn @ B3 @ A3 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_459_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_460_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_461_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_462_mult_Oassoc,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% mult.assoc
thf(fact_463_mult_Oassoc,axiom,
! [A: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
= ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).
% mult.assoc
thf(fact_464_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_465_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_466_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_467_mult_Osafe__commute,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
( ( syntax380677951125133566nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ B )
= ( times_3573771949741848930nteger @ B @ A ) ) ) ).
% mult.safe_commute
thf(fact_468_mult_Osafe__commute,axiom,
! [X: assn,Y: assn,A: assn,B: assn] :
( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A )
=> ( ( times_times_assn @ A @ B )
= ( times_times_assn @ B @ A ) ) ) ).
% mult.safe_commute
thf(fact_469_mult_Osafe__commute,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( ( syntax3730441303064801268at_rat @ ( times_times_rat @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ B )
= ( times_times_rat @ B @ A ) ) ) ).
% mult.safe_commute
thf(fact_470_mult_Osafe__commute,axiom,
! [X: nat,Y: nat,A: nat,B: nat] :
( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A )
=> ( ( times_times_nat @ A @ B )
= ( times_times_nat @ B @ A ) ) ) ).
% mult.safe_commute
thf(fact_471_mult_Osafe__commute,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ B )
= ( times_times_int @ B @ A ) ) ) ).
% mult.safe_commute
thf(fact_472_sup__bot_Osemilattice__neutr__axioms,axiom,
semila6683195626996185257at_nat @ sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_473_sup__bot_Osemilattice__neutr__axioms,axiom,
semila3675781407871293737at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_474_sup__bot_Osemilattice__neutr__axioms,axiom,
semila858173770281612099at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_475_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1065760912024005978_set_o @ sup_sup_set_o @ bot_bot_set_o ).
% sup_bot.semilattice_neutr_axioms
thf(fact_476_sup__bot_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.semilattice_neutr_axioms
thf(fact_477_sup__bot_Osemilattice__neutr__axioms,axiom,
semila6287294981380917632et_int @ sup_sup_set_int @ bot_bot_set_int ).
% sup_bot.semilattice_neutr_axioms
thf(fact_478_sup__bot_Omonoid__axioms,axiom,
monoid3323479476634721864at_nat @ sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat ).
% sup_bot.monoid_axioms
thf(fact_479_sup__bot_Omonoid__axioms,axiom,
monoid534952638934301128at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).
% sup_bot.monoid_axioms
thf(fact_480_sup__bot_Omonoid__axioms,axiom,
monoid1519094961268053602at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).
% sup_bot.monoid_axioms
thf(fact_481_sup__bot_Omonoid__axioms,axiom,
monoid_set_o @ sup_sup_set_o @ bot_bot_set_o ).
% sup_bot.monoid_axioms
thf(fact_482_sup__bot_Omonoid__axioms,axiom,
monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.monoid_axioms
thf(fact_483_sup__bot_Omonoid__axioms,axiom,
monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).
% sup_bot.monoid_axioms
thf(fact_484_one__assn__def,axiom,
( one_one_assn
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_485_boolean__algebra_Ocomplement__unique,axiom,
! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( ( inf_in3604695632404883862_int_o @ A @ X )
= bot_bo8147686125503663512_int_o )
=> ( ( ( sup_su8463660629351352368_int_o @ A @ X )
= top_to1578927101902068148_int_o )
=> ( ( ( inf_in3604695632404883862_int_o @ A @ Y )
= bot_bo8147686125503663512_int_o )
=> ( ( ( sup_su8463660629351352368_int_o @ A @ Y )
= top_to1578927101902068148_int_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_486_boolean__algebra_Ocomplement__unique,axiom,
! [A: list_char > $o,X: list_char > $o,Y: list_char > $o] :
( ( ( inf_inf_list_char_o @ A @ X )
= bot_bot_list_char_o )
=> ( ( ( sup_sup_list_char_o @ A @ X )
= top_top_list_char_o )
=> ( ( ( inf_inf_list_char_o @ A @ Y )
= bot_bot_list_char_o )
=> ( ( ( sup_sup_list_char_o @ A @ Y )
= top_top_list_char_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_487_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( ( inf_in1697001100524423349at_nat @ A @ X )
= bot_bo8422036546324065075at_nat )
=> ( ( ( sup_su3035147773818789531at_nat @ A @ X )
= top_to8764741339697478167at_nat )
=> ( ( ( inf_in1697001100524423349at_nat @ A @ Y )
= bot_bo8422036546324065075at_nat )
=> ( ( ( sup_su3035147773818789531at_nat @ A @ Y )
= top_to8764741339697478167at_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_488_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( ( inf_in7913087082777306421at_nat @ A @ X )
= bot_bo228742789529271731at_nat )
=> ( ( ( sup_su5525570899277871387at_nat @ A @ X )
= top_to6833984726390702231at_nat )
=> ( ( ( inf_in7913087082777306421at_nat @ A @ Y )
= bot_bo228742789529271731at_nat )
=> ( ( ( sup_su5525570899277871387at_nat @ A @ Y )
= top_to6833984726390702231at_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_489_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A @ X )
= bot_bo2099793752762293965at_nat )
=> ( ( ( sup_su6327502436637775413at_nat @ A @ X )
= top_to4669805908274784177at_nat )
=> ( ( ( inf_in2572325071724192079at_nat @ A @ Y )
= bot_bo2099793752762293965at_nat )
=> ( ( ( sup_su6327502436637775413at_nat @ A @ Y )
= top_to4669805908274784177at_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_490_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_list_nat,X: set_list_nat,Y: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ X )
= bot_bot_set_list_nat )
=> ( ( ( sup_sup_set_list_nat @ A @ X )
= top_top_set_list_nat )
=> ( ( ( inf_inf_set_list_nat @ A @ Y )
= bot_bot_set_list_nat )
=> ( ( ( sup_sup_set_list_nat @ A @ Y )
= top_top_set_list_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_491_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_o,X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ A @ X )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ X )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A @ Y )
= top_top_set_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_492_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_rat,X: set_rat,Y: set_rat] :
( ( ( inf_inf_set_rat @ A @ X )
= bot_bot_set_rat )
=> ( ( ( sup_sup_set_rat @ A @ X )
= top_top_set_rat )
=> ( ( ( inf_inf_set_rat @ A @ Y )
= bot_bot_set_rat )
=> ( ( ( sup_sup_set_rat @ A @ Y )
= top_top_set_rat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_493_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_494_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_int,X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ A @ X )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ A @ X )
= top_top_set_int )
=> ( ( ( inf_inf_set_int @ A @ Y )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ A @ Y )
= top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_495_mod__star__conv,axiom,
! [A2: assn,B2: assn,H3: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A2 @ B2 ) @ H3 )
= ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As2: set_nat] :
( ( H3
= ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As2 ) ) )
& ( ( inf_inf_set_nat @ As1 @ As2 )
= bot_bot_set_nat )
& ( rep_assn @ A2 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
& ( rep_assn @ B2 @ ( produc7507926704131184380et_nat @ Hr @ As2 ) ) ) ) ) ).
% mod_star_conv
thf(fact_496_star__assnI,axiom,
! [P: assn,H3: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,As3: set_nat] :
( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
=> ( ( ( inf_inf_set_nat @ As @ As3 )
= bot_bot_set_nat )
=> ( rep_assn @ ( times_times_assn @ P @ Q2 ) @ ( produc7507926704131184380et_nat @ H3 @ ( sup_sup_set_nat @ As @ As3 ) ) ) ) ) ) ).
% star_assnI
thf(fact_497_sup__bot_Ocomm__monoid__axioms,axiom,
comm_m4707310049023135612at_nat @ sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat ).
% sup_bot.comm_monoid_axioms
thf(fact_498_sup__bot_Ocomm__monoid__axioms,axiom,
comm_m2457627138559323900at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).
% sup_bot.comm_monoid_axioms
thf(fact_499_sup__bot_Ocomm__monoid__axioms,axiom,
comm_m5709050211238513046at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).
% sup_bot.comm_monoid_axioms
thf(fact_500_sup__bot_Ocomm__monoid__axioms,axiom,
comm_monoid_set_o @ sup_sup_set_o @ bot_bot_set_o ).
% sup_bot.comm_monoid_axioms
thf(fact_501_sup__bot_Ocomm__monoid__axioms,axiom,
comm_monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).
% sup_bot.comm_monoid_axioms
thf(fact_502_sup__bot_Ocomm__monoid__axioms,axiom,
comm_monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).
% sup_bot.comm_monoid_axioms
thf(fact_503_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
= ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_504_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( uminus7117520113953359693_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
= ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_505_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( uminus7154032370948614053char_o @ ( sup_sup_list_char_o @ X @ Y ) )
= ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ ( uminus7154032370948614053char_o @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_506_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( uminus2330091110623919550at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
= ( inf_in1697001100524423349at_nat @ ( uminus2330091110623919550at_nat @ X ) @ ( uminus2330091110623919550at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_507_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
= ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( uminus935396558254630718at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_508_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_nat,Y: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_509_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: product_unit,Y: product_unit] :
( ( uminus2952777764628376836t_unit @ ( sup_sup_Product_unit @ X @ Y ) )
= ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_510_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_511_add_Oinverse__inverse,axiom,
! [A: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_512_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_513_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_514_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_515_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_516_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( uminus2952777764628376836t_unit @ X )
= ( uminus2952777764628376836t_unit @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_517_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: product_unit] :
( ( uminus2952777764628376836t_unit @ ( uminus2952777764628376836t_unit @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_518_mult_Oright__neutral,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
= A ) ).
% mult.right_neutral
thf(fact_519_mult_Oright__neutral,axiom,
! [A: assn] :
( ( times_times_assn @ A @ one_one_assn )
= A ) ).
% mult.right_neutral
thf(fact_520_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_521_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_522_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_523_mult__1,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
= A ) ).
% mult_1
thf(fact_524_mult__1,axiom,
! [A: assn] :
( ( times_times_assn @ one_one_assn @ A )
= A ) ).
% mult_1
thf(fact_525_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_526_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_527_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_528_inf__top__left,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ X )
= X ) ).
% inf_top_left
thf(fact_529_inf__top__left,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ X )
= X ) ).
% inf_top_left
thf(fact_530_inf__top__left,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ top_top_list_char_o @ X )
= X ) ).
% inf_top_left
thf(fact_531_inf__top__left,axiom,
! [X: set_list_nat] :
( ( inf_inf_set_list_nat @ top_top_set_list_nat @ X )
= X ) ).
% inf_top_left
thf(fact_532_inf__top__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X )
= X ) ).
% inf_top_left
thf(fact_533_inf__top__left,axiom,
! [X: set_rat] :
( ( inf_inf_set_rat @ top_top_set_rat @ X )
= X ) ).
% inf_top_left
thf(fact_534_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_535_inf__top__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ top_top_set_int @ X )
= X ) ).
% inf_top_left
thf(fact_536_inf__top__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ top_to4669805908274784177at_nat )
= X ) ).
% inf_top_right
thf(fact_537_inf__top__right,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ top_to1578927101902068148_int_o )
= X ) ).
% inf_top_right
thf(fact_538_inf__top__right,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ top_top_list_char_o )
= X ) ).
% inf_top_right
thf(fact_539_inf__top__right,axiom,
! [X: set_list_nat] :
( ( inf_inf_set_list_nat @ X @ top_top_set_list_nat )
= X ) ).
% inf_top_right
thf(fact_540_inf__top__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% inf_top_right
thf(fact_541_inf__top__right,axiom,
! [X: set_rat] :
( ( inf_inf_set_rat @ X @ top_top_set_rat )
= X ) ).
% inf_top_right
thf(fact_542_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_543_inf__top__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ top_top_set_int )
= X ) ).
% inf_top_right
thf(fact_544_inf__eq__top__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ X @ Y )
= top_to4669805908274784177at_nat )
= ( ( X = top_to4669805908274784177at_nat )
& ( Y = top_to4669805908274784177at_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_545_inf__eq__top__iff,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( ( inf_in3604695632404883862_int_o @ X @ Y )
= top_to1578927101902068148_int_o )
= ( ( X = top_to1578927101902068148_int_o )
& ( Y = top_to1578927101902068148_int_o ) ) ) ).
% inf_eq_top_iff
thf(fact_546_inf__eq__top__iff,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( ( inf_inf_list_char_o @ X @ Y )
= top_top_list_char_o )
= ( ( X = top_top_list_char_o )
& ( Y = top_top_list_char_o ) ) ) ).
% inf_eq_top_iff
thf(fact_547_inf__eq__top__iff,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ( inf_inf_set_list_nat @ X @ Y )
= top_top_set_list_nat )
= ( ( X = top_top_set_list_nat )
& ( Y = top_top_set_list_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_548_inf__eq__top__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= top_top_set_o )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_549_inf__eq__top__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( inf_inf_set_rat @ X @ Y )
= top_top_set_rat )
= ( ( X = top_top_set_rat )
& ( Y = top_top_set_rat ) ) ) ).
% inf_eq_top_iff
thf(fact_550_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_551_inf__eq__top__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ X @ Y )
= top_top_set_int )
= ( ( X = top_top_set_int )
& ( Y = top_top_set_int ) ) ) ).
% inf_eq_top_iff
thf(fact_552_top__eq__inf__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( top_to4669805908274784177at_nat
= ( inf_in2572325071724192079at_nat @ X @ Y ) )
= ( ( X = top_to4669805908274784177at_nat )
& ( Y = top_to4669805908274784177at_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_553_top__eq__inf__iff,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( top_to1578927101902068148_int_o
= ( inf_in3604695632404883862_int_o @ X @ Y ) )
= ( ( X = top_to1578927101902068148_int_o )
& ( Y = top_to1578927101902068148_int_o ) ) ) ).
% top_eq_inf_iff
thf(fact_554_top__eq__inf__iff,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( top_top_list_char_o
= ( inf_inf_list_char_o @ X @ Y ) )
= ( ( X = top_top_list_char_o )
& ( Y = top_top_list_char_o ) ) ) ).
% top_eq_inf_iff
thf(fact_555_top__eq__inf__iff,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( top_top_set_list_nat
= ( inf_inf_set_list_nat @ X @ Y ) )
= ( ( X = top_top_set_list_nat )
& ( Y = top_top_set_list_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_556_top__eq__inf__iff,axiom,
! [X: set_o,Y: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X @ Y ) )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_557_top__eq__inf__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( top_top_set_rat
= ( inf_inf_set_rat @ X @ Y ) )
= ( ( X = top_top_set_rat )
& ( Y = top_top_set_rat ) ) ) ).
% top_eq_inf_iff
thf(fact_558_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_559_top__eq__inf__iff,axiom,
! [X: set_int,Y: set_int] :
( ( top_top_set_int
= ( inf_inf_set_int @ X @ Y ) )
= ( ( X = top_top_set_int )
& ( Y = top_top_set_int ) ) ) ).
% top_eq_inf_iff
thf(fact_560_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A @ B )
= top_to4669805908274784177at_nat )
= ( ( A = top_to4669805908274784177at_nat )
& ( B = top_to4669805908274784177at_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_561_inf__top_Oeq__neutr__iff,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ( inf_in3604695632404883862_int_o @ A @ B )
= top_to1578927101902068148_int_o )
= ( ( A = top_to1578927101902068148_int_o )
& ( B = top_to1578927101902068148_int_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_562_inf__top_Oeq__neutr__iff,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( ( inf_inf_list_char_o @ A @ B )
= top_top_list_char_o )
= ( ( A = top_top_list_char_o )
& ( B = top_top_list_char_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_563_inf__top_Oeq__neutr__iff,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ B )
= top_top_set_list_nat )
= ( ( A = top_top_set_list_nat )
& ( B = top_top_set_list_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_564_inf__top_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_565_inf__top_Oeq__neutr__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( ( inf_inf_set_rat @ A @ B )
= top_top_set_rat )
= ( ( A = top_top_set_rat )
& ( B = top_top_set_rat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_566_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_567_inf__top_Oeq__neutr__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( inf_inf_set_int @ A @ B )
= top_top_set_int )
= ( ( A = top_top_set_int )
& ( B = top_top_set_int ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_568_inf__top_Oleft__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_569_inf__top_Oleft__neutral,axiom,
! [A: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_570_inf__top_Oleft__neutral,axiom,
! [A: list_char > $o] :
( ( inf_inf_list_char_o @ top_top_list_char_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_571_inf__top_Oleft__neutral,axiom,
! [A: set_list_nat] :
( ( inf_inf_set_list_nat @ top_top_set_list_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_572_inf__top_Oleft__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_573_inf__top_Oleft__neutral,axiom,
! [A: set_rat] :
( ( inf_inf_set_rat @ top_top_set_rat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_574_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_575_inf__top_Oleft__neutral,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ top_top_set_int @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_576_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( top_to4669805908274784177at_nat
= ( inf_in2572325071724192079at_nat @ A @ B ) )
= ( ( A = top_to4669805908274784177at_nat )
& ( B = top_to4669805908274784177at_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_577_inf__top_Oneutr__eq__iff,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( top_to1578927101902068148_int_o
= ( inf_in3604695632404883862_int_o @ A @ B ) )
= ( ( A = top_to1578927101902068148_int_o )
& ( B = top_to1578927101902068148_int_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_578_inf__top_Oneutr__eq__iff,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( top_top_list_char_o
= ( inf_inf_list_char_o @ A @ B ) )
= ( ( A = top_top_list_char_o )
& ( B = top_top_list_char_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_579_inf__top_Oneutr__eq__iff,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( top_top_set_list_nat
= ( inf_inf_set_list_nat @ A @ B ) )
= ( ( A = top_top_set_list_nat )
& ( B = top_top_set_list_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_580_inf__top_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A @ B ) )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_581_inf__top_Oneutr__eq__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( top_top_set_rat
= ( inf_inf_set_rat @ A @ B ) )
= ( ( A = top_top_set_rat )
& ( B = top_top_set_rat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_582_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_583_inf__top_Oneutr__eq__iff,axiom,
! [A: set_int,B: set_int] :
( ( top_top_set_int
= ( inf_inf_set_int @ A @ B ) )
= ( ( A = top_top_set_int )
& ( B = top_top_set_int ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_584_inf__top_Oright__neutral,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A @ top_to4669805908274784177at_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_585_inf__top_Oright__neutral,axiom,
! [A: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ A @ top_to1578927101902068148_int_o )
= A ) ).
% inf_top.right_neutral
thf(fact_586_inf__top_Oright__neutral,axiom,
! [A: list_char > $o] :
( ( inf_inf_list_char_o @ A @ top_top_list_char_o )
= A ) ).
% inf_top.right_neutral
thf(fact_587_inf__top_Oright__neutral,axiom,
! [A: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ top_top_set_list_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_588_inf__top_Oright__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% inf_top.right_neutral
thf(fact_589_inf__top_Oright__neutral,axiom,
! [A: set_rat] :
( ( inf_inf_set_rat @ A @ top_top_set_rat )
= A ) ).
% inf_top.right_neutral
thf(fact_590_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_591_inf__top_Oright__neutral,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ A @ top_top_set_int )
= A ) ).
% inf_top.right_neutral
thf(fact_592_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ top_to8764741339697478167at_nat @ X )
= top_to8764741339697478167at_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_593_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ top_to6833984726390702231at_nat @ X )
= top_to6833984726390702231at_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_594_boolean__algebra_Odisj__one__left,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ top_top_set_list_nat @ X )
= top_top_set_list_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_595_boolean__algebra_Odisj__one__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_596_boolean__algebra_Odisj__one__left,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ X )
= top_top_set_rat ) ).
% boolean_algebra.disj_one_left
thf(fact_597_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_598_boolean__algebra_Odisj__one__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ top_top_set_int @ X )
= top_top_set_int ) ).
% boolean_algebra.disj_one_left
thf(fact_599_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ top_to8764741339697478167at_nat )
= top_to8764741339697478167at_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_600_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ top_to6833984726390702231at_nat )
= top_to6833984726390702231at_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_601_boolean__algebra_Odisj__one__right,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ top_top_set_list_nat )
= top_top_set_list_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_602_boolean__algebra_Odisj__one__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_603_boolean__algebra_Odisj__one__right,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ X @ top_top_set_rat )
= top_top_set_rat ) ).
% boolean_algebra.disj_one_right
thf(fact_604_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_605_boolean__algebra_Odisj__one__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% boolean_algebra.disj_one_right
thf(fact_606_sup__top__left,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ top_to8764741339697478167at_nat @ X )
= top_to8764741339697478167at_nat ) ).
% sup_top_left
thf(fact_607_sup__top__left,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ top_to6833984726390702231at_nat @ X )
= top_to6833984726390702231at_nat ) ).
% sup_top_left
thf(fact_608_sup__top__left,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ top_top_set_list_nat @ X )
= top_top_set_list_nat ) ).
% sup_top_left
thf(fact_609_sup__top__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% sup_top_left
thf(fact_610_sup__top__left,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ X )
= top_top_set_rat ) ).
% sup_top_left
thf(fact_611_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_612_sup__top__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ top_top_set_int @ X )
= top_top_set_int ) ).
% sup_top_left
thf(fact_613_sup__top__right,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ top_to8764741339697478167at_nat )
= top_to8764741339697478167at_nat ) ).
% sup_top_right
thf(fact_614_sup__top__right,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ top_to6833984726390702231at_nat )
= top_to6833984726390702231at_nat ) ).
% sup_top_right
thf(fact_615_sup__top__right,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ top_top_set_list_nat )
= top_top_set_list_nat ) ).
% sup_top_right
thf(fact_616_sup__top__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_617_sup__top__right,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ X @ top_top_set_rat )
= top_top_set_rat ) ).
% sup_top_right
thf(fact_618_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_619_sup__top__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% sup_top_right
thf(fact_620_boolean__algebra_Ocompl__one,axiom,
( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
= bot_bo2099793752762293965at_nat ) ).
% boolean_algebra.compl_one
thf(fact_621_boolean__algebra_Ocompl__one,axiom,
( ( uminus3195874150345416415st_nat @ top_top_set_list_nat )
= bot_bot_set_list_nat ) ).
% boolean_algebra.compl_one
thf(fact_622_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_623_boolean__algebra_Ocompl__one,axiom,
( ( uminus2201863774496077783et_rat @ top_top_set_rat )
= bot_bot_set_rat ) ).
% boolean_algebra.compl_one
thf(fact_624_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_625_boolean__algebra_Ocompl__one,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% boolean_algebra.compl_one
thf(fact_626_boolean__algebra_Ocompl__one,axiom,
( ( uminus2952777764628376836t_unit @ top_top_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.compl_one
thf(fact_627_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
= top_to4669805908274784177at_nat ) ).
% boolean_algebra.compl_zero
thf(fact_628_boolean__algebra_Ocompl__zero,axiom,
( ( uminus3195874150345416415st_nat @ bot_bot_set_list_nat )
= top_top_set_list_nat ) ).
% boolean_algebra.compl_zero
thf(fact_629_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_630_boolean__algebra_Ocompl__zero,axiom,
( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
= top_top_set_rat ) ).
% boolean_algebra.compl_zero
thf(fact_631_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_632_boolean__algebra_Ocompl__zero,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% boolean_algebra.compl_zero
thf(fact_633_boolean__algebra_Ocompl__zero,axiom,
( ( uminus2952777764628376836t_unit @ bot_bot_Product_unit )
= top_top_Product_unit ) ).
% boolean_algebra.compl_zero
thf(fact_634_inf__compl__bot__left1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
= bot_bo8147686125503663512_int_o ) ).
% inf_compl_bot_left1
thf(fact_635_inf__compl__bot__left1,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ ( inf_inf_list_char_o @ X @ Y ) )
= bot_bot_list_char_o ) ).
% inf_compl_bot_left1
thf(fact_636_inf__compl__bot__left1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
= bot_bo2099793752762293965at_nat ) ).
% inf_compl_bot_left1
thf(fact_637_inf__compl__bot__left1,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left1
thf(fact_638_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_639_inf__compl__bot__left1,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
= bot_bot_set_int ) ).
% inf_compl_bot_left1
thf(fact_640_inf__compl__bot__left1,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( inf_inf_Product_unit @ X @ Y ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_left1
thf(fact_641_inf__compl__bot__left2,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ Y ) )
= bot_bo8147686125503663512_int_o ) ).
% inf_compl_bot_left2
thf(fact_642_inf__compl__bot__left2,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ Y ) )
= bot_bot_list_char_o ) ).
% inf_compl_bot_left2
thf(fact_643_inf__compl__bot__left2,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ Y ) )
= bot_bo2099793752762293965at_nat ) ).
% inf_compl_bot_left2
thf(fact_644_inf__compl__bot__left2,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left2
thf(fact_645_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_646_inf__compl__bot__left2,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
= bot_bot_set_int ) ).
% inf_compl_bot_left2
thf(fact_647_inf__compl__bot__left2,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_left2
thf(fact_648_inf__compl__bot__right,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ ( uminus7117520113953359693_int_o @ X ) ) )
= bot_bo8147686125503663512_int_o ) ).
% inf_compl_bot_right
thf(fact_649_inf__compl__bot__right,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ ( uminus7154032370948614053char_o @ X ) ) )
= bot_bot_list_char_o ) ).
% inf_compl_bot_right
thf(fact_650_inf__compl__bot__right,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ ( uminus6524753893492686040at_nat @ X ) ) )
= bot_bo2099793752762293965at_nat ) ).
% inf_compl_bot_right
thf(fact_651_inf__compl__bot__right,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
= bot_bot_set_o ) ).
% inf_compl_bot_right
thf(fact_652_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_653_inf__compl__bot__right,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
= bot_bot_set_int ) ).
% inf_compl_bot_right
thf(fact_654_inf__compl__bot__right,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_right
thf(fact_655_boolean__algebra_Oconj__cancel__left,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ X )
= bot_bo8147686125503663512_int_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_656_boolean__algebra_Oconj__cancel__left,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ X )
= bot_bot_list_char_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_657_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ X )
= bot_bo2099793752762293965at_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_658_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_659_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_660_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
= bot_bot_set_int ) ).
% boolean_algebra.conj_cancel_left
thf(fact_661_boolean__algebra_Oconj__cancel__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_cancel_left
thf(fact_662_boolean__algebra_Oconj__cancel__right,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ ( uminus7117520113953359693_int_o @ X ) )
= bot_bo8147686125503663512_int_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_663_boolean__algebra_Oconj__cancel__right,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ ( uminus7154032370948614053char_o @ X ) )
= bot_bot_list_char_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_664_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ X ) )
= bot_bo2099793752762293965at_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_665_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_666_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_667_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
= bot_bot_set_int ) ).
% boolean_algebra.conj_cancel_right
thf(fact_668_boolean__algebra_Oconj__cancel__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_cancel_right
thf(fact_669_sup__compl__top__left1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ ( sup_su3035147773818789531at_nat @ X @ Y ) )
= top_to8764741339697478167at_nat ) ).
% sup_compl_top_left1
thf(fact_670_sup__compl__top__left1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
= top_to6833984726390702231at_nat ) ).
% sup_compl_top_left1
thf(fact_671_sup__compl__top__left1,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ X ) @ ( sup_sup_set_list_nat @ X @ Y ) )
= top_top_set_list_nat ) ).
% sup_compl_top_left1
thf(fact_672_sup__compl__top__left1,axiom,
! [X: set_o,Y: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ ( sup_sup_set_o @ X @ Y ) )
= top_top_set_o ) ).
% sup_compl_top_left1
thf(fact_673_sup__compl__top__left1,axiom,
! [X: set_rat,Y: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ ( sup_sup_set_rat @ X @ Y ) )
= top_top_set_rat ) ).
% sup_compl_top_left1
thf(fact_674_sup__compl__top__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( sup_sup_set_nat @ X @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_675_sup__compl__top__left1,axiom,
! [X: set_int,Y: set_int] :
( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( sup_sup_set_int @ X @ Y ) )
= top_top_set_int ) ).
% sup_compl_top_left1
thf(fact_676_sup__compl__top__left1,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( sup_sup_Product_unit @ X @ Y ) )
= top_top_Product_unit ) ).
% sup_compl_top_left1
thf(fact_677_sup__compl__top__left2,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ Y ) )
= top_to8764741339697478167at_nat ) ).
% sup_compl_top_left2
thf(fact_678_sup__compl__top__left2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ Y ) )
= top_to6833984726390702231at_nat ) ).
% sup_compl_top_left2
thf(fact_679_sup__compl__top__left2,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ X ) @ Y ) )
= top_top_set_list_nat ) ).
% sup_compl_top_left2
thf(fact_680_sup__compl__top__left2,axiom,
! [X: set_o,Y: set_o] :
( ( sup_sup_set_o @ X @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
= top_top_set_o ) ).
% sup_compl_top_left2
thf(fact_681_sup__compl__top__left2,axiom,
! [X: set_rat,Y: set_rat] :
( ( sup_sup_set_rat @ X @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ Y ) )
= top_top_set_rat ) ).
% sup_compl_top_left2
thf(fact_682_sup__compl__top__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_683_sup__compl__top__left2,axiom,
! [X: set_int,Y: set_int] :
( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
= top_top_set_int ) ).
% sup_compl_top_left2
thf(fact_684_sup__compl__top__left2,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
= top_top_Product_unit ) ).
% sup_compl_top_left2
thf(fact_685_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ X )
= top_to8764741339697478167at_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_686_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ X )
= top_to6833984726390702231at_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_687_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ X ) @ X )
= top_top_set_list_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_688_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ X )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_left
thf(fact_689_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ X )
= top_top_set_rat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_690_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_691_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
= top_top_set_int ) ).
% boolean_algebra.disj_cancel_left
thf(fact_692_boolean__algebra_Odisj__cancel__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
= top_top_Product_unit ) ).
% boolean_algebra.disj_cancel_left
thf(fact_693_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X @ ( uminus2330091110623919550at_nat @ X ) )
= top_to8764741339697478167at_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_694_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X @ ( uminus935396558254630718at_nat @ X ) )
= top_to6833984726390702231at_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_695_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_list_nat] :
( ( sup_sup_set_list_nat @ X @ ( uminus3195874150345416415st_nat @ X ) )
= top_top_set_list_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_696_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ ( uminus_uminus_set_o @ X ) )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_right
thf(fact_697_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ X @ ( uminus2201863774496077783et_rat @ X ) )
= top_top_set_rat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_698_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_699_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
= top_top_set_int ) ).
% boolean_algebra.disj_cancel_right
thf(fact_700_boolean__algebra_Odisj__cancel__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
= top_top_Product_unit ) ).
% boolean_algebra.disj_cancel_right
thf(fact_701_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
= ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_702_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( uminus7117520113953359693_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
= ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_703_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( uminus7154032370948614053char_o @ ( inf_inf_list_char_o @ X @ Y ) )
= ( sup_sup_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ ( uminus7154032370948614053char_o @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_704_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( uminus2330091110623919550at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) )
= ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ ( uminus2330091110623919550at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_705_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) )
= ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( uminus935396558254630718at_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_706_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_nat,Y: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_707_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: product_unit,Y: product_unit] :
( ( uminus2952777764628376836t_unit @ ( inf_inf_Product_unit @ X @ Y ) )
= ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_708_mult_Omonoid__axioms,axiom,
monoid_Code_integer @ times_3573771949741848930nteger @ one_one_Code_integer ).
% mult.monoid_axioms
thf(fact_709_mult_Omonoid__axioms,axiom,
monoid_assn @ times_times_assn @ one_one_assn ).
% mult.monoid_axioms
thf(fact_710_mult_Omonoid__axioms,axiom,
monoid_rat @ times_times_rat @ one_one_rat ).
% mult.monoid_axioms
thf(fact_711_mult_Omonoid__axioms,axiom,
monoid_nat @ times_times_nat @ one_one_nat ).
% mult.monoid_axioms
thf(fact_712_mult_Omonoid__axioms,axiom,
monoid_int @ times_times_int @ one_one_int ).
% mult.monoid_axioms
thf(fact_713_mult_Ocomm__monoid__axioms,axiom,
comm_m1654649897431166824nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).
% mult.comm_monoid_axioms
thf(fact_714_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_assn @ times_times_assn @ one_one_assn ).
% mult.comm_monoid_axioms
thf(fact_715_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_rat @ times_times_rat @ one_one_rat ).
% mult.comm_monoid_axioms
thf(fact_716_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ times_times_nat @ one_one_nat ).
% mult.comm_monoid_axioms
thf(fact_717_mult_Ocomm__monoid__axioms,axiom,
comm_monoid_int @ times_times_int @ one_one_int ).
% mult.comm_monoid_axioms
thf(fact_718_semilattice__neutr_Oaxioms_I2_J,axiom,
! [F2: nat > nat > nat,Z: nat] :
( ( semila9081495762789891438tr_nat @ F2 @ Z )
=> ( comm_monoid_nat @ F2 @ Z ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_719_monoid_Oleft__neutral,axiom,
! [F2: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F2 @ Z )
=> ( ( F2 @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_720_monoid_Oright__neutral,axiom,
! [F2: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F2 @ Z )
=> ( ( F2 @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_721_one__assn__raw_Ocases,axiom,
! [X: produc3658429121746597890et_nat] :
~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( X
!= ( produc7507926704131184380et_nat @ H @ As4 ) ) ).
% one_assn_raw.cases
thf(fact_722_comm__monoid_Ocomm__neutral,axiom,
! [F2: nat > nat > nat,Z: nat,A: nat] :
( ( comm_monoid_nat @ F2 @ Z )
=> ( ( F2 @ A @ Z )
= A ) ) ).
% comm_monoid.comm_neutral
thf(fact_723_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_724_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% equation_minus_iff
thf(fact_725_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_726_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_727_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_728_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_729_one__reorient,axiom,
! [X: code_integer] :
( ( one_one_Code_integer = X )
= ( X = one_one_Code_integer ) ) ).
% one_reorient
thf(fact_730_one__reorient,axiom,
! [X: assn] :
( ( one_one_assn = X )
= ( X = one_one_assn ) ) ).
% one_reorient
thf(fact_731_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_732_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_733_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_734_sup__cancel__left1,axiom,
! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ A ) @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ B ) )
= top_to8764741339697478167at_nat ) ).
% sup_cancel_left1
thf(fact_735_sup__cancel__left1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ A ) @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ B ) )
= top_to6833984726390702231at_nat ) ).
% sup_cancel_left1
thf(fact_736_sup__cancel__left1,axiom,
! [X: set_list_nat,A: set_list_nat,B: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ X @ A ) @ ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ X ) @ B ) )
= top_top_set_list_nat ) ).
% sup_cancel_left1
thf(fact_737_sup__cancel__left1,axiom,
! [X: set_o,A: set_o,B: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ X @ A ) @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
= top_top_set_o ) ).
% sup_cancel_left1
thf(fact_738_sup__cancel__left1,axiom,
! [X: set_rat,A: set_rat,B: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ X @ A ) @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ B ) )
= top_top_set_rat ) ).
% sup_cancel_left1
thf(fact_739_sup__cancel__left1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ A ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_740_sup__cancel__left1,axiom,
! [X: set_int,A: set_int,B: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ X @ A ) @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
= top_top_set_int ) ).
% sup_cancel_left1
thf(fact_741_sup__cancel__left1,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X @ A ) @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
= top_top_Product_unit ) ).
% sup_cancel_left1
thf(fact_742_sup__cancel__left2,axiom,
! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ X ) @ A ) @ ( sup_su3035147773818789531at_nat @ X @ B ) )
= top_to8764741339697478167at_nat ) ).
% sup_cancel_left2
thf(fact_743_sup__cancel__left2,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ A ) @ ( sup_su5525570899277871387at_nat @ X @ B ) )
= top_to6833984726390702231at_nat ) ).
% sup_cancel_left2
thf(fact_744_sup__cancel__left2,axiom,
! [X: set_list_nat,A: set_list_nat,B: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ X ) @ A ) @ ( sup_sup_set_list_nat @ X @ B ) )
= top_top_set_list_nat ) ).
% sup_cancel_left2
thf(fact_745_sup__cancel__left2,axiom,
! [X: set_o,A: set_o,B: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( sup_sup_set_o @ X @ B ) )
= top_top_set_o ) ).
% sup_cancel_left2
thf(fact_746_sup__cancel__left2,axiom,
! [X: set_rat,A: set_rat,B: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ A ) @ ( sup_sup_set_rat @ X @ B ) )
= top_top_set_rat ) ).
% sup_cancel_left2
thf(fact_747_sup__cancel__left2,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( sup_sup_set_nat @ X @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_748_sup__cancel__left2,axiom,
! [X: set_int,A: set_int,B: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ X ) @ A ) @ ( sup_sup_set_int @ X @ B ) )
= top_top_set_int ) ).
% sup_cancel_left2
thf(fact_749_sup__cancel__left2,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A ) @ ( sup_sup_Product_unit @ X @ B ) )
= top_top_Product_unit ) ).
% sup_cancel_left2
thf(fact_750_inf__top_Ocomm__monoid__axioms,axiom,
comm_m5709050211238513046at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).
% inf_top.comm_monoid_axioms
thf(fact_751_inf__top_Ocomm__monoid__axioms,axiom,
comm_m5839176995973721103_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).
% inf_top.comm_monoid_axioms
thf(fact_752_inf__top_Ocomm__monoid__axioms,axiom,
comm_m8764728085499324515char_o @ inf_inf_list_char_o @ top_top_list_char_o ).
% inf_top.comm_monoid_axioms
thf(fact_753_inf__top_Ocomm__monoid__axioms,axiom,
comm_m4511514676032769697st_nat @ inf_inf_set_list_nat @ top_top_set_list_nat ).
% inf_top.comm_monoid_axioms
thf(fact_754_inf__top_Ocomm__monoid__axioms,axiom,
comm_monoid_set_o @ inf_inf_set_o @ top_top_set_o ).
% inf_top.comm_monoid_axioms
thf(fact_755_inf__top_Ocomm__monoid__axioms,axiom,
comm_monoid_set_rat @ inf_inf_set_rat @ top_top_set_rat ).
% inf_top.comm_monoid_axioms
thf(fact_756_inf__top_Ocomm__monoid__axioms,axiom,
comm_monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.comm_monoid_axioms
thf(fact_757_inf__top_Ocomm__monoid__axioms,axiom,
comm_monoid_set_int @ inf_inf_set_int @ top_top_set_int ).
% inf_top.comm_monoid_axioms
thf(fact_758_inf__top_Omonoid__axioms,axiom,
monoid1519094961268053602at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).
% inf_top.monoid_axioms
thf(fact_759_inf__top_Omonoid__axioms,axiom,
monoid6910981727789844035_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).
% inf_top.monoid_axioms
thf(fact_760_inf__top_Omonoid__axioms,axiom,
monoid_list_char_o @ inf_inf_list_char_o @ top_top_list_char_o ).
% inf_top.monoid_axioms
thf(fact_761_inf__top_Omonoid__axioms,axiom,
monoid_set_list_nat @ inf_inf_set_list_nat @ top_top_set_list_nat ).
% inf_top.monoid_axioms
thf(fact_762_inf__top_Omonoid__axioms,axiom,
monoid_set_o @ inf_inf_set_o @ top_top_set_o ).
% inf_top.monoid_axioms
thf(fact_763_inf__top_Omonoid__axioms,axiom,
monoid_set_rat @ inf_inf_set_rat @ top_top_set_rat ).
% inf_top.monoid_axioms
thf(fact_764_inf__top_Omonoid__axioms,axiom,
monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.monoid_axioms
thf(fact_765_inf__top_Omonoid__axioms,axiom,
monoid_set_int @ inf_inf_set_int @ top_top_set_int ).
% inf_top.monoid_axioms
thf(fact_766_comm__monoid__mult__class_Omult__1,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_767_comm__monoid__mult__class_Omult__1,axiom,
! [A: assn] :
( ( times_times_assn @ one_one_assn @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_768_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_769_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_770_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_771_mult_Ocomm__neutral,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
= A ) ).
% mult.comm_neutral
thf(fact_772_mult_Ocomm__neutral,axiom,
! [A: assn] :
( ( times_times_assn @ A @ one_one_assn )
= A ) ).
% mult.comm_neutral
thf(fact_773_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_774_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_775_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_776_inf__top_Osemilattice__neutr__axioms,axiom,
semila858173770281612099at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).
% inf_top.semilattice_neutr_axioms
thf(fact_777_inf__top_Osemilattice__neutr__axioms,axiom,
semila6538661356715008162_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).
% inf_top.semilattice_neutr_axioms
thf(fact_778_inf__top_Osemilattice__neutr__axioms,axiom,
semila3963277552210239504char_o @ inf_inf_list_char_o @ top_top_list_char_o ).
% inf_top.semilattice_neutr_axioms
thf(fact_779_inf__top_Osemilattice__neutr__axioms,axiom,
semila7136768128396663732st_nat @ inf_inf_set_list_nat @ top_top_set_list_nat ).
% inf_top.semilattice_neutr_axioms
thf(fact_780_inf__top_Osemilattice__neutr__axioms,axiom,
semila1065760912024005978_set_o @ inf_inf_set_o @ top_top_set_o ).
% inf_top.semilattice_neutr_axioms
thf(fact_781_inf__top_Osemilattice__neutr__axioms,axiom,
semila6956917442496717612et_rat @ inf_inf_set_rat @ top_top_set_rat ).
% inf_top.semilattice_neutr_axioms
thf(fact_782_inf__top_Osemilattice__neutr__axioms,axiom,
semila1241773964035338532et_nat @ inf_inf_set_nat @ top_top_set_nat ).
% inf_top.semilattice_neutr_axioms
thf(fact_783_inf__top_Osemilattice__neutr__axioms,axiom,
semila6287294981380917632et_int @ inf_inf_set_int @ top_top_set_int ).
% inf_top.semilattice_neutr_axioms
thf(fact_784_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X @ top_to4669805908274784177at_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_785_boolean__algebra_Oconj__one__right,axiom,
! [X: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X @ top_to1578927101902068148_int_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_786_boolean__algebra_Oconj__one__right,axiom,
! [X: list_char > $o] :
( ( inf_inf_list_char_o @ X @ top_top_list_char_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_787_boolean__algebra_Oconj__one__right,axiom,
! [X: set_list_nat] :
( ( inf_inf_set_list_nat @ X @ top_top_set_list_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_788_boolean__algebra_Oconj__one__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_789_boolean__algebra_Oconj__one__right,axiom,
! [X: set_rat] :
( ( inf_inf_set_rat @ X @ top_top_set_rat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_790_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_791_boolean__algebra_Oconj__one__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ top_top_set_int )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_792_mod__h__bot__indep,axiom,
! [P: assn,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
= ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ bot_bot_set_nat ) ) ) ).
% mod_h_bot_indep
thf(fact_793_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ X @ Y )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ X @ Y )
= top_top_Product_unit )
=> ( ( uminus2952777764628376836t_unit @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_794_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ X @ Y )
= top_top_set_o )
=> ( ( uminus_uminus_set_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_795_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( inf_inf_set_rat @ X @ Y )
= bot_bot_set_rat )
=> ( ( ( sup_sup_set_rat @ X @ Y )
= top_top_set_rat )
=> ( ( uminus2201863774496077783et_rat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_796_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_797_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ X @ Y )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ X @ Y )
= top_top_set_int )
=> ( ( uminus1532241313380277803et_int @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_798_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ( inf_inf_set_list_nat @ X @ Y )
= bot_bot_set_list_nat )
=> ( ( ( sup_sup_set_list_nat @ X @ Y )
= top_top_set_list_nat )
=> ( ( uminus3195874150345416415st_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_799_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( ( inf_inf_list_char_o @ X @ Y )
= bot_bot_list_char_o )
=> ( ( ( sup_sup_list_char_o @ X @ Y )
= top_top_list_char_o )
=> ( ( uminus7154032370948614053char_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_800_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ X @ Y )
= bot_bo2099793752762293965at_nat )
=> ( ( ( sup_su6327502436637775413at_nat @ X @ Y )
= top_to4669805908274784177at_nat )
=> ( ( uminus6524753893492686040at_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_801_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( ( inf_in3604695632404883862_int_o @ X @ Y )
= bot_bo8147686125503663512_int_o )
=> ( ( ( sup_su8463660629351352368_int_o @ X @ Y )
= top_to1578927101902068148_int_o )
=> ( ( uminus7117520113953359693_int_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_802_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( ( inf_in1697001100524423349at_nat @ X @ Y )
= bot_bo8422036546324065075at_nat )
=> ( ( ( sup_su3035147773818789531at_nat @ X @ Y )
= top_to8764741339697478167at_nat )
=> ( ( uminus2330091110623919550at_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_803_one__assn__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( As4 = bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_804_one__assn__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( one_assn_raw @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( As4 != bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_805_one__assn__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( Y
= ( As4 != bot_bot_set_nat ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_806_one__assn__raw_Osimps,axiom,
! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( As = bot_bot_set_nat ) ) ).
% one_assn_raw.simps
thf(fact_807_times__assn__raw_Osimps,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( times_assn_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( ? [As1: set_nat,As2: set_nat] :
( ( As
= ( sup_sup_set_nat @ As1 @ As2 ) )
& ( ( inf_inf_set_nat @ As1 @ As2 )
= bot_bot_set_nat )
& ( P @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
& ( Q2 @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_808_times__assn__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( Y
= ( ~ ? [As1: set_nat,As2: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As1 @ As2 ) )
& ( ( inf_inf_set_nat @ As1 @ As2 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_809_times__assn__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ~ ? [As12: set_nat,As22: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As12 @ As22 ) )
& ( ( inf_inf_set_nat @ As12 @ As22 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_810_times__assn__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ? [As13: set_nat,As23: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As13 @ As23 ) )
& ( ( inf_inf_set_nat @ As13 @ As23 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_811_assn__one__left,axiom,
! [P: assn] :
( ( times_times_assn @ one_one_assn @ P )
= P ) ).
% assn_one_left
thf(fact_812_inf__cancel__left1,axiom,
! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ A ) @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ B ) )
= bot_bo8147686125503663512_int_o ) ).
% inf_cancel_left1
thf(fact_813_inf__cancel__left1,axiom,
! [X: list_char > $o,A: list_char > $o,B: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ X @ A ) @ ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ B ) )
= bot_bot_list_char_o ) ).
% inf_cancel_left1
thf(fact_814_inf__cancel__left1,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ A ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ B ) )
= bot_bo2099793752762293965at_nat ) ).
% inf_cancel_left1
thf(fact_815_inf__cancel__left1,axiom,
! [X: set_o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left1
thf(fact_816_inf__cancel__left1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_817_inf__cancel__left1,axiom,
! [X: set_int,A: set_int,B: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
= bot_bot_set_int ) ).
% inf_cancel_left1
thf(fact_818_inf__cancel__left1,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ A ) @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
= bot_bot_Product_unit ) ).
% inf_cancel_left1
thf(fact_819_inf__cancel__left2,axiom,
! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ A ) @ ( inf_in3604695632404883862_int_o @ X @ B ) )
= bot_bo8147686125503663512_int_o ) ).
% inf_cancel_left2
thf(fact_820_inf__cancel__left2,axiom,
! [X: list_char > $o,A: list_char > $o,B: list_char > $o] :
( ( inf_inf_list_char_o @ ( inf_inf_list_char_o @ ( uminus7154032370948614053char_o @ X ) @ A ) @ ( inf_inf_list_char_o @ X @ B ) )
= bot_bot_list_char_o ) ).
% inf_cancel_left2
thf(fact_821_inf__cancel__left2,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ A ) @ ( inf_in2572325071724192079at_nat @ X @ B ) )
= bot_bo2099793752762293965at_nat ) ).
% inf_cancel_left2
thf(fact_822_inf__cancel__left2,axiom,
! [X: set_o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( inf_inf_set_o @ X @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left2
thf(fact_823_inf__cancel__left2,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( inf_inf_set_nat @ X @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_824_inf__cancel__left2,axiom,
! [X: set_int,A: set_int,B: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A ) @ ( inf_inf_set_int @ X @ B ) )
= bot_bot_set_int ) ).
% inf_cancel_left2
thf(fact_825_inf__cancel__left2,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A ) @ ( inf_inf_Product_unit @ X @ B ) )
= bot_bot_Product_unit ) ).
% inf_cancel_left2
thf(fact_826_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_827_mult__minus1,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1
thf(fact_828_mult__minus1,axiom,
! [Z: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1
thf(fact_829_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_830_mult__minus1__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1_right
thf(fact_831_mult__minus1__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1_right
thf(fact_832_Un__Int__eq_I1_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_833_Un__Int__eq_I1_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_834_Un__Int__eq_I1_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_835_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_836_Un__Int__eq_I2_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_837_Un__Int__eq_I2_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_838_Un__Int__eq_I2_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_839_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_840_Un__Int__eq_I3_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_841_Un__Int__eq_I3_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ S @ ( sup_su3035147773818789531at_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_842_Un__Int__eq_I3_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_843_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_844_Un__Int__eq_I4_J,axiom,
! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ T @ ( sup_su6327502436637775413at_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_845_Un__Int__eq_I4_J,axiom,
! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ T @ ( sup_su3035147773818789531at_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_846_Un__Int__eq_I4_J,axiom,
! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ T @ ( sup_su5525570899277871387at_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_847_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_848_Int__Un__eq_I1_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_849_Int__Un__eq_I1_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_850_Int__Un__eq_I1_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_851_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_852_Int__Un__eq_I2_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_853_Int__Un__eq_I2_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_854_Int__Un__eq_I2_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_855_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_856_Int__Un__eq_I3_J,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_857_Int__Un__eq_I3_J,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ S @ ( inf_in1697001100524423349at_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_858_Int__Un__eq_I3_J,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_859_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_860_Int__Un__eq_I4_J,axiom,
! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ T @ ( inf_in2572325071724192079at_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_861_Int__Un__eq_I4_J,axiom,
! [T: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ T @ ( inf_in1697001100524423349at_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_862_Int__Un__eq_I4_J,axiom,
! [T: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ T @ ( inf_in7913087082777306421at_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_863_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_864_Un__empty,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ( sup_su3035147773818789531at_nat @ A2 @ B2 )
= bot_bo8422036546324065075at_nat )
= ( ( A2 = bot_bo8422036546324065075at_nat )
& ( B2 = bot_bo8422036546324065075at_nat ) ) ) ).
% Un_empty
thf(fact_865_Un__empty,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
= bot_bo228742789529271731at_nat )
= ( ( A2 = bot_bo228742789529271731at_nat )
& ( B2 = bot_bo228742789529271731at_nat ) ) ) ).
% Un_empty
thf(fact_866_Un__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( sup_su6327502436637775413at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ( A2 = bot_bo2099793752762293965at_nat )
& ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).
% Un_empty
thf(fact_867_Un__empty,axiom,
! [A2: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_868_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_869_Un__empty,axiom,
! [A2: set_int,B2: set_int] :
( ( ( sup_sup_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ( A2 = bot_bot_set_int )
& ( B2 = bot_bot_set_int ) ) ) ).
% Un_empty
thf(fact_870_Compl__disjoint2,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ A2 )
= bot_bo2099793752762293965at_nat ) ).
% Compl_disjoint2
thf(fact_871_Compl__disjoint2,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_872_Compl__disjoint2,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_873_Compl__disjoint2,axiom,
! [A2: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ A2 )
= bot_bot_set_int ) ).
% Compl_disjoint2
thf(fact_874_empty__iff,axiom,
! [C: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ C @ bot_bo1488462491386950373nt_int ) ).
% empty_iff
thf(fact_875_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_876_empty__iff,axiom,
! [C: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ C @ bot_bo7824918357723582233_nat_o ) ).
% empty_iff
thf(fact_877_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_878_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_879_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_880_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_881_all__not__in__conv,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ( ! [X3: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ X3 @ A2 ) )
= ( A2 = bot_bo1488462491386950373nt_int ) ) ).
% all_not_in_conv
thf(fact_882_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_883_all__not__in__conv,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( ! [X3: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X3 @ A2 ) )
= ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% all_not_in_conv
thf(fact_884_all__not__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A2 ) )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_885_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_886_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_887_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X3: int] :
~ ( member_int @ X3 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_888_Collect__empty__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ( ( collec5210948495886036740nt_int @ P )
= bot_bo1488462491386950373nt_int )
= ( ! [X3: set_Pr958786334691620121nt_int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_889_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_890_Collect__empty__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ( collec939566748876313656_nat_o @ P )
= bot_bo7824918357723582233_nat_o )
= ( ! [X3: produc3658429121746597890et_nat > $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_891_Collect__empty__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( ! [X3: product_prod_nat_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_892_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_893_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_894_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_895_empty__Collect__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ( bot_bo1488462491386950373nt_int
= ( collec5210948495886036740nt_int @ P ) )
= ( ! [X3: set_Pr958786334691620121nt_int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_896_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_897_empty__Collect__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( bot_bo7824918357723582233_nat_o
= ( collec939566748876313656_nat_o @ P ) )
= ( ! [X3: produc3658429121746597890et_nat > $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_898_empty__Collect__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ P ) )
= ( ! [X3: product_prod_nat_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_899_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_900_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_901_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_902_IntI,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ A2 )
=> ( ( member2340774599025711042nt_int @ C @ B2 )
=> ( member2340774599025711042nt_int @ C @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_903_IntI,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_904_IntI,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ A2 )
=> ( ( member_int @ C @ B2 )
=> ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_905_IntI,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ A2 )
=> ( ( member6576561426505652726_nat_o @ C @ B2 )
=> ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_906_IntI,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ A2 )
=> ( ( member8440522571783428010at_nat @ C @ B2 )
=> ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_907_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_908_Int__iff,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) )
= ( ( member2340774599025711042nt_int @ C @ A2 )
& ( member2340774599025711042nt_int @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_909_Int__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
& ( member_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_910_Int__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( ( member_int @ C @ A2 )
& ( member_int @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_911_Int__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) )
= ( ( member6576561426505652726_nat_o @ C @ A2 )
& ( member6576561426505652726_nat_o @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_912_Int__iff,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( ( member8440522571783428010at_nat @ C @ A2 )
& ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_913_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_914_Int__UNIV,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= top_to4669805908274784177at_nat )
= ( ( A2 = top_to4669805908274784177at_nat )
& ( B2 = top_to4669805908274784177at_nat ) ) ) ).
% Int_UNIV
thf(fact_915_Int__UNIV,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A2 @ B2 )
= top_top_set_list_nat )
= ( ( A2 = top_top_set_list_nat )
& ( B2 = top_top_set_list_nat ) ) ) ).
% Int_UNIV
thf(fact_916_Int__UNIV,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= top_top_set_o )
= ( ( A2 = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_917_Int__UNIV,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( inf_inf_set_rat @ A2 @ B2 )
= top_top_set_rat )
= ( ( A2 = top_top_set_rat )
& ( B2 = top_top_set_rat ) ) ) ).
% Int_UNIV
thf(fact_918_Int__UNIV,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_919_Int__UNIV,axiom,
! [A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ A2 @ B2 )
= top_top_set_int )
= ( ( A2 = top_top_set_int )
& ( B2 = top_top_set_int ) ) ) ).
% Int_UNIV
thf(fact_920_UnCI,axiom,
! [C: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int] :
( ( ~ ( member2340774599025711042nt_int @ C @ B2 )
=> ( member2340774599025711042nt_int @ C @ A2 ) )
=> ( member2340774599025711042nt_int @ C @ ( sup_su2047564715030645325nt_int @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_921_UnCI,axiom,
! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ A2 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_922_UnCI,axiom,
! [C: int,B2: set_int,A2: set_int] :
( ( ~ ( member_int @ C @ B2 )
=> ( member_int @ C @ A2 ) )
=> ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_923_UnCI,axiom,
! [C: produc3658429121746597890et_nat > $o,B2: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o] :
( ( ~ ( member6576561426505652726_nat_o @ C @ B2 )
=> ( member6576561426505652726_nat_o @ C @ A2 ) )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_924_UnCI,axiom,
! [C: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( ~ ( member6689249552917799696at_nat @ C @ B2 )
=> ( member6689249552917799696at_nat @ C @ A2 ) )
=> ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_925_UnCI,axiom,
! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( ~ ( member8757157785044589968at_nat @ C @ B2 )
=> ( member8757157785044589968at_nat @ C @ A2 ) )
=> ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_926_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_927_Un__iff,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( sup_su2047564715030645325nt_int @ A2 @ B2 ) )
= ( ( member2340774599025711042nt_int @ C @ A2 )
| ( member2340774599025711042nt_int @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_928_Un__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
| ( member_set_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_929_Un__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( ( member_int @ C @ A2 )
| ( member_int @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_930_Un__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A2 @ B2 ) )
= ( ( member6576561426505652726_nat_o @ C @ A2 )
| ( member6576561426505652726_nat_o @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_931_Un__iff,axiom,
! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( ( member6689249552917799696at_nat @ C @ A2 )
| ( member6689249552917799696at_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_932_Un__iff,axiom,
! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( ( member8757157785044589968at_nat @ C @ A2 )
| ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_933_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_934_ComplI,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ~ ( member2340774599025711042nt_int @ C @ A2 )
=> ( member2340774599025711042nt_int @ C @ ( uminus6423885277529793776nt_int @ A2 ) ) ) ).
% ComplI
thf(fact_935_ComplI,axiom,
! [C: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_936_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_937_ComplI,axiom,
! [C: int,A2: set_int] :
( ~ ( member_int @ C @ A2 )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% ComplI
thf(fact_938_ComplI,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ C @ A2 )
=> ( member6576561426505652726_nat_o @ C @ ( uminus5254974436814262692_nat_o @ A2 ) ) ) ).
% ComplI
thf(fact_939_Compl__iff,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( uminus6423885277529793776nt_int @ A2 ) )
= ( ~ ( member2340774599025711042nt_int @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_940_Compl__iff,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
= ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_941_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_942_Compl__iff,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
= ( ~ ( member_int @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_943_Compl__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( uminus5254974436814262692_nat_o @ A2 ) )
= ( ~ ( member6576561426505652726_nat_o @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_944_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec5210948495886036740nt_int
@ ^ [S2: set_Pr958786334691620121nt_int] : P )
= top_to6034159466715884489nt_int ) )
& ( ~ P
=> ( ( collec5210948495886036740nt_int
@ ^ [S2: set_Pr958786334691620121nt_int] : P )
= bot_bo1488462491386950373nt_int ) ) ) ).
% Collect_const
thf(fact_945_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_set_nat
@ ^ [S2: set_nat] : P )
= top_top_set_set_nat ) )
& ( ~ P
=> ( ( collect_set_nat
@ ^ [S2: set_nat] : P )
= bot_bot_set_set_nat ) ) ) ).
% Collect_const
thf(fact_946_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec939566748876313656_nat_o
@ ^ [S2: produc3658429121746597890et_nat > $o] : P )
= top_to4869887016220417533_nat_o ) )
& ( ~ P
=> ( ( collec939566748876313656_nat_o
@ ^ [S2: produc3658429121746597890et_nat > $o] : P )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% Collect_const
thf(fact_947_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec3392354462482085612at_nat
@ ^ [S2: product_prod_nat_nat] : P )
= top_to4669805908274784177at_nat ) )
& ( ~ P
=> ( ( collec3392354462482085612at_nat
@ ^ [S2: product_prod_nat_nat] : P )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_const
thf(fact_948_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_list_nat
@ ^ [S2: list_nat] : P )
= top_top_set_list_nat ) )
& ( ~ P
=> ( ( collect_list_nat
@ ^ [S2: list_nat] : P )
= bot_bot_set_list_nat ) ) ) ).
% Collect_const
thf(fact_949_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_950_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_rat
@ ^ [S2: rat] : P )
= top_top_set_rat ) )
& ( ~ P
=> ( ( collect_rat
@ ^ [S2: rat] : P )
= bot_bot_set_rat ) ) ) ).
% Collect_const
thf(fact_951_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_952_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_int
@ ^ [S2: int] : P )
= top_top_set_int ) )
& ( ~ P
=> ( ( collect_int
@ ^ [S2: int] : P )
= bot_bot_set_int ) ) ) ).
% Collect_const
thf(fact_953_Compl__disjoint,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
= bot_bo2099793752762293965at_nat ) ).
% Compl_disjoint
thf(fact_954_Compl__disjoint,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_955_Compl__disjoint,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_956_Compl__disjoint,axiom,
! [A2: set_int] :
( ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= bot_bot_set_int ) ).
% Compl_disjoint
thf(fact_957_ComplD,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( uminus6423885277529793776nt_int @ A2 ) )
=> ~ ( member2340774599025711042nt_int @ C @ A2 ) ) ).
% ComplD
thf(fact_958_ComplD,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
=> ~ ( member_set_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_959_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_960_ComplD,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
=> ~ ( member_int @ C @ A2 ) ) ).
% ComplD
thf(fact_961_ComplD,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( uminus5254974436814262692_nat_o @ A2 ) )
=> ~ ( member6576561426505652726_nat_o @ C @ A2 ) ) ).
% ComplD
thf(fact_962_Compl__eq,axiom,
( uminus6423885277529793776nt_int
= ( ^ [A4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ X3 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_963_Compl__eq,axiom,
( uminus613421341184616069et_nat
= ( ^ [A4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
~ ( member_set_nat @ X3 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_964_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
~ ( member_nat @ X3 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_965_Compl__eq,axiom,
( uminus1532241313380277803et_int
= ( ^ [A4: set_int] :
( collect_int
@ ^ [X3: int] :
~ ( member_int @ X3 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_966_Compl__eq,axiom,
( uminus5254974436814262692_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X3 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_967_Collect__neg__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
~ ( P @ X3 ) )
= ( uminus6423885277529793776nt_int @ ( collec5210948495886036740nt_int @ P ) ) ) ).
% Collect_neg_eq
thf(fact_968_Collect__neg__eq,axiom,
! [P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
~ ( P @ X3 ) )
= ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_969_Collect__neg__eq,axiom,
! [P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) )
= ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_970_Collect__neg__eq,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X3: int] :
~ ( P @ X3 ) )
= ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% Collect_neg_eq
thf(fact_971_Collect__neg__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
~ ( P @ X3 ) )
= ( uminus5254974436814262692_nat_o @ ( collec939566748876313656_nat_o @ P ) ) ) ).
% Collect_neg_eq
thf(fact_972_uminus__set__def,axiom,
( uminus6423885277529793776nt_int
= ( ^ [A4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ( uminus8147837162492574189_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_973_uminus__set__def,axiom,
( uminus613421341184616069et_nat
= ( ^ [A4: set_set_nat] :
( collect_set_nat
@ ( uminus6401447641752708672_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_974_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A4: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_975_uminus__set__def,axiom,
( uminus1532241313380277803et_int
= ( ^ [A4: set_int] :
( collect_int
@ ( uminus_uminus_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_976_uminus__set__def,axiom,
( uminus5254974436814262692_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ( uminus8928876914783121465at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_977_emptyE,axiom,
! [A: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ A @ bot_bo1488462491386950373nt_int ) ).
% emptyE
thf(fact_978_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_979_emptyE,axiom,
! [A: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% emptyE
thf(fact_980_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_981_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_982_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_983_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_984_equals0D,axiom,
! [A2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
( ( A2 = bot_bo1488462491386950373nt_int )
=> ~ ( member2340774599025711042nt_int @ A @ A2 ) ) ).
% equals0D
thf(fact_985_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_986_equals0D,axiom,
! [A2: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o] :
( ( A2 = bot_bo7824918357723582233_nat_o )
=> ~ ( member6576561426505652726_nat_o @ A @ A2 ) ) ).
% equals0D
thf(fact_987_equals0D,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A2 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_988_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_989_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_990_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_991_equals0I,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ! [Y2: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ Y2 @ A2 )
=> ( A2 = bot_bo1488462491386950373nt_int ) ) ).
% equals0I
thf(fact_992_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_993_equals0I,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ! [Y2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ Y2 @ A2 )
=> ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% equals0I
thf(fact_994_equals0I,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ! [Y2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y2 @ A2 )
=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_995_equals0I,axiom,
! [A2: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_996_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_997_equals0I,axiom,
! [A2: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_998_ex__in__conv,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ( ? [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A2 ) )
= ( A2 != bot_bo1488462491386950373nt_int ) ) ).
% ex_in_conv
thf(fact_999_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_1000_ex__in__conv,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( ? [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A2 ) )
= ( A2 != bot_bo7824918357723582233_nat_o ) ) ).
% ex_in_conv
thf(fact_1001_ex__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
= ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_1002_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_1003_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1004_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X3: int] : ( member_int @ X3 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_1005_bot__set__def,axiom,
( bot_bo1488462491386950373nt_int
= ( collec5210948495886036740nt_int @ bot_bo2686080419298087992_int_o ) ) ).
% bot_set_def
thf(fact_1006_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_1007_bot__set__def,axiom,
( bot_bo7824918357723582233_nat_o
= ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).
% bot_set_def
thf(fact_1008_bot__set__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% bot_set_def
thf(fact_1009_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_1010_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1011_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_1012_Compl__UNIV__eq,axiom,
( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
= bot_bo2099793752762293965at_nat ) ).
% Compl_UNIV_eq
thf(fact_1013_Compl__UNIV__eq,axiom,
( ( uminus3195874150345416415st_nat @ top_top_set_list_nat )
= bot_bot_set_list_nat ) ).
% Compl_UNIV_eq
thf(fact_1014_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_1015_Compl__UNIV__eq,axiom,
( ( uminus2201863774496077783et_rat @ top_top_set_rat )
= bot_bot_set_rat ) ).
% Compl_UNIV_eq
thf(fact_1016_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1017_Compl__UNIV__eq,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% Compl_UNIV_eq
thf(fact_1018_Compl__empty__eq,axiom,
( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
= top_to4669805908274784177at_nat ) ).
% Compl_empty_eq
thf(fact_1019_Compl__empty__eq,axiom,
( ( uminus3195874150345416415st_nat @ bot_bot_set_list_nat )
= top_top_set_list_nat ) ).
% Compl_empty_eq
thf(fact_1020_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_1021_Compl__empty__eq,axiom,
( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
= top_top_set_rat ) ).
% Compl_empty_eq
thf(fact_1022_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1023_Compl__empty__eq,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% Compl_empty_eq
thf(fact_1024_empty__not__UNIV,axiom,
bot_bo2099793752762293965at_nat != top_to4669805908274784177at_nat ).
% empty_not_UNIV
thf(fact_1025_empty__not__UNIV,axiom,
bot_bot_set_list_nat != top_top_set_list_nat ).
% empty_not_UNIV
thf(fact_1026_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_1027_empty__not__UNIV,axiom,
bot_bot_set_rat != top_top_set_rat ).
% empty_not_UNIV
thf(fact_1028_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1029_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
thf(fact_1030_IntE,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) )
=> ~ ( ( member2340774599025711042nt_int @ C @ A2 )
=> ~ ( member2340774599025711042nt_int @ C @ B2 ) ) ) ).
% IntE
thf(fact_1031_IntE,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ~ ( member_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1032_IntE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ~ ( member_int @ C @ B2 ) ) ) ).
% IntE
thf(fact_1033_IntE,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) )
=> ~ ( ( member6576561426505652726_nat_o @ C @ A2 )
=> ~ ( member6576561426505652726_nat_o @ C @ B2 ) ) ) ).
% IntE
thf(fact_1034_IntE,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ~ ( ( member8440522571783428010at_nat @ C @ A2 )
=> ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1035_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1036_IntD1,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) )
=> ( member2340774599025711042nt_int @ C @ A2 ) ) ).
% IntD1
thf(fact_1037_IntD1,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1038_IntD1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( member_int @ C @ A2 ) ) ).
% IntD1
thf(fact_1039_IntD1,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) )
=> ( member6576561426505652726_nat_o @ C @ A2 ) ) ).
% IntD1
thf(fact_1040_IntD1,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1041_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1042_IntD2,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) )
=> ( member2340774599025711042nt_int @ C @ B2 ) ) ).
% IntD2
thf(fact_1043_IntD2,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( member_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1044_IntD2,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( member_int @ C @ B2 ) ) ).
% IntD2
thf(fact_1045_IntD2,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) )
=> ( member6576561426505652726_nat_o @ C @ B2 ) ) ).
% IntD2
thf(fact_1046_IntD2,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1047_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1048_Int__assoc,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1049_Int__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1050_Int__absorb,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1051_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1052_Int__commute,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_1053_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_1054_Int__UNIV__left,axiom,
! [B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1055_Int__UNIV__left,axiom,
! [B2: set_list_nat] :
( ( inf_inf_set_list_nat @ top_top_set_list_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1056_Int__UNIV__left,axiom,
! [B2: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1057_Int__UNIV__left,axiom,
! [B2: set_rat] :
( ( inf_inf_set_rat @ top_top_set_rat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1058_Int__UNIV__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1059_Int__UNIV__left,axiom,
! [B2: set_int] :
( ( inf_inf_set_int @ top_top_set_int @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1060_Int__UNIV__right,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ top_to4669805908274784177at_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1061_Int__UNIV__right,axiom,
! [A2: set_list_nat] :
( ( inf_inf_set_list_nat @ A2 @ top_top_set_list_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1062_Int__UNIV__right,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ top_top_set_o )
= A2 ) ).
% Int_UNIV_right
thf(fact_1063_Int__UNIV__right,axiom,
! [A2: set_rat] :
( ( inf_inf_set_rat @ A2 @ top_top_set_rat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1064_Int__UNIV__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1065_Int__UNIV__right,axiom,
! [A2: set_int] :
( ( inf_inf_set_int @ A2 @ top_top_set_int )
= A2 ) ).
% Int_UNIV_right
thf(fact_1066_Int__left__absorb,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1067_Int__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1068_Int__left__commute,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
= ( inf_in2572325071724192079at_nat @ B2 @ ( inf_in2572325071724192079at_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1069_Int__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1070_UnE,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ ( sup_su2047564715030645325nt_int @ A2 @ B2 ) )
=> ( ~ ( member2340774599025711042nt_int @ C @ A2 )
=> ( member2340774599025711042nt_int @ C @ B2 ) ) ) ).
% UnE
thf(fact_1071_UnE,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_1072_UnE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) )
=> ( ~ ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% UnE
thf(fact_1073_UnE,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A2 @ B2 ) )
=> ( ~ ( member6576561426505652726_nat_o @ C @ A2 )
=> ( member6576561426505652726_nat_o @ C @ B2 ) ) ) ).
% UnE
thf(fact_1074_UnE,axiom,
! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
=> ( ~ ( member6689249552917799696at_nat @ C @ A2 )
=> ( member6689249552917799696at_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_1075_UnE,axiom,
! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
=> ( ~ ( member8757157785044589968at_nat @ C @ A2 )
=> ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_1076_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_1077_UnI1,axiom,
! [C: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ A2 )
=> ( member2340774599025711042nt_int @ C @ ( sup_su2047564715030645325nt_int @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1078_UnI1,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1079_UnI1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ A2 )
=> ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1080_UnI1,axiom,
! [C: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ A2 )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1081_UnI1,axiom,
! [C: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( member6689249552917799696at_nat @ C @ A2 )
=> ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1082_UnI1,axiom,
! [C: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ C @ A2 )
=> ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1083_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1084_UnI2,axiom,
! [C: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ C @ B2 )
=> ( member2340774599025711042nt_int @ C @ ( sup_su2047564715030645325nt_int @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1085_UnI2,axiom,
! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1086_UnI2,axiom,
! [C: int,B2: set_int,A2: set_int] :
( ( member_int @ C @ B2 )
=> ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1087_UnI2,axiom,
! [C: produc3658429121746597890et_nat > $o,B2: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ B2 )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1088_UnI2,axiom,
! [C: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( member6689249552917799696at_nat @ C @ B2 )
=> ( member6689249552917799696at_nat @ C @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1089_UnI2,axiom,
! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ C @ B2 )
=> ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1090_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1091_bex__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,P: produc4166570645942440679at_nat > $o] :
( ( ? [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1092_bex__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
( ( ? [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1093_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1094_ball__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,P: produc4166570645942440679at_nat > $o] :
( ( ! [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1095_ball__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
( ( ! [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1096_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1097_Un__assoc,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ C2 )
= ( sup_su3035147773818789531at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1098_Un__assoc,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C2 )
= ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1099_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_1100_Un__absorb,axiom,
! [A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1101_Un__absorb,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1102_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1103_Un__commute,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1104_Un__commute,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1105_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1106_Un__UNIV__left,axiom,
! [B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ top_to8764741339697478167at_nat @ B2 )
= top_to8764741339697478167at_nat ) ).
% Un_UNIV_left
thf(fact_1107_Un__UNIV__left,axiom,
! [B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ top_to6833984726390702231at_nat @ B2 )
= top_to6833984726390702231at_nat ) ).
% Un_UNIV_left
thf(fact_1108_Un__UNIV__left,axiom,
! [B2: set_list_nat] :
( ( sup_sup_set_list_nat @ top_top_set_list_nat @ B2 )
= top_top_set_list_nat ) ).
% Un_UNIV_left
thf(fact_1109_Un__UNIV__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B2 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_1110_Un__UNIV__left,axiom,
! [B2: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ B2 )
= top_top_set_rat ) ).
% Un_UNIV_left
thf(fact_1111_Un__UNIV__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B2 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1112_Un__UNIV__left,axiom,
! [B2: set_int] :
( ( sup_sup_set_int @ top_top_set_int @ B2 )
= top_top_set_int ) ).
% Un_UNIV_left
thf(fact_1113_Un__UNIV__right,axiom,
! [A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ top_to8764741339697478167at_nat )
= top_to8764741339697478167at_nat ) ).
% Un_UNIV_right
thf(fact_1114_Un__UNIV__right,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ top_to6833984726390702231at_nat )
= top_to6833984726390702231at_nat ) ).
% Un_UNIV_right
thf(fact_1115_Un__UNIV__right,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ A2 @ top_top_set_list_nat )
= top_top_set_list_nat ) ).
% Un_UNIV_right
thf(fact_1116_Un__UNIV__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_1117_Un__UNIV__right,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ A2 @ top_top_set_rat )
= top_top_set_rat ) ).
% Un_UNIV_right
thf(fact_1118_Un__UNIV__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1119_Un__UNIV__right,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ top_top_set_int )
= top_top_set_int ) ).
% Un_UNIV_right
thf(fact_1120_Un__left__absorb,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_1121_Un__left__absorb,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_1122_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_1123_Compl__partition,axiom,
! [A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( uminus2330091110623919550at_nat @ A2 ) )
= top_to8764741339697478167at_nat ) ).
% Compl_partition
thf(fact_1124_Compl__partition,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( uminus935396558254630718at_nat @ A2 ) )
= top_to6833984726390702231at_nat ) ).
% Compl_partition
thf(fact_1125_Compl__partition,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ A2 @ ( uminus3195874150345416415st_nat @ A2 ) )
= top_top_set_list_nat ) ).
% Compl_partition
thf(fact_1126_Compl__partition,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= top_top_set_o ) ).
% Compl_partition
thf(fact_1127_Compl__partition,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ A2 @ ( uminus2201863774496077783et_rat @ A2 ) )
= top_top_set_rat ) ).
% Compl_partition
thf(fact_1128_Compl__partition,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_1129_Compl__partition,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= top_top_set_int ) ).
% Compl_partition
thf(fact_1130_Un__left__commute,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) )
= ( sup_su3035147773818789531at_nat @ B2 @ ( sup_su3035147773818789531at_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1131_Un__left__commute,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
= ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1132_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1133_Compl__partition2,axiom,
! [A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A2 ) @ A2 )
= top_to8764741339697478167at_nat ) ).
% Compl_partition2
thf(fact_1134_Compl__partition2,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ A2 )
= top_to6833984726390702231at_nat ) ).
% Compl_partition2
thf(fact_1135_Compl__partition2,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ A2 ) @ A2 )
= top_top_set_list_nat ) ).
% Compl_partition2
thf(fact_1136_Compl__partition2,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
= top_top_set_o ) ).
% Compl_partition2
thf(fact_1137_Compl__partition2,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ A2 ) @ A2 )
= top_top_set_rat ) ).
% Compl_partition2
thf(fact_1138_Compl__partition2,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_1139_Compl__partition2,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ A2 )
= top_top_set_int ) ).
% Compl_partition2
thf(fact_1140_Set_Oempty__def,axiom,
( bot_bo1488462491386950373nt_int
= ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : $false ) ) ).
% Set.empty_def
thf(fact_1141_Set_Oempty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $false ) ) ).
% Set.empty_def
thf(fact_1142_Set_Oempty__def,axiom,
( bot_bo7824918357723582233_nat_o
= ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : $false ) ) ).
% Set.empty_def
thf(fact_1143_Set_Oempty__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : $false ) ) ).
% Set.empty_def
thf(fact_1144_Set_Oempty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X3: $o] : $false ) ) ).
% Set.empty_def
thf(fact_1145_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% Set.empty_def
thf(fact_1146_Set_Oempty__def,axiom,
( bot_bot_set_int
= ( collect_int
@ ^ [X3: int] : $false ) ) ).
% Set.empty_def
thf(fact_1147_Int__def,axiom,
( inf_in8396524679539076455nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ A4 )
& ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1148_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1149_Int__def,axiom,
( inf_inf_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ( member_int @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1150_Int__def,axiom,
( inf_in1906310914598751387_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A4 )
& ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1151_Int__def,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1152_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1153_Int__Collect,axiom,
! [X: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( member2340774599025711042nt_int @ X @ ( inf_in8396524679539076455nt_int @ A2 @ ( collec5210948495886036740nt_int @ P ) ) )
= ( ( member2340774599025711042nt_int @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1154_Int__Collect,axiom,
! [X: set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1155_Int__Collect,axiom,
! [X: int,A2: set_int,P: int > $o] :
( ( member_int @ X @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) )
= ( ( member_int @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1156_Int__Collect,axiom,
! [X: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) ) )
= ( ( member6576561426505652726_nat_o @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1157_Int__Collect,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) ) )
= ( ( member8440522571783428010at_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1158_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1159_inf__set__def,axiom,
( inf_in8396524679539076455nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ( inf_in5102985939729578038_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1160_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1161_inf__set__def,axiom,
( inf_inf_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ( inf_inf_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 )
@ ^ [X3: int] : ( member_int @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1162_inf__set__def,axiom,
( inf_in1906310914598751387_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ( inf_in9018898317438245890at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1163_inf__set__def,axiom,
( inf_in2269163501485487111nt_int
= ( ^ [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( collec213857154873943460nt_int
@ ( inf_in3604695632404883862_int_o
@ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A4 )
@ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1164_inf__set__def,axiom,
( inf_in6177012701088388009t_char
= ( ^ [A4: set_list_char,B4: set_list_char] :
( collect_list_char
@ ( inf_inf_list_char_o
@ ^ [X3: list_char] : ( member_list_char @ X3 @ A4 )
@ ^ [X3: list_char] : ( member_list_char @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1165_inf__set__def,axiom,
( inf_in2572325071724192079at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ( inf_in5163264567034779214_nat_o
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A4 )
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1166_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1167_Collect__conj__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in8396524679539076455nt_int @ ( collec5210948495886036740nt_int @ P ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1168_Collect__conj__eq,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1169_Collect__conj__eq,axiom,
! [P: int > $o,Q2: int > $o] :
( ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1170_Collect__conj__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in1906310914598751387_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1171_Collect__conj__eq,axiom,
! [P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1172_Collect__conj__eq,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1173_Un__def,axiom,
( sup_su2047564715030645325nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ A4 )
| ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1174_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
| ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1175_Un__def,axiom,
( sup_sup_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
| ( member_int @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1176_Un__def,axiom,
( sup_su5209123915105501825_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A4 )
| ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1177_Un__def,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
( collec5204685387357076818at_nat
@ ^ [X3: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X3 @ A4 )
| ( member6689249552917799696at_nat @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1178_Un__def,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X3 @ A4 )
| ( member8757157785044589968at_nat @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1179_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
| ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1180_sup__set__def,axiom,
( sup_su2047564715030645325nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ( sup_su1852724690005176016_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1181_sup__set__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( sup_sup_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1182_sup__set__def,axiom,
( sup_sup_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ( sup_sup_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 )
@ ^ [X3: int] : ( member_int @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1183_sup__set__def,axiom,
( sup_su5209123915105501825_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ( sup_su8003631835303541148at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1184_sup__set__def,axiom,
( sup_su3035147773818789531at_nat
= ( ^ [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
( collec5204685387357076818at_nat
@ ( sup_su6256023009775730178_nat_o
@ ^ [X3: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X3 @ A4 )
@ ^ [X3: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1185_sup__set__def,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
( collec6321179662152712658at_nat
@ ( sup_su2080679670758317954_nat_o
@ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A4 )
@ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1186_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1187_Collect__imp__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_su2047564715030645325nt_int @ ( uminus6423885277529793776nt_int @ ( collec5210948495886036740nt_int @ P ) ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1188_Collect__imp__eq,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1189_Collect__imp__eq,axiom,
! [P: int > $o,Q2: int > $o] :
( ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1190_Collect__imp__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_su5209123915105501825_nat_o @ ( uminus5254974436814262692_nat_o @ ( collec939566748876313656_nat_o @ P ) ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1191_Collect__imp__eq,axiom,
! [P: produc4166570645942440679at_nat > $o,Q2: produc4166570645942440679at_nat > $o] :
( ( collec5204685387357076818at_nat
@ ^ [X3: produc4166570645942440679at_nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ ( collec5204685387357076818at_nat @ P ) ) @ ( collec5204685387357076818at_nat @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1192_Collect__imp__eq,axiom,
! [P: produc3843707927480180839at_nat > $o,Q2: produc3843707927480180839at_nat > $o] :
( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1193_Collect__imp__eq,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_1194_Collect__disj__eq,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su2047564715030645325nt_int @ ( collec5210948495886036740nt_int @ P ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1195_Collect__disj__eq,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1196_Collect__disj__eq,axiom,
! [P: int > $o,Q2: int > $o] :
( ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1197_Collect__disj__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su5209123915105501825_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1198_Collect__disj__eq,axiom,
! [P: produc4166570645942440679at_nat > $o,Q2: produc4166570645942440679at_nat > $o] :
( ( collec5204685387357076818at_nat
@ ^ [X3: produc4166570645942440679at_nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su3035147773818789531at_nat @ ( collec5204685387357076818at_nat @ P ) @ ( collec5204685387357076818at_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1199_Collect__disj__eq,axiom,
! [P: produc3843707927480180839at_nat > $o,Q2: produc3843707927480180839at_nat > $o] :
( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1200_Collect__disj__eq,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_1201_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1202_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_1203_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_1204_Int__emptyI,axiom,
! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A2 )
=> ~ ( member2340774599025711042nt_int @ X2 @ B2 ) )
=> ( ( inf_in8396524679539076455nt_int @ A2 @ B2 )
= bot_bo1488462491386950373nt_int ) ) ).
% Int_emptyI
thf(fact_1205_Int__emptyI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ( member_set_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_1206_Int__emptyI,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ~ ( member6576561426505652726_nat_o @ X2 @ B2 ) )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ B2 )
= bot_bo7824918357723582233_nat_o ) ) ).
% Int_emptyI
thf(fact_1207_Int__emptyI,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ~ ( member8440522571783428010at_nat @ X2 @ B2 ) )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat ) ) ).
% Int_emptyI
thf(fact_1208_Int__emptyI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ~ ( member_o @ X2 @ B2 ) )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1209_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1210_Int__emptyI,axiom,
! [A2: set_int,B2: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ~ ( member_int @ X2 @ B2 ) )
=> ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int ) ) ).
% Int_emptyI
thf(fact_1211_disjoint__iff,axiom,
! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( ( inf_in8396524679539076455nt_int @ A2 @ B2 )
= bot_bo1488462491386950373nt_int )
= ( ! [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ A2 )
=> ~ ( member2340774599025711042nt_int @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1212_disjoint__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ~ ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1213_disjoint__iff,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A2 @ B2 )
= bot_bo7824918357723582233_nat_o )
= ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A2 )
=> ~ ( member6576561426505652726_nat_o @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1214_disjoint__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ~ ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1215_disjoint__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1216_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1217_disjoint__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ~ ( member_int @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1218_Int__empty__left,axiom,
! [B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B2 )
= bot_bo2099793752762293965at_nat ) ).
% Int_empty_left
thf(fact_1219_Int__empty__left,axiom,
! [B2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_1220_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1221_Int__empty__left,axiom,
! [B2: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ B2 )
= bot_bot_set_int ) ).
% Int_empty_left
thf(fact_1222_Int__empty__right,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= bot_bo2099793752762293965at_nat ) ).
% Int_empty_right
thf(fact_1223_Int__empty__right,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_1224_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1225_Int__empty__right,axiom,
! [A2: set_int] :
( ( inf_inf_set_int @ A2 @ bot_bot_set_int )
= bot_bot_set_int ) ).
% Int_empty_right
thf(fact_1226_disjoint__iff__not__equal,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1227_disjoint__iff__not__equal,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( X3 = ~ Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1228_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1229_disjoint__iff__not__equal,axiom,
! [A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1230_Un__empty__left,axiom,
! [B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1231_Un__empty__left,axiom,
! [B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1232_Un__empty__left,axiom,
! [B2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1233_Un__empty__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1234_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1235_Un__empty__left,axiom,
! [B2: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1236_Un__empty__right,axiom,
! [A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ bot_bo8422036546324065075at_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1237_Un__empty__right,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1238_Un__empty__right,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1239_Un__empty__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Un_empty_right
thf(fact_1240_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1241_Un__empty__right,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Un_empty_right
thf(fact_1242_Compl__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_1243_Compl__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( uminus2330091110623919550at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( inf_in1697001100524423349at_nat @ ( uminus2330091110623919550at_nat @ A2 ) @ ( uminus2330091110623919550at_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_1244_Compl__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_1245_Compl__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% Compl_Un
thf(fact_1246_Compl__Int,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_1247_Compl__Int,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( uminus2330091110623919550at_nat @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A2 ) @ ( uminus2330091110623919550at_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_1248_Compl__Int,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_1249_Compl__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% Compl_Int
thf(fact_1250_Un__Int__crazy,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A2 ) )
= ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1251_Un__Int__crazy,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) @ ( inf_in1697001100524423349at_nat @ B2 @ C2 ) ) @ ( inf_in1697001100524423349at_nat @ C2 @ A2 ) )
= ( inf_in1697001100524423349at_nat @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) @ ( sup_su3035147773818789531at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1252_Un__Int__crazy,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A2 ) )
= ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1253_Un__Int__crazy,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1254_Int__Un__distrib,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1255_Int__Un__distrib,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) @ ( inf_in1697001100524423349at_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1256_Int__Un__distrib,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1257_Int__Un__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1258_Un__Int__distrib,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) @ ( sup_su6327502436637775413at_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1259_Un__Int__distrib,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( inf_in1697001100524423349at_nat @ B2 @ C2 ) )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ ( sup_su3035147773818789531at_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1260_Un__Int__distrib,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ ( sup_su5525570899277871387at_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1261_Un__Int__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1262_Int__Un__distrib2,axiom,
! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) @ A2 )
= ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ A2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1263_Int__Un__distrib2,axiom,
! [B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) @ A2 )
= ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B2 @ A2 ) @ ( inf_in1697001100524423349at_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1264_Int__Un__distrib2,axiom,
! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A2 )
= ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ A2 ) @ ( inf_in7913087082777306421at_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1265_Int__Un__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1266_Un__Int__distrib2,axiom,
! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) @ A2 )
= ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ A2 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1267_Un__Int__distrib2,axiom,
! [B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B2 @ C2 ) @ A2 )
= ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B2 @ A2 ) @ ( sup_su3035147773818789531at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1268_Un__Int__distrib2,axiom,
! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) @ A2 )
= ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ A2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1269_Un__Int__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A2 ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1270_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1271_mult__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1272_mult__minus__right,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1273_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1274_minus__mult__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( times_3573771949741848930nteger @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1275_minus__mult__minus,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( times_times_rat @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1276_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1277_mult__minus__left,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1278_mult__minus__left,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1279_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_1280_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( ( times_3573771949741848930nteger @ X @ X )
= one_one_Code_integer )
= ( ( X = one_one_Code_integer )
| ( X
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% square_eq_1_iff
thf(fact_1281_square__eq__1__iff,axiom,
! [X: rat] :
( ( ( times_times_rat @ X @ X )
= one_one_rat )
= ( ( X = one_one_rat )
| ( X
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% square_eq_1_iff
thf(fact_1282_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1283_verit__minus__simplify_I4_J,axiom,
! [B: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1284_verit__minus__simplify_I4_J,axiom,
! [B: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1285_old_Oprod_Oinject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_1286_old_Oprod_Oinject,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ A @ B )
= ( produc5001842942810119800et_nat @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_1287_old_Oprod_Oinject,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B5: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ A @ B )
= ( produc2245416461498447860et_nat @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_1288_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_1289_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_1290_prod_Oinject,axiom,
! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ X1 @ X22 )
= ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_1291_prod_Oinject,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
= ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_1292_prod_Oinject,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
= ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_1293_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
= ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_1294_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
= ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_1295_lambda__one,axiom,
( ( ^ [X3: code_integer] : X3 )
= ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).
% lambda_one
thf(fact_1296_lambda__one,axiom,
( ( ^ [X3: assn] : X3 )
= ( times_times_assn @ one_one_assn ) ) ).
% lambda_one
thf(fact_1297_lambda__one,axiom,
( ( ^ [X3: rat] : X3 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_1298_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1299_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1300_disjointI,axiom,
! [A: set_se6260736226359567993nt_int,B: set_se6260736226359567993nt_int] :
( ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A )
=> ~ ( member2340774599025711042nt_int @ X2 @ B ) )
=> ( ( inf_in8396524679539076455nt_int @ A @ B )
= bot_bo1488462491386950373nt_int ) ) ).
% disjointI
thf(fact_1301_disjointI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ~ ( member_set_nat @ X2 @ B ) )
=> ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% disjointI
thf(fact_1302_disjointI,axiom,
! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ~ ( member6576561426505652726_nat_o @ X2 @ B ) )
=> ( ( inf_in1906310914598751387_nat_o @ A @ B )
= bot_bo7824918357723582233_nat_o ) ) ).
% disjointI
thf(fact_1303_disjointI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A )
=> ~ ( member8440522571783428010at_nat @ X2 @ B ) )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= bot_bo2099793752762293965at_nat ) ) ).
% disjointI
thf(fact_1304_disjointI,axiom,
! [A: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% disjointI
thf(fact_1305_disjointI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% disjointI
thf(fact_1306_disjointI,axiom,
! [A: set_int,B: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ~ ( member_int @ X2 @ B ) )
=> ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int ) ) ).
% disjointI
thf(fact_1307_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_1308_minus__mult__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_mult_commute
thf(fact_1309_minus__mult__commute,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_mult_commute
thf(fact_1310_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1311_square__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ A )
= ( times_3573771949741848930nteger @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1312_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ A )
= ( times_times_rat @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1313_UNIV__I,axiom,
! [X: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X @ top_to6034159466715884489nt_int ) ).
% UNIV_I
thf(fact_1314_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_1315_UNIV__I,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ top_to4869887016220417533_nat_o ) ).
% UNIV_I
thf(fact_1316_UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat @ X @ top_top_set_list_nat ) ).
% UNIV_I
thf(fact_1317_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_1318_UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_I
thf(fact_1319_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1320_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_1321_inf1I,axiom,
! [A2: product_prod_int_int > $o,X: product_prod_int_int,B2: product_prod_int_int > $o] :
( ( A2 @ X )
=> ( ( B2 @ X )
=> ( inf_in3604695632404883862_int_o @ A2 @ B2 @ X ) ) ) ).
% inf1I
thf(fact_1322_inf1I,axiom,
! [A2: list_char > $o,X: list_char,B2: list_char > $o] :
( ( A2 @ X )
=> ( ( B2 @ X )
=> ( inf_inf_list_char_o @ A2 @ B2 @ X ) ) ) ).
% inf1I
thf(fact_1323_UNIV__def,axiom,
( top_to6034159466715884489nt_int
= ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : $true ) ) ).
% UNIV_def
thf(fact_1324_UNIV__def,axiom,
( top_top_set_set_nat
= ( collect_set_nat
@ ^ [X3: set_nat] : $true ) ) ).
% UNIV_def
thf(fact_1325_UNIV__def,axiom,
( top_to4869887016220417533_nat_o
= ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : $true ) ) ).
% UNIV_def
thf(fact_1326_UNIV__def,axiom,
( top_top_set_list_nat
= ( collect_list_nat
@ ^ [X3: list_nat] : $true ) ) ).
% UNIV_def
thf(fact_1327_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X3: $o] : $true ) ) ).
% UNIV_def
thf(fact_1328_UNIV__def,axiom,
( top_top_set_rat
= ( collect_rat
@ ^ [X3: rat] : $true ) ) ).
% UNIV_def
thf(fact_1329_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X3: nat] : $true ) ) ).
% UNIV_def
thf(fact_1330_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X3: int] : $true ) ) ).
% UNIV_def
thf(fact_1331_UNIV__eq__I,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ! [X2: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( top_to6034159466715884489nt_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_1332_UNIV__eq__I,axiom,
! [A2: set_set_nat] :
( ! [X2: set_nat] : ( member_set_nat @ X2 @ A2 )
=> ( top_top_set_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_1333_UNIV__eq__I,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ! [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( top_to4869887016220417533_nat_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_1334_UNIV__eq__I,axiom,
! [A2: set_list_nat] :
( ! [X2: list_nat] : ( member_list_nat @ X2 @ A2 )
=> ( top_top_set_list_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_1335_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X2: $o] : ( member_o @ X2 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_1336_UNIV__eq__I,axiom,
! [A2: set_rat] :
( ! [X2: rat] : ( member_rat @ X2 @ A2 )
=> ( top_top_set_rat = A2 ) ) ).
% UNIV_eq_I
thf(fact_1337_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_1338_UNIV__eq__I,axiom,
! [A2: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A2 )
=> ( top_top_set_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_1339_top__set__def,axiom,
( top_to6034159466715884489nt_int
= ( collec5210948495886036740nt_int @ top_to6187514276045680724_int_o ) ) ).
% top_set_def
thf(fact_1340_top__set__def,axiom,
( top_top_set_set_nat
= ( collect_set_nat @ top_top_set_nat_o ) ) ).
% top_set_def
thf(fact_1341_top__set__def,axiom,
( top_to4869887016220417533_nat_o
= ( collec939566748876313656_nat_o @ top_to2462227307520033056at_o_o ) ) ).
% top_set_def
thf(fact_1342_top__set__def,axiom,
( top_top_set_list_nat
= ( collect_list_nat @ top_top_list_nat_o ) ) ).
% top_set_def
thf(fact_1343_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_1344_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_1345_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1346_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_1347_UNIV__witness,axiom,
? [X2: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X2 @ top_to6034159466715884489nt_int ) ).
% UNIV_witness
thf(fact_1348_UNIV__witness,axiom,
? [X2: set_nat] : ( member_set_nat @ X2 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_1349_UNIV__witness,axiom,
? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ top_to4869887016220417533_nat_o ) ).
% UNIV_witness
thf(fact_1350_UNIV__witness,axiom,
? [X2: list_nat] : ( member_list_nat @ X2 @ top_top_set_list_nat ) ).
% UNIV_witness
thf(fact_1351_UNIV__witness,axiom,
? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_1352_UNIV__witness,axiom,
? [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_1353_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1354_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_1355_inf1D2,axiom,
! [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o,X: product_prod_int_int] :
( ( inf_in3604695632404883862_int_o @ A2 @ B2 @ X )
=> ( B2 @ X ) ) ).
% inf1D2
thf(fact_1356_inf1D2,axiom,
! [A2: list_char > $o,B2: list_char > $o,X: list_char] :
( ( inf_inf_list_char_o @ A2 @ B2 @ X )
=> ( B2 @ X ) ) ).
% inf1D2
thf(fact_1357_inf1D1,axiom,
! [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o,X: product_prod_int_int] :
( ( inf_in3604695632404883862_int_o @ A2 @ B2 @ X )
=> ( A2 @ X ) ) ).
% inf1D1
thf(fact_1358_inf1D1,axiom,
! [A2: list_char > $o,B2: list_char > $o,X: list_char] :
( ( inf_inf_list_char_o @ A2 @ B2 @ X )
=> ( A2 @ X ) ) ).
% inf1D1
thf(fact_1359_inf1E,axiom,
! [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o,X: product_prod_int_int] :
( ( inf_in3604695632404883862_int_o @ A2 @ B2 @ X )
=> ~ ( ( A2 @ X )
=> ~ ( B2 @ X ) ) ) ).
% inf1E
thf(fact_1360_inf1E,axiom,
! [A2: list_char > $o,B2: list_char > $o,X: list_char] :
( ( inf_inf_list_char_o @ A2 @ B2 @ X )
=> ~ ( ( A2 @ X )
=> ~ ( B2 @ X ) ) ) ).
% inf1E
thf(fact_1361_times__assn__raw_Ocases,axiom,
! [X: produc2732055786443039994et_nat] :
~ ! [P3: produc3658429121746597890et_nat > $o,Q3: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As4: set_nat] :
( X
!= ( produc2245416461498447860et_nat @ P3 @ ( produc5001842942810119800et_nat @ Q3 @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_1362_pairself_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F3: int > option6357759511663192854e_term,A6: int,B6: int] :
( X
!= ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ A6 @ B6 ) ) ) ).
% pairself.cases
thf(fact_1363_old_Oprod_Oexhaust,axiom,
! [Y: produc1908205239877642774nteger] :
~ ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
( Y
!= ( produc8603105652947943368nteger @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_1364_old_Oprod_Oexhaust,axiom,
! [Y: produc3925858234332021118et_nat] :
~ ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( Y
!= ( produc5001842942810119800et_nat @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_1365_old_Oprod_Oexhaust,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_1366_old_Oprod_Oexhaust,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
( Y
!= ( produc5700946648718959541nt_int @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_1367_old_Oprod_Oexhaust,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( Y
!= ( produc4305682042979456191nt_int @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_1368_bex2I,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S )
=> ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S )
=> ( P @ A @ B ) )
=> ? [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_1369_bex2I,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
=> ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
=> ( P @ A @ B ) )
=> ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_1370_bex2I,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
=> ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
=> ( P @ A @ B ) )
=> ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_1371_bex2I,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S )
=> ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S )
=> ( P @ A @ B ) )
=> ? [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_1372_bex2I,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S )
=> ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S )
=> ( P @ A @ B ) )
=> ? [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_1373_surj__pair,axiom,
! [P4: produc1908205239877642774nteger] :
? [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( P4
= ( produc8603105652947943368nteger @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1374_surj__pair,axiom,
! [P4: produc3925858234332021118et_nat] :
? [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( P4
= ( produc5001842942810119800et_nat @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1375_surj__pair,axiom,
! [P4: produc2732055786443039994et_nat] :
? [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( P4
= ( produc2245416461498447860et_nat @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1376_surj__pair,axiom,
! [P4: produc2285326912895808259nt_int] :
? [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( P4
= ( produc5700946648718959541nt_int @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1377_surj__pair,axiom,
! [P4: produc7773217078559923341nt_int] :
? [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_1378_prod__cases,axiom,
! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
( ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A6 @ B6 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1379_prod__cases,axiom,
! [P: produc3925858234332021118et_nat > $o,P4: produc3925858234332021118et_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A6 @ B6 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1380_prod__cases,axiom,
! [P: produc2732055786443039994et_nat > $o,P4: produc2732055786443039994et_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A6 @ B6 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1381_prod__cases,axiom,
! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
( ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A6 @ B6 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1382_prod__cases,axiom,
! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_1383_Pair__inject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_1384_Pair__inject,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ A @ B )
= ( produc5001842942810119800et_nat @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_1385_Pair__inject,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B5: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ A @ B )
= ( produc2245416461498447860et_nat @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_1386_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_1387_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_1388_prod__cases3,axiom,
! [Y: produc1908205239877642774nteger] :
~ ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: code_integer,C3: code_integer] :
( Y
!= ( produc8603105652947943368nteger @ A6 @ ( produc1086072967326762835nteger @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_1389_prod__cases3,axiom,
! [Y: produc3925858234332021118et_nat] :
~ ! [A6: produc3658429121746597890et_nat > $o,B6: heap_e7401611519738050253t_unit,C3: set_nat] :
( Y
!= ( produc5001842942810119800et_nat @ A6 @ ( produc7507926704131184380et_nat @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_1390_prod__cases3,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A6 @ ( produc5001842942810119800et_nat @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_1391_prod__cases3,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: int,C3: int] :
( Y
!= ( produc5700946648718959541nt_int @ A6 @ ( product_Pair_int_int @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_1392_prod__cases3,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A6: int > option6357759511663192854e_term,B6: int,C3: int] :
( Y
!= ( produc4305682042979456191nt_int @ A6 @ ( product_Pair_int_int @ B6 @ C3 ) ) ) ).
% prod_cases3
thf(fact_1393_prod__cases4,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A6 @ ( produc5001842942810119800et_nat @ B6 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) ) ).
% prod_cases4
thf(fact_1394_prod__induct3,axiom,
! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
( ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A6 @ ( produc1086072967326762835nteger @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_1395_prod__induct3,axiom,
! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: heap_e7401611519738050253t_unit,C3: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A6 @ ( produc7507926704131184380et_nat @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_1396_prod__induct3,axiom,
! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A6 @ ( produc5001842942810119800et_nat @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_1397_prod__induct3,axiom,
! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
( ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A6 @ ( product_Pair_int_int @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_1398_prod__induct3,axiom,
! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
( ! [A6: int > option6357759511663192854e_term,B6: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A6 @ ( product_Pair_int_int @ B6 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_1399_prod__induct4,axiom,
! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A6 @ ( produc5001842942810119800et_nat @ B6 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_1400_memb__imp__not__empty,axiom,
! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ X @ S )
=> ( S != bot_bo1488462491386950373nt_int ) ) ).
% memb_imp_not_empty
thf(fact_1401_memb__imp__not__empty,axiom,
! [X: set_nat,S: set_set_nat] :
( ( member_set_nat @ X @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_1402_memb__imp__not__empty,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ S )
=> ( S != bot_bo7824918357723582233_nat_o ) ) ).
% memb_imp_not_empty
thf(fact_1403_memb__imp__not__empty,axiom,
! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ S )
=> ( S != bot_bo2099793752762293965at_nat ) ) ).
% memb_imp_not_empty
thf(fact_1404_memb__imp__not__empty,axiom,
! [X: $o,S: set_o] :
( ( member_o @ X @ S )
=> ( S != bot_bot_set_o ) ) ).
% memb_imp_not_empty
thf(fact_1405_memb__imp__not__empty,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_1406_memb__imp__not__empty,axiom,
! [X: int,S: set_int] :
( ( member_int @ X @ S )
=> ( S != bot_bot_set_int ) ) ).
% memb_imp_not_empty
thf(fact_1407_set__notEmptyE,axiom,
! [S: set_se6260736226359567993nt_int] :
( ( S != bot_bo1488462491386950373nt_int )
=> ~ ! [X2: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1408_set__notEmptyE,axiom,
! [S: set_set_nat] :
( ( S != bot_bot_set_set_nat )
=> ~ ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1409_set__notEmptyE,axiom,
! [S: set_Pr4532377907799695533_nat_o] :
( ( S != bot_bo7824918357723582233_nat_o )
=> ~ ! [X2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1410_set__notEmptyE,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ( S != bot_bo2099793752762293965at_nat )
=> ~ ! [X2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1411_set__notEmptyE,axiom,
! [S: set_o] :
( ( S != bot_bot_set_o )
=> ~ ! [X2: $o] :
~ ( member_o @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1412_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1413_set__notEmptyE,axiom,
! [S: set_int] :
( ( S != bot_bot_set_int )
=> ~ ! [X2: int] :
~ ( member_int @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_1414_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1415_verit__negate__coefficient_I3_J,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
=> ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1416_verit__negate__coefficient_I3_J,axiom,
! [A: rat,B: rat] :
( ( A = B )
=> ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1417_sup__Un__eq,axiom,
! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
( ( sup_su1852724690005176016_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ R )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ S ) )
= ( ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ ( sup_su2047564715030645325nt_int @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1418_sup__Un__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1419_sup__Un__eq,axiom,
! [R: set_int,S: set_int] :
( ( sup_sup_int_o
@ ^ [X3: int] : ( member_int @ X3 @ R )
@ ^ [X3: int] : ( member_int @ X3 @ S ) )
= ( ^ [X3: int] : ( member_int @ X3 @ ( sup_sup_set_int @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1420_sup__Un__eq,axiom,
! [R: set_Pr4532377907799695533_nat_o,S: set_Pr4532377907799695533_nat_o] :
( ( sup_su8003631835303541148at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ ( sup_su5209123915105501825_nat_o @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1421_sup__Un__eq,axiom,
! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
( ( sup_su6256023009775730178_nat_o
@ ^ [X3: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X3 @ R )
@ ^ [X3: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X3 @ S ) )
= ( ^ [X3: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X3 @ ( sup_su3035147773818789531at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1422_sup__Un__eq,axiom,
! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
( ( sup_su2080679670758317954_nat_o
@ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ R )
@ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ S ) )
= ( ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1423_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1424_inf__Int__eq,axiom,
! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
( ( inf_in5102985939729578038_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ R )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ S ) )
= ( ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ ( inf_in8396524679539076455nt_int @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1425_inf__Int__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ ( inf_inf_set_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1426_inf__Int__eq,axiom,
! [R: set_int,S: set_int] :
( ( inf_inf_int_o
@ ^ [X3: int] : ( member_int @ X3 @ R )
@ ^ [X3: int] : ( member_int @ X3 @ S ) )
= ( ^ [X3: int] : ( member_int @ X3 @ ( inf_inf_set_int @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1427_inf__Int__eq,axiom,
! [R: set_Pr4532377907799695533_nat_o,S: set_Pr4532377907799695533_nat_o] :
( ( inf_in9018898317438245890at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ ( inf_in1906310914598751387_nat_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1428_inf__Int__eq,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( inf_in5163264567034779214_nat_o
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ R )
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ S ) )
= ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1429_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1430_inf__Int__eq,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( inf_in3604695632404883862_int_o
@ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ R )
@ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ S ) )
= ( ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ ( inf_in2269163501485487111nt_int @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1431_inf__Int__eq,axiom,
! [R: set_list_char,S: set_list_char] :
( ( inf_inf_list_char_o
@ ^ [X3: list_char] : ( member_list_char @ X3 @ R )
@ ^ [X3: list_char] : ( member_list_char @ X3 @ S ) )
= ( ^ [X3: list_char] : ( member_list_char @ X3 @ ( inf_in6177012701088388009t_char @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1432_iso__tuple__UNIV__I,axiom,
! [X: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X @ top_to6034159466715884489nt_int ) ).
% iso_tuple_UNIV_I
thf(fact_1433_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_1434_iso__tuple__UNIV__I,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ top_to4869887016220417533_nat_o ) ).
% iso_tuple_UNIV_I
thf(fact_1435_iso__tuple__UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat @ X @ top_top_set_list_nat ) ).
% iso_tuple_UNIV_I
thf(fact_1436_iso__tuple__UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_1437_iso__tuple__UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_1438_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_1439_iso__tuple__UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_1440_Collect__empty__eq__bot,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ( ( collec5210948495886036740nt_int @ P )
= bot_bo1488462491386950373nt_int )
= ( P = bot_bo2686080419298087992_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1441_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1442_Collect__empty__eq__bot,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ( collec939566748876313656_nat_o @ P )
= bot_bo7824918357723582233_nat_o )
= ( P = bot_bo7963750851167320836at_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1443_Collect__empty__eq__bot,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( P = bot_bo482883023278783056_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1444_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1445_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1446_Collect__empty__eq__bot,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( P = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1447_bot__empty__eq,axiom,
( bot_bo2686080419298087992_int_o
= ( ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ bot_bo1488462491386950373nt_int ) ) ) ).
% bot_empty_eq
thf(fact_1448_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1449_bot__empty__eq,axiom,
( bot_bo7963750851167320836at_o_o
= ( ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% bot_empty_eq
thf(fact_1450_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1451_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1452_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1453_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_1454_type__copy__ex__RepI,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,F4: $o > $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( ? [X4: $o] : ( F4 @ X4 ) )
= ( ? [B3: product_unit] : ( F4 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1455_type__copy__ex__RepI,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,F4: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ top_to4869887016220417533_nat_o )
=> ( ( ? [X4: produc3658429121746597890et_nat > $o] : ( F4 @ X4 ) )
= ( ? [B3: assn] : ( F4 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1456_type__copy__ex__RepI,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,F4: list_char > $o] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ top_to3190553582279535303t_char )
=> ( ( ? [X4: list_char] : ( F4 @ X4 ) )
= ( ? [B3: literal] : ( F4 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1457_type__copy__ex__RepI,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,F4: set_Pr958786334691620121nt_int > $o] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ top_to6034159466715884489nt_int )
=> ( ( ? [X4: set_Pr958786334691620121nt_int] : ( F4 @ X4 ) )
= ( ? [B3: rat] : ( F4 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1458_type__copy__ex__RepI,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,F4: set_Pr1261947904930325089at_nat > $o] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ top_to7629004291339433233at_nat )
=> ( ( ? [X4: set_Pr1261947904930325089at_nat] : ( F4 @ X4 ) )
= ( ? [B3: int] : ( F4 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_1459_type__copy__obj__one__point__absE,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,S3: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ~ ! [X2: $o] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1460_type__copy__obj__one__point__absE,axiom,
! [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,S3: assn] :
( ( type_d3909072315231072503_nat_o @ Rep @ Abs @ top_to4869887016220417533_nat_o )
=> ~ ! [X2: produc3658429121746597890et_nat > $o] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1461_type__copy__obj__one__point__absE,axiom,
! [Rep: literal > list_char,Abs: list_char > literal,S3: literal] :
( ( type_d4752411451802217481t_char @ Rep @ Abs @ top_to3190553582279535303t_char )
=> ~ ! [X2: list_char] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1462_type__copy__obj__one__point__absE,axiom,
! [Rep: rat > set_Pr958786334691620121nt_int,Abs: set_Pr958786334691620121nt_int > rat,S3: rat] :
( ( type_d8554052265237484179nt_int @ Rep @ Abs @ top_to6034159466715884489nt_int )
=> ~ ! [X2: set_Pr958786334691620121nt_int] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1463_type__copy__obj__one__point__absE,axiom,
! [Rep: int > set_Pr1261947904930325089at_nat,Abs: set_Pr1261947904930325089at_nat > int,S3: int] :
( ( type_d8752208705193531015at_nat @ Rep @ Abs @ top_to7629004291339433233at_nat )
=> ~ ! [X2: set_Pr1261947904930325089at_nat] :
( S3
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_1464_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1465_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_1466_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_inc_simps(4)
thf(fact_1467_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea7165431671056489859t_unit @ inf_inf_Product_unit @ sup_sup_Product_unit @ uminus2952777764628376836t_unit @ bot_bot_Product_unit @ top_top_Product_unit ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1468_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea379910186789422830_set_o @ inf_inf_set_o @ sup_sup_set_o @ uminus_uminus_set_o @ bot_bot_set_o @ top_top_set_o ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1469_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea6493995471900120728et_rat @ inf_inf_set_rat @ sup_sup_set_rat @ uminus2201863774496077783et_rat @ bot_bot_set_rat @ top_top_set_rat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1470_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1471_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea5824373010784320748et_int @ inf_inf_set_int @ sup_sup_set_int @ uminus1532241313380277803et_int @ bot_bot_set_int @ top_top_set_int ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1472_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea9074081411689916704st_nat @ inf_inf_set_list_nat @ sup_sup_set_list_nat @ uminus3195874150345416415st_nat @ bot_bot_set_list_nat @ top_top_set_list_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1473_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea934052484356091300char_o @ inf_inf_list_char_o @ sup_sup_list_char_o @ uminus7154032370948614053char_o @ bot_bot_list_char_o @ top_top_list_char_o ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1474_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea949959532490082519at_nat @ inf_in2572325071724192079at_nat @ sup_su6327502436637775413at_nat @ uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat @ top_to4669805908274784177at_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1475_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea1849928419393854478_int_o @ inf_in3604695632404883862_int_o @ sup_su8463660629351352368_int_o @ uminus7117520113953359693_int_o @ bot_bo8147686125503663512_int_o @ top_to1578927101902068148_int_o ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1476_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea51845251928480829at_nat @ inf_in1697001100524423349at_nat @ sup_su3035147773818789531at_nat @ uminus2330091110623919550at_nat @ bot_bo8422036546324065075at_nat @ top_to8764741339697478167at_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1477_top__empty__eq,axiom,
( top_to6187514276045680724_int_o
= ( ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ top_to6034159466715884489nt_int ) ) ) ).
% top_empty_eq
thf(fact_1478_top__empty__eq,axiom,
( top_top_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1479_top__empty__eq,axiom,
( top_to2462227307520033056at_o_o
= ( ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ top_to4869887016220417533_nat_o ) ) ) ).
% top_empty_eq
thf(fact_1480_top__empty__eq,axiom,
( top_top_list_nat_o
= ( ^ [X3: list_nat] : ( member_list_nat @ X3 @ top_top_set_list_nat ) ) ) ).
% top_empty_eq
thf(fact_1481_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_1482_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_1483_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1484_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_1485_sup__Un__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
( ( sup_su234547053653886311eger_o
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R )
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( sup_su4591284015454442744nteger @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1486_sup__Un__eq2,axiom,
! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
( ( sup_su1630790145277462993_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( sup_su7128418612487073120et_nat @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1487_sup__Un__eq2,axiom,
! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
( ( sup_su6535292691877529429_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( sup_su8975264963432250076et_nat @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1488_sup__Un__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ( sup_su4182031696650224058_int_o
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( sup_su3382966977382714213nt_int @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1489_sup__Un__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ( sup_su600977994968626096_int_o
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( sup_su3298353300217089135nt_int @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1490_sup__Un__eq2,axiom,
! [R: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
( ( sup_su2200014604384089602_nat_o
@ ^ [X3: multis2468970476368604999at_nat,Y3: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: multis2468970476368604999at_nat,Y3: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: multis2468970476368604999at_nat,Y3: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X3 @ Y3 ) @ ( sup_su3035147773818789531at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1491_sup__Un__eq2,axiom,
! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
( ( sup_su7519161239522478338_nat_o
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1492_inf__Int__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
( ( inf_in3563340378267265229eger_o
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R )
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( inf_in3408396081536963678nteger @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1493_inf__Int__eq2,axiom,
! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
( ( inf_in3295504058751909687_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( inf_in1768905781608824518et_nat @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1494_inf__Int__eq2,axiom,
! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
( ( inf_in2641120393918057659_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( inf_in3088352823822785602et_nat @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1495_inf__Int__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ( inf_in8045883074377559328_int_o
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( inf_in2576786181675586251nt_int @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1496_inf__Int__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ( inf_in8201999774870019094_int_o
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( inf_in8806012763027673301nt_int @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1497_inf__Int__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_1498_bot__empty__eq2,axiom,
( bot_bo3000040243691356879eger_o
= ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ bot_bo5443222936135328352nteger ) ) ) ).
% bot_empty_eq2
thf(fact_1499_bot__empty__eq2,axiom,
( bot_bo5580076615179976505_nat_o
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ bot_bo1481135142794719944et_nat ) ) ) ).
% bot_empty_eq2
thf(fact_1500_bot__empty__eq2,axiom,
( bot_bo3790638025767943357_nat_o
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ bot_bo5635537948650799172et_nat ) ) ) ).
% bot_empty_eq2
thf(fact_1501_bot__empty__eq2,axiom,
( bot_bo8662317086119403298_int_o
= ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ bot_bo572930865798478029nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_1502_bot__empty__eq2,axiom,
( bot_bo1403522918969695512_int_o
= ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_1503_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_1504_pred__equals__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
( ( ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_1505_pred__equals__eq2,axiom,
! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
( ( ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_1506_pred__equals__eq2,axiom,
! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
( ( ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_1507_pred__equals__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ( ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_1508_pred__equals__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ( ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_1509_top__empty__eq2,axiom,
( top_to8112647782992986859eger_o
= ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ top_to7512759353274530428nteger ) ) ) ).
% top_empty_eq2
thf(fact_1510_top__empty__eq2,axiom,
( top_to2428096842796733269_nat_o
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ top_to8753217654552796900et_nat ) ) ) ).
% top_empty_eq2
thf(fact_1511_top__empty__eq2,axiom,
( top_to7190503160269336793_nat_o
= ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ top_to1599102959340997728et_nat ) ) ) ).
% top_empty_eq2
thf(fact_1512_top__empty__eq2,axiom,
( top_to6261655714344447806_int_o
= ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ top_to3181862456159035625nt_int ) ) ) ).
% top_empty_eq2
thf(fact_1513_top__empty__eq2,axiom,
( top_to6513043852502318900_int_o
= ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ top_to2069137843433766899nt_int ) ) ) ).
% top_empty_eq2
thf(fact_1514_times__assn__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) )
=> ? [As13: set_nat,As23: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As13 @ As23 ) )
& ( ( inf_inf_set_nat @ As13 @ As23 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_1515_times__assn__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) )
=> ~ ? [As12: set_nat,As22: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As12 @ As22 ) )
& ( ( inf_inf_set_nat @ As12 @ As22 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_1516_times__assn__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( Y
= ( ? [As1: set_nat,As2: set_nat] :
( ( As4
= ( sup_sup_set_nat @ As1 @ As2 ) )
& ( ( inf_inf_set_nat @ As1 @ As2 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) )
=> ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_1517_Set_Ois__empty__def,axiom,
( is_emp1662574758705540307at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat] : ( A4 = bot_bo2099793752762293965at_nat ) ) ) ).
% Set.is_empty_def
thf(fact_1518_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_1519_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_1520_Set_Ois__empty__def,axiom,
( is_empty_int
= ( ^ [A4: set_int] : ( A4 = bot_bot_set_int ) ) ) ).
% Set.is_empty_def
thf(fact_1521_wand__assnI,axiom,
! [H3: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,R: assn] :
( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( inf_inf_set_nat @ As @ As5 )
= bot_bot_set_nat )
=> ( ( relH @ As @ H3 @ H5 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
=> ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H5 @ As5 ) )
=> ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q2 @ R ) @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% wand_assnI
thf(fact_1522_vimage__if,axiom,
! [C: nat,A2: set_nat,D2: nat,B2: set_o] :
( ( ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : ( if_nat @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_o ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : ( if_nat @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : ( if_nat @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus_uminus_set_o @ B2 ) ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : ( if_nat @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_o ) ) ) ) ) ).
% vimage_if
thf(fact_1523_vimage__if,axiom,
! [C: int,A2: set_int,D2: int,B2: set_o] :
( ( ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : ( if_int @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_o ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : ( if_int @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : ( if_int @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus_uminus_set_o @ B2 ) ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : ( if_int @ ( member_o @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_o ) ) ) ) ) ).
% vimage_if
thf(fact_1524_vimage__if,axiom,
! [C: nat,A2: set_nat,D2: nat,B2: set_rat] :
( ( ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : ( if_nat @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_rat ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : ( if_nat @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : ( if_nat @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus2201863774496077783et_rat @ B2 ) ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : ( if_nat @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_rat ) ) ) ) ) ).
% vimage_if
thf(fact_1525_vimage__if,axiom,
! [C: int,A2: set_int,D2: int,B2: set_rat] :
( ( ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : ( if_int @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_rat ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : ( if_int @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : ( if_int @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus2201863774496077783et_rat @ B2 ) ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : ( if_int @ ( member_rat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_rat ) ) ) ) ) ).
% vimage_if
thf(fact_1526_vimage__if,axiom,
! [C: nat,A2: set_nat,D2: nat,B2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_nat ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus5710092332889474511et_nat @ B2 ) ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_nat ) ) ) ) ) ).
% vimage_if
thf(fact_1527_vimage__if,axiom,
! [C: int,A2: set_int,D2: int,B2: set_nat] :
( ( ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_nat ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus5710092332889474511et_nat @ B2 ) ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_nat ) ) ) ) ) ).
% vimage_if
thf(fact_1528_vimage__if,axiom,
! [C: nat,A2: set_nat,D2: nat,B2: set_int] :
( ( ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_int ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus1532241313380277803et_int @ B2 ) ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_int ) ) ) ) ) ).
% vimage_if
thf(fact_1529_vimage__if,axiom,
! [C: int,A2: set_int,D2: int,B2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_int ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus1532241313380277803et_int @ B2 ) ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_int ) ) ) ) ) ).
% vimage_if
thf(fact_1530_vimage__if,axiom,
! [C: nat,A2: set_nat,D2: nat,B2: set_set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_set_nat_nat
@ ^ [X3: set_nat] : ( if_nat @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_set_nat ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_set_nat_nat
@ ^ [X3: set_nat] : ( if_nat @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( ( member_nat @ D2 @ A2 )
=> ( ( vimage_set_nat_nat
@ ^ [X3: set_nat] : ( if_nat @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus613421341184616069et_nat @ B2 ) ) )
& ( ~ ( member_nat @ D2 @ A2 )
=> ( ( vimage_set_nat_nat
@ ^ [X3: set_nat] : ( if_nat @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_set_nat ) ) ) ) ) ).
% vimage_if
thf(fact_1531_vimage__if,axiom,
! [C: int,A2: set_int,D2: int,B2: set_set_nat] :
( ( ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_set_nat_int
@ ^ [X3: set_nat] : ( if_int @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= top_top_set_set_nat ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_set_nat_int
@ ^ [X3: set_nat] : ( if_int @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= B2 ) ) ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( ( member_int @ D2 @ A2 )
=> ( ( vimage_set_nat_int
@ ^ [X3: set_nat] : ( if_int @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= ( uminus613421341184616069et_nat @ B2 ) ) )
& ( ~ ( member_int @ D2 @ A2 )
=> ( ( vimage_set_nat_int
@ ^ [X3: set_nat] : ( if_int @ ( member_set_nat @ X3 @ B2 ) @ C @ D2 )
@ A2 )
= bot_bot_set_set_nat ) ) ) ) ) ).
% vimage_if
thf(fact_1532_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d5972188401318659438st_nat @ bNF_id_bnf_list_nat @ bNF_id_bnf_list_nat @ top_top_set_list_nat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_1533_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_definition_o_o @ bNF_id_bnf_o @ bNF_id_bnf_o @ top_top_set_o ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_1534_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d5298809244756387038at_rat @ bNF_id_bnf_rat @ bNF_id_bnf_rat @ top_top_set_rat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_1535_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d6250493948777748686at_nat @ bNF_id_bnf_nat @ bNF_id_bnf_nat @ top_top_set_nat ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_1536_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
type_d7247357190169752966nt_int @ bNF_id_bnf_int @ bNF_id_bnf_int @ top_top_set_int ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_1537_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
= one_one_Code_integer ) ).
% dbl_dec_simps(3)
thf(fact_1538_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_1539_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_1540_wand__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As5 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H5 @ As5 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As4 @ As5 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_1541_wand__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As6 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H6 )
& ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As4 @ As6 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_1542_vimage__Collect__eq,axiom,
! [F2: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1543_vimage__Collect__eq,axiom,
! [F2: int > nat,P: nat > $o] :
( ( vimage_int_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_int
@ ^ [Y3: int] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1544_vimage__Collect__eq,axiom,
! [F2: nat > int,P: int > $o] :
( ( vimage_nat_int @ F2 @ ( collect_int @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1545_vimage__Collect__eq,axiom,
! [F2: int > int,P: int > $o] :
( ( vimage_int_int @ F2 @ ( collect_int @ P ) )
= ( collect_int
@ ^ [Y3: int] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1546_vimage__Collect__eq,axiom,
! [F2: nat > set_nat,P: set_nat > $o] :
( ( vimage_nat_set_nat @ F2 @ ( collect_set_nat @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1547_vimage__Collect__eq,axiom,
! [F2: int > set_nat,P: set_nat > $o] :
( ( vimage_int_set_nat @ F2 @ ( collect_set_nat @ P ) )
= ( collect_int
@ ^ [Y3: int] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1548_vimage__Collect__eq,axiom,
! [F2: set_nat > nat,P: nat > $o] :
( ( vimage_set_nat_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_set_nat
@ ^ [Y3: set_nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1549_vimage__Collect__eq,axiom,
! [F2: set_nat > int,P: int > $o] :
( ( vimage_set_nat_int @ F2 @ ( collect_int @ P ) )
= ( collect_set_nat
@ ^ [Y3: set_nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1550_vimage__Collect__eq,axiom,
! [F2: set_nat > set_nat,P: set_nat > $o] :
( ( vimage4765135879290611145et_nat @ F2 @ ( collect_set_nat @ P ) )
= ( collect_set_nat
@ ^ [Y3: set_nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1551_vimage__Collect__eq,axiom,
! [F2: nat > set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( vimage2935056327143036492nt_int @ F2 @ ( collec5210948495886036740nt_int @ P ) )
= ( collect_nat
@ ^ [Y3: nat] : ( P @ ( F2 @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1552_vimage__empty,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1553_vimage__empty,axiom,
! [F2: nat > $o] :
( ( vimage_nat_o @ F2 @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1554_vimage__empty,axiom,
! [F2: int > $o] :
( ( vimage_int_o @ F2 @ bot_bot_set_o )
= bot_bot_set_int ) ).
% vimage_empty
thf(fact_1555_vimage__empty,axiom,
! [F2: $o > nat] :
( ( vimage_o_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1556_vimage__empty,axiom,
! [F2: nat > nat] :
( ( vimage_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1557_vimage__empty,axiom,
! [F2: int > nat] :
( ( vimage_int_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_int ) ).
% vimage_empty
thf(fact_1558_vimage__empty,axiom,
! [F2: $o > int] :
( ( vimage_o_int @ F2 @ bot_bot_set_int )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1559_vimage__empty,axiom,
! [F2: nat > int] :
( ( vimage_nat_int @ F2 @ bot_bot_set_int )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1560_vimage__empty,axiom,
! [F2: int > int] :
( ( vimage_int_int @ F2 @ bot_bot_set_int )
= bot_bot_set_int ) ).
% vimage_empty
thf(fact_1561_vimage__empty,axiom,
! [F2: $o > product_prod_nat_nat] :
( ( vimage2606541041029201316at_nat @ F2 @ bot_bo2099793752762293965at_nat )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1562_vimage__UNIV,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1563_vimage__UNIV,axiom,
! [F2: rat > $o] :
( ( vimage_rat_o @ F2 @ top_top_set_o )
= top_top_set_rat ) ).
% vimage_UNIV
thf(fact_1564_vimage__UNIV,axiom,
! [F2: nat > $o] :
( ( vimage_nat_o @ F2 @ top_top_set_o )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1565_vimage__UNIV,axiom,
! [F2: int > $o] :
( ( vimage_int_o @ F2 @ top_top_set_o )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_1566_vimage__UNIV,axiom,
! [F2: $o > rat] :
( ( vimage_o_rat @ F2 @ top_top_set_rat )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1567_vimage__UNIV,axiom,
! [F2: rat > rat] :
( ( vimage_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat ) ).
% vimage_UNIV
thf(fact_1568_vimage__UNIV,axiom,
! [F2: nat > rat] :
( ( vimage_nat_rat @ F2 @ top_top_set_rat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_1569_vimage__UNIV,axiom,
! [F2: int > rat] :
( ( vimage_int_rat @ F2 @ top_top_set_rat )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_1570_vimage__UNIV,axiom,
! [F2: $o > nat] :
( ( vimage_o_nat @ F2 @ top_top_set_nat )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1571_vimage__UNIV,axiom,
! [F2: rat > nat] :
( ( vimage_rat_nat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% vimage_UNIV
thf(fact_1572_vimage__Int,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( vimage2449269961533847803at_nat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( inf_in2572325071724192079at_nat @ ( vimage2449269961533847803at_nat @ F2 @ A2 ) @ ( vimage2449269961533847803at_nat @ F2 @ B2 ) ) ) ).
% vimage_Int
thf(fact_1573_vimage__Int,axiom,
! [F2: nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( vimage8013328719654469172at_nat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ F2 @ A2 ) @ ( vimage8013328719654469172at_nat @ F2 @ B2 ) ) ) ).
% vimage_Int
thf(fact_1574_vimage__Int,axiom,
! [F2: product_prod_nat_nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage4653281326611754070at_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_in2572325071724192079at_nat @ ( vimage4653281326611754070at_nat @ F2 @ A2 ) @ ( vimage4653281326611754070at_nat @ F2 @ B2 ) ) ) ).
% vimage_Int
thf(fact_1575_vimage__Int,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage_nat_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ A2 ) @ ( vimage_nat_nat @ F2 @ B2 ) ) ) ).
% vimage_Int
thf(fact_1576_vimage__Un,axiom,
! [F2: produc4166570645942440679at_nat > produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( vimage3529983231074640251at_nat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage3529983231074640251at_nat @ F2 @ A2 ) @ ( vimage3529983231074640251at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1577_vimage__Un,axiom,
! [F2: produc3843707927480180839at_nat > produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( vimage2716263466903273467at_nat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( vimage2716263466903273467at_nat @ F2 @ A2 ) @ ( vimage2716263466903273467at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1578_vimage__Un,axiom,
! [F2: nat > produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( vimage7262694317370708122at_nat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage7262694317370708122at_nat @ F2 @ A2 ) @ ( vimage7262694317370708122at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1579_vimage__Un,axiom,
! [F2: produc4166570645942440679at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( vimage5412736291816116987at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage5412736291816116987at_nat @ F2 @ A2 ) @ ( vimage5412736291816116987at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1580_vimage__Un,axiom,
! [F2: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( vimage6896080417876799867at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( vimage6896080417876799867at_nat @ F2 @ A2 ) @ ( vimage6896080417876799867at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1581_vimage__Un,axiom,
! [F2: nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( vimage6435164912253009178at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage6435164912253009178at_nat @ F2 @ A2 ) @ ( vimage6435164912253009178at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1582_vimage__Un,axiom,
! [F2: produc4166570645942440679at_nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage7952471809483217980at_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage7952471809483217980at_nat @ F2 @ A2 ) @ ( vimage7952471809483217980at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1583_vimage__Un,axiom,
! [F2: produc3843707927480180839at_nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage7134676753939176892at_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( vimage7134676753939176892at_nat @ F2 @ A2 ) @ ( vimage7134676753939176892at_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1584_vimage__Un,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage_nat_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F2 @ A2 ) @ ( vimage_nat_nat @ F2 @ B2 ) ) ) ).
% vimage_Un
thf(fact_1585_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_1586_relH__dist__union,axiom,
! [As: set_nat,As3: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( relH @ ( sup_sup_set_nat @ As @ As3 ) @ H3 @ H4 )
= ( ( relH @ As @ H3 @ H4 )
& ( relH @ As3 @ H3 @ H4 ) ) ) ).
% relH_dist_union
thf(fact_1587_bool__assn__proper_I5_J,axiom,
! [P: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H2: produc3658429121746597890et_nat] :
( ( in_range @ H2 )
& ~ ( P @ H2 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_1588_in__range__empty,axiom,
! [H3: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ).
% in_range_empty
thf(fact_1589_vimage__const,axiom,
! [C: nat,A2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : C
@ A2 )
= top_top_set_o ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( vimage_o_nat
@ ^ [X3: $o] : C
@ A2 )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_1590_vimage__const,axiom,
! [C: int,A2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : C
@ A2 )
= top_top_set_o ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( vimage_o_int
@ ^ [X3: $o] : C
@ A2 )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_1591_vimage__const,axiom,
! [C: nat,A2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : C
@ A2 )
= top_top_set_rat ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( vimage_rat_nat
@ ^ [X3: rat] : C
@ A2 )
= bot_bot_set_rat ) ) ) ).
% vimage_const
thf(fact_1592_vimage__const,axiom,
! [C: int,A2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : C
@ A2 )
= top_top_set_rat ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( vimage_rat_int
@ ^ [X3: rat] : C
@ A2 )
= bot_bot_set_rat ) ) ) ).
% vimage_const
thf(fact_1593_vimage__const,axiom,
! [C: nat,A2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : C
@ A2 )
= top_top_set_nat ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( vimage_nat_nat
@ ^ [X3: nat] : C
@ A2 )
= bot_bot_set_nat ) ) ) ).
% vimage_const
thf(fact_1594_vimage__const,axiom,
! [C: int,A2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : C
@ A2 )
= top_top_set_nat ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( vimage_nat_int
@ ^ [X3: nat] : C
@ A2 )
= bot_bot_set_nat ) ) ) ).
% vimage_const
thf(fact_1595_vimage__const,axiom,
! [C: nat,A2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : C
@ A2 )
= top_top_set_int ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( vimage_int_nat
@ ^ [X3: int] : C
@ A2 )
= bot_bot_set_int ) ) ) ).
% vimage_const
thf(fact_1596_vimage__const,axiom,
! [C: int,A2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : C
@ A2 )
= top_top_set_int ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( vimage_int_int
@ ^ [X3: int] : C
@ A2 )
= bot_bot_set_int ) ) ) ).
% vimage_const
thf(fact_1597_vimage__const,axiom,
! [C: nat,A2: set_nat] :
( ( ( member_nat @ C @ A2 )
=> ( ( vimage_list_nat_nat
@ ^ [X3: list_nat] : C
@ A2 )
= top_top_set_list_nat ) )
& ( ~ ( member_nat @ C @ A2 )
=> ( ( vimage_list_nat_nat
@ ^ [X3: list_nat] : C
@ A2 )
= bot_bot_set_list_nat ) ) ) ).
% vimage_const
thf(fact_1598_vimage__const,axiom,
! [C: int,A2: set_int] :
( ( ( member_int @ C @ A2 )
=> ( ( vimage_list_nat_int
@ ^ [X3: list_nat] : C
@ A2 )
= top_top_set_list_nat ) )
& ( ~ ( member_int @ C @ A2 )
=> ( ( vimage_list_nat_int
@ ^ [X3: list_nat] : C
@ A2 )
= bot_bot_set_list_nat ) ) ) ).
% vimage_const
thf(fact_1599_in__range__dist__union,axiom,
! [H3: heap_e7401611519738050253t_unit,As: set_nat,As3: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ ( sup_sup_set_nat @ As @ As3 ) ) )
= ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
& ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ).
% in_range_dist_union
thf(fact_1600_relH__sym,axiom,
! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( relH @ As @ H4 @ H3 ) ) ).
% relH_sym
thf(fact_1601_relH__trans,axiom,
! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H12 @ H23 )
=> ( ( relH @ As @ H23 @ H32 )
=> ( relH @ As @ H12 @ H32 ) ) ) ).
% relH_trans
thf(fact_1602_vimage__def,axiom,
( vimage_nat_nat
= ( ^ [F: nat > nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] : ( member_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1603_vimage__def,axiom,
( vimage_nat_int
= ( ^ [F: nat > int,B4: set_int] :
( collect_nat
@ ^ [X3: nat] : ( member_int @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1604_vimage__def,axiom,
( vimage_int_nat
= ( ^ [F: int > nat,B4: set_nat] :
( collect_int
@ ^ [X3: int] : ( member_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1605_vimage__def,axiom,
( vimage_int_int
= ( ^ [F: int > int,B4: set_int] :
( collect_int
@ ^ [X3: int] : ( member_int @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1606_vimage__def,axiom,
( vimage_set_nat_nat
= ( ^ [F: set_nat > nat,B4: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( member_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1607_vimage__def,axiom,
( vimage_set_nat_int
= ( ^ [F: set_nat > int,B4: set_int] :
( collect_set_nat
@ ^ [X3: set_nat] : ( member_int @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1608_vimage__def,axiom,
( vimage_nat_set_nat
= ( ^ [F: nat > set_nat,B4: set_set_nat] :
( collect_nat
@ ^ [X3: nat] : ( member_set_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1609_vimage__def,axiom,
( vimage_int_set_nat
= ( ^ [F: int > set_nat,B4: set_set_nat] :
( collect_int
@ ^ [X3: int] : ( member_set_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1610_vimage__def,axiom,
( vimage4765135879290611145et_nat
= ( ^ [F: set_nat > set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1611_vimage__def,axiom,
( vimage928727498438615150nt_nat
= ( ^ [F: set_Pr958786334691620121nt_int > nat,B4: set_nat] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member_nat @ ( F @ X3 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_1612_relH__in__rangeI_I2_J,axiom,
! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_1613_relH__in__rangeI_I1_J,axiom,
! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H3 @ H4 )
=> ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_1614_relH__refl,axiom,
! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( relH @ As @ H3 @ H3 ) ) ).
% relH_refl
thf(fact_1615_proper__iff,axiom,
! [P: produc3658429121746597890et_nat > $o,As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
( ( proper @ P )
=> ( ( relH @ As @ H3 @ H4 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
=> ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( P @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ).
% proper_iff
thf(fact_1616_proper__def,axiom,
( proper
= ( ^ [P2: produc3658429121746597890et_nat > $o] :
! [H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As7 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As7 ) ) )
& ( ( ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As7 ) )
& ( relH @ As7 @ H2 @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
=> ( P2 @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) ) ) ) ) ).
% proper_def
thf(fact_1617_properD2,axiom,
! [P: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat,H4: heap_e7401611519738050253t_unit] :
( ( proper @ P )
=> ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( ( relH @ As @ H3 @ H4 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
=> ( P @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ).
% properD2
thf(fact_1618_properI,axiom,
! [P: produc3658429121746597890et_nat > $o] :
( ! [As4: set_nat,H: heap_e7401611519738050253t_unit] :
( ( P @ ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) ) )
=> ( ! [As4: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( ( P @ ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( relH @ As4 @ H @ H5 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
=> ( P @ ( produc7507926704131184380et_nat @ H5 @ As4 ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_1619_models__in__range,axiom,
! [P: assn,H3: produc3658429121746597890et_nat] :
( ( rep_assn @ P @ H3 )
=> ( in_range @ H3 ) ) ).
% models_in_range
thf(fact_1620_top__assn__def,axiom,
( top_top_assn
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_1621_properD1,axiom,
! [P: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( proper @ P )
=> ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% properD1
thf(fact_1622_mod__relH,axiom,
! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,P: assn] :
( ( relH @ As @ H3 @ H4 )
=> ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ).
% mod_relH
thf(fact_1623_uminus__assn__def,axiom,
( uminus_uminus_assn
= ( ^ [P2: assn] :
( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( in_range @ H2 )
& ~ ( rep_assn @ P2 @ H2 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_1624_wand__raw_Osimps,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( wand_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
& ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
( ( ( ( inf_inf_set_nat @ As @ As8 )
= bot_bot_set_nat )
& ( relH @ As @ H3 @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
& ( P @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
=> ( Q2 @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As8 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_1625_wand__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( Y
= ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As8 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As4 @ As8 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_1626_wand__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( Y
= ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H7: heap_e7401611519738050253t_unit,As8: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As8 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H7 @ As8 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As4 @ As8 ) ) ) ) ) )
=> ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_1627_wand__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) )
=> ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As6 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H6 )
& ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As4 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_1628_wand__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
& ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
( ( ( ( inf_inf_set_nat @ As4 @ As5 )
= bot_bot_set_nat )
& ( relH @ As4 @ H @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As4 ) )
& ( X @ ( produc7507926704131184380et_nat @ H5 @ As5 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As4 @ As5 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_1629_bijective__Empty,axiom,
bijective_nat_nat @ bot_bo2099793752762293965at_nat ).
% bijective_Empty
thf(fact_1630_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1631_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1632_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1633_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1634_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1635_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1636_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1637_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_1638_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_1639_pairself_Opelims,axiom,
! [X: int > option6357759511663192854e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
( ( ( pairse3534876335208825186e_term @ X @ Xa )
= Y )
=> ( ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ Xa ) )
=> ~ ! [A6: int,B6: int] :
( ( Xa
= ( product_Pair_int_int @ A6 @ B6 ) )
=> ( ( Y
= ( produc5950683997804057413e_term @ ( X @ A6 ) @ ( X @ B6 ) ) )
=> ~ ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ ( product_Pair_int_int @ A6 @ B6 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_1640_rel__restrict__compl,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: set_nat] :
( ( inf_in2572325071724192079at_nat @ ( rel_restrict_nat @ R @ A2 ) @ ( rel_restrict_nat @ R @ ( uminus5710092332889474511et_nat @ A2 ) ) )
= bot_bo2099793752762293965at_nat ) ).
% rel_restrict_compl
thf(fact_1641_Id__on__empty,axiom,
( ( id_on_2554058798563519774at_nat @ bot_bo2099793752762293965at_nat )
= bot_bo5327735625951526323at_nat ) ).
% Id_on_empty
thf(fact_1642_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_1643_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Id_on_empty
thf(fact_1644_Id__on__empty,axiom,
( ( id_on_int @ bot_bot_set_int )
= bot_bo1796632182523588997nt_int ) ).
% Id_on_empty
thf(fact_1645_mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ C )
= ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1646_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1647_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1648_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1649_mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ C @ A )
= ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1650_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1651_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1652_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1653_mult__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% mult_eq_0_iff
thf(fact_1654_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_1655_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1656_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1657_mult__zero__right,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ A @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% mult_zero_right
thf(fact_1658_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1659_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1660_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1661_mult__zero__left,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% mult_zero_left
thf(fact_1662_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1663_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1664_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1665_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1666_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1667_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Z ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1668_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1669_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: code_natural] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ W ) @ Z ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1670_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1671_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1672_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1673_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1674_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N ) )
= ( numera5444537566228673987atural @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1675_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1676_neg__equal__zero,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_1677_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_1678_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1679_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_1680_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_1681_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1682_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_1683_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_1684_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1685_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A ) )
= ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1686_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1687_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1688_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_1689_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_1690_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_1691_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_1692_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_1693_rel__restrict__empty,axiom,
! [R: set_Pr8693737435421807431at_nat] :
( ( rel_re36662636952045539at_nat @ R @ bot_bo2099793752762293965at_nat )
= R ) ).
% rel_restrict_empty
thf(fact_1694_rel__restrict__empty,axiom,
! [R: set_Product_prod_o_o] :
( ( rel_restrict_o @ R @ bot_bot_set_o )
= R ) ).
% rel_restrict_empty
thf(fact_1695_rel__restrict__empty,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( rel_restrict_nat @ R @ bot_bot_set_nat )
= R ) ).
% rel_restrict_empty
thf(fact_1696_rel__restrict__empty,axiom,
! [R: set_Pr958786334691620121nt_int] :
( ( rel_restrict_int @ R @ bot_bot_set_int )
= R ) ).
% rel_restrict_empty
thf(fact_1697_Id__onI,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ A @ A2 )
=> ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ A @ A ) @ ( id_on_6410342593070439734nt_int @ A2 ) ) ) ).
% Id_onI
thf(fact_1698_Id__onI,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ A ) @ ( id_on_set_nat @ A2 ) ) ) ).
% Id_onI
thf(fact_1699_Id__onI,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A2 ) ) ) ).
% Id_onI
thf(fact_1700_Id__onI,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ ( id_on_int @ A2 ) ) ) ).
% Id_onI
thf(fact_1701_Id__onI,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ A ) @ ( id_on_7438695062414216554_nat_o @ A2 ) ) ) ).
% Id_onI
thf(fact_1702_mult__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ C )
= C )
= ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_right2
thf(fact_1703_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_1704_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1705_mult__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( C
= ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_right1
thf(fact_1706_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_1707_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1708_mult__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( ( ( times_3573771949741848930nteger @ C @ A )
= C )
= ( ( C = zero_z3403309356797280102nteger )
| ( A = one_one_Code_integer ) ) ) ).
% mult_cancel_left2
thf(fact_1709_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_1710_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1711_mult__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( C
= ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_left1
thf(fact_1712_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_1713_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1714_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1715_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1716_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1717_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1718_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1719_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1720_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1721_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1722_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1723_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
= one_one_Code_integer ) ).
% dbl_inc_simps(2)
thf(fact_1724_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_1725_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1726_zero__reorient,axiom,
! [X: code_integer] :
( ( zero_z3403309356797280102nteger = X )
= ( X = zero_z3403309356797280102nteger ) ) ).
% zero_reorient
thf(fact_1727_zero__reorient,axiom,
! [X: multis2468970476368604999at_nat] :
( ( zero_z1048942125864253310at_nat = X )
= ( X = zero_z1048942125864253310at_nat ) ) ).
% zero_reorient
thf(fact_1728_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_1729_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1730_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_1731_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1732_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1733_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1734_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1735_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1736_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_rat @ M )
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1737_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_1738_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_1739_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
!= ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_1740_mult__right__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( ( times_3573771949741848930nteger @ A @ C )
= ( times_3573771949741848930nteger @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1741_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1742_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1743_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1744_mult__left__cancel,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( ( times_3573771949741848930nteger @ C @ A )
= ( times_3573771949741848930nteger @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1745_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1746_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1747_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1748_no__zero__divisors,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( ( times_3573771949741848930nteger @ A @ B )
!= zero_z3403309356797280102nteger ) ) ) ).
% no_zero_divisors
thf(fact_1749_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_1750_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1751_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1752_divisors__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( A = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% divisors_zero
thf(fact_1753_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_1754_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1755_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1756_mult__not__zero,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ B )
!= zero_z3403309356797280102nteger )
=> ( ( A != zero_z3403309356797280102nteger )
& ( B != zero_z3403309356797280102nteger ) ) ) ).
% mult_not_zero
thf(fact_1757_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_1758_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1759_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1760_zero__neq__one,axiom,
zero_z3403309356797280102nteger != one_one_Code_integer ).
% zero_neq_one
thf(fact_1761_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_1762_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1763_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1764_rel__restrictI,axiom,
! [X: set_Pr958786334691620121nt_int,R: set_se6260736226359567993nt_int,Y: set_Pr958786334691620121nt_int,E: set_Pr8057934254988874055nt_int] :
( ~ ( member2340774599025711042nt_int @ X @ R )
=> ( ~ ( member2340774599025711042nt_int @ Y @ R )
=> ( ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ X @ Y ) @ E )
=> ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ X @ Y ) @ ( rel_re4349003428253852667nt_int @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_1765_rel__restrictI,axiom,
! [X: set_nat,R: set_set_nat,Y: set_nat,E: set_Pr5488025237498180813et_nat] :
( ~ ( member_set_nat @ X @ R )
=> ( ~ ( member_set_nat @ Y @ R )
=> ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ E )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( rel_restrict_set_nat @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_1766_rel__restrictI,axiom,
! [X: nat,R: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
( ~ ( member_nat @ X @ R )
=> ( ~ ( member_nat @ Y @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_1767_rel__restrictI,axiom,
! [X: int,R: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
( ~ ( member_int @ X @ R )
=> ( ~ ( member_int @ Y @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_1768_rel__restrictI,axiom,
! [X: produc3658429121746597890et_nat > $o,R: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ R )
=> ( ~ ( member6576561426505652726_nat_o @ Y @ R )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ E )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ E @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_1769_rel__restrict__notR_I1_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_Pr8057934254988874055nt_int,R: set_se6260736226359567993nt_int] :
( ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ X @ Y ) @ ( rel_re4349003428253852667nt_int @ A2 @ R ) )
=> ~ ( member2340774599025711042nt_int @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_1770_rel__restrict__notR_I1_J,axiom,
! [X: set_nat,Y: set_nat,A2: set_Pr5488025237498180813et_nat,R: set_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( rel_restrict_set_nat @ A2 @ R ) )
=> ~ ( member_set_nat @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_1771_rel__restrict__notR_I1_J,axiom,
! [X: nat,Y: nat,A2: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A2 @ R ) )
=> ~ ( member_nat @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_1772_rel__restrict__notR_I1_J,axiom,
! [X: int,Y: int,A2: set_Pr958786334691620121nt_int,R: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A2 @ R ) )
=> ~ ( member_int @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_1773_rel__restrict__notR_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A2: set_Pr2161125870931222855_nat_o,R: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A2 @ R ) )
=> ~ ( member6576561426505652726_nat_o @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_1774_rel__restrict__notR_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_Pr8057934254988874055nt_int,R: set_se6260736226359567993nt_int] :
( ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ X @ Y ) @ ( rel_re4349003428253852667nt_int @ A2 @ R ) )
=> ~ ( member2340774599025711042nt_int @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_1775_rel__restrict__notR_I2_J,axiom,
! [X: set_nat,Y: set_nat,A2: set_Pr5488025237498180813et_nat,R: set_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( rel_restrict_set_nat @ A2 @ R ) )
=> ~ ( member_set_nat @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_1776_rel__restrict__notR_I2_J,axiom,
! [X: nat,Y: nat,A2: set_Pr1261947904930325089at_nat,R: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A2 @ R ) )
=> ~ ( member_nat @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_1777_rel__restrict__notR_I2_J,axiom,
! [X: int,Y: int,A2: set_Pr958786334691620121nt_int,R: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A2 @ R ) )
=> ~ ( member_int @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_1778_rel__restrict__notR_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A2: set_Pr2161125870931222855_nat_o,R: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A2 @ R ) )
=> ~ ( member6576561426505652726_nat_o @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_1779_rel__restrict__union,axiom,
! [R: set_Pr4759807026041935047at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( rel_re2197483123211154633at_nat @ R @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( rel_re2197483123211154633at_nat @ ( rel_re2197483123211154633at_nat @ R @ A2 ) @ B2 ) ) ).
% rel_restrict_union
thf(fact_1780_rel__restrict__union,axiom,
! [R: set_Pr5564308138774400199at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( rel_re2767227651283840073at_nat @ R @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( rel_re2767227651283840073at_nat @ ( rel_re2767227651283840073at_nat @ R @ A2 ) @ B2 ) ) ).
% rel_restrict_union
thf(fact_1781_rel__restrict__union,axiom,
! [R: set_Pr1261947904930325089at_nat,A2: set_nat,B2: set_nat] :
( ( rel_restrict_nat @ R @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( rel_restrict_nat @ ( rel_restrict_nat @ R @ A2 ) @ B2 ) ) ).
% rel_restrict_union
thf(fact_1782_Id__on__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ X @ Y ) @ ( id_on_6410342593070439734nt_int @ A2 ) )
= ( ( X = Y )
& ( member2340774599025711042nt_int @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_1783_Id__on__iff,axiom,
! [X: set_nat,Y: set_nat,A2: set_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( id_on_set_nat @ A2 ) )
= ( ( X = Y )
& ( member_set_nat @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_1784_Id__on__iff,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A2 ) )
= ( ( X = Y )
& ( member_nat @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_1785_Id__on__iff,axiom,
! [X: int,Y: int,A2: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( id_on_int @ A2 ) )
= ( ( X = Y )
& ( member_int @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_1786_Id__on__iff,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( id_on_7438695062414216554_nat_o @ A2 ) )
= ( ( X = Y )
& ( member6576561426505652726_nat_o @ X @ A2 ) ) ) ).
% Id_on_iff
thf(fact_1787_Id__on__eqI,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( A = B )
=> ( ( member2340774599025711042nt_int @ A @ A2 )
=> ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ A @ B ) @ ( id_on_6410342593070439734nt_int @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_1788_Id__on__eqI,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( A = B )
=> ( ( member_set_nat @ A @ A2 )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( id_on_set_nat @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_1789_Id__on__eqI,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_1790_Id__on__eqI,axiom,
! [A: int,B: int,A2: set_int] :
( ( A = B )
=> ( ( member_int @ A @ A2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( id_on_int @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_1791_Id__on__eqI,axiom,
! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( A = B )
=> ( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ B ) @ ( id_on_7438695062414216554_nat_o @ A2 ) ) ) ) ).
% Id_on_eqI
thf(fact_1792_Id__onE,axiom,
! [C: produc412284730452459111nt_int,A2: set_se6260736226359567993nt_int] :
( ( member5325734588016868240nt_int @ C @ ( id_on_6410342593070439734nt_int @ A2 ) )
=> ~ ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( C
!= ( produc782388162306416855nt_int @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_1793_Id__onE,axiom,
! [C: produc7819656566062154093et_nat,A2: set_set_nat] :
( ( member8277197624267554838et_nat @ C @ ( id_on_set_nat @ A2 ) )
=> ~ ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( C
!= ( produc4532415448927165861et_nat @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_1794_Id__onE,axiom,
! [C: product_prod_nat_nat,A2: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A2 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( C
!= ( product_Pair_nat_nat @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_1795_Id__onE,axiom,
! [C: product_prod_int_int,A2: set_int] :
( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A2 ) )
=> ~ ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( C
!= ( product_Pair_int_int @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_1796_Id__onE,axiom,
! [C: produc4928098042776334183_nat_o,A2: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ C @ ( id_on_7438695062414216554_nat_o @ A2 ) )
=> ~ ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( C
!= ( produc7368190662567826135_nat_o @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_1797_lambda__zero,axiom,
( ( ^ [H2: code_integer] : zero_z3403309356797280102nteger )
= ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger ) ) ).
% lambda_zero
thf(fact_1798_lambda__zero,axiom,
( ( ^ [H2: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_1799_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_1800_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_1801_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1802_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1803_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1804_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_1805_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ N )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_1806_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_1807_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1808_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_1809_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_1810_bijective__def,axiom,
( biject9051520373387432658nteger
= ( ^ [R2: set_Pr1281608226676607948nteger] :
( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Z3: produc8923325533196201883nteger] :
( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R2 )
& ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Z3 ) @ R2 ) )
=> ( Y3 = Z3 ) )
& ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc6241069584506657477e_term > option6357759511663192854e_term,Z3: produc8923325533196201883nteger] :
( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Z3 ) @ R2 )
& ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ Y3 @ Z3 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_1811_bijective__def,axiom,
( biject2615096655818420098et_nat
= ( ^ [R2: set_Pr3286484037609594932et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R2 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z3 ) @ R2 ) )
=> ( Y3 = Z3 ) )
& ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat] :
( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z3 ) @ R2 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ Z3 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_1812_bijective__def,axiom,
( biject1468766312547416318et_nat
= ( ^ [R2: set_Pr8536935166611901872et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R2 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z3 ) @ R2 ) )
=> ( Y3 = Z3 ) )
& ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z3: produc3925858234332021118et_nat] :
( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z3 ) @ R2 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ Z3 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_1813_bijective__def,axiom,
( biject383251550997737151nt_int
= ( ^ [R2: set_Pr9222295170931077689nt_int] :
( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Z3: product_prod_int_int] :
( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R2 )
& ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Z3 ) @ R2 ) )
=> ( Y3 = Z3 ) )
& ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: produc8551481072490612790e_term > option6357759511663192854e_term,Z3: product_prod_int_int] :
( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Z3 ) @ R2 )
& ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y3 @ Z3 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_1814_bijective__def,axiom,
( biject576505603616484041nt_int
= ( ^ [R2: set_Pr1872883991513573699nt_int] :
( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Z3: product_prod_int_int] :
( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R2 )
& ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Z3 ) @ R2 ) )
=> ( Y3 = Z3 ) )
& ! [X3: int > option6357759511663192854e_term,Y3: int > option6357759511663192854e_term,Z3: product_prod_int_int] :
( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Z3 ) @ R2 )
& ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y3 @ Z3 ) @ R2 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_1815_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1816_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1817_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1818_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1819_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1820_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1821_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_1822_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_1823_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(172)
thf(fact_1824_numeral__times__minus__swap,axiom,
! [W: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1825_numeral__times__minus__swap,axiom,
! [W: num,X: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
= ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1826_numeral__times__minus__swap,axiom,
! [W: num,X: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
= ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1827_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1828_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1829_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1830_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1831_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1832_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1833_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1834_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1835_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1836_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
= zero_z3403309356797280102nteger )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1837_eq__numeral__iff__iszero_I11_J,axiom,
! [X: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
= zero_zero_rat )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ X ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1838_divides__aux__eq,axiom,
! [Q4: code_integer,R3: code_integer] :
( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R3 ) )
= ( R3 = zero_z3403309356797280102nteger ) ) ).
% divides_aux_eq
thf(fact_1839_divides__aux__eq,axiom,
! [Q4: nat,R3: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R3 ) )
= ( R3 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_1840_divides__aux__eq,axiom,
! [Q4: int,R3: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R3 ) )
= ( R3 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_1841_minus__assn__def,axiom,
( minus_minus_assn
= ( ^ [A3: assn,B3: assn] : ( inf_inf_assn @ A3 @ ( uminus_uminus_assn @ B3 ) ) ) ) ).
% minus_assn_def
thf(fact_1842_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ A @ A )
= zero_z3403309356797280102nteger ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1843_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multis2468970476368604999at_nat] :
( ( minus_4286766774447292334at_nat @ A @ A )
= zero_z1048942125864253310at_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1844_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1845_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1846_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1847_diff__zero,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% diff_zero
thf(fact_1848_diff__zero,axiom,
! [A: multis2468970476368604999at_nat] :
( ( minus_4286766774447292334at_nat @ A @ zero_z1048942125864253310at_nat )
= A ) ).
% diff_zero
thf(fact_1849_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_1850_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1851_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1852_zero__diff,axiom,
! [A: multis2468970476368604999at_nat] :
( ( minus_4286766774447292334at_nat @ zero_z1048942125864253310at_nat @ A )
= zero_z1048942125864253310at_nat ) ).
% zero_diff
thf(fact_1853_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1854_diff__0__right,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% diff_0_right
thf(fact_1855_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_1856_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1857_diff__self,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ A @ A )
= zero_z3403309356797280102nteger ) ).
% diff_self
thf(fact_1858_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_1859_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1860_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1861_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1862_minus__diff__eq,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
= ( minus_minus_rat @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1863_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1864_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1865_div__by__1,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
= A ) ).
% div_by_1
thf(fact_1866_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_1867_div__by__1,axiom,
! [A: code_natural] :
( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
= A ) ).
% div_by_1
thf(fact_1868_diff__numeral__special_I9_J,axiom,
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(9)
thf(fact_1869_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_1870_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1871_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1872_left__diff__distrib__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1873_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1874_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1875_right__diff__distrib__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1876_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1877_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1878_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1879_verit__minus__simplify_I3_J,axiom,
! [B: rat] :
( ( minus_minus_rat @ zero_zero_rat @ B )
= ( uminus_uminus_rat @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1880_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1881_diff__0,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
= ( uminus1351360451143612070nteger @ A ) ) ).
% diff_0
thf(fact_1882_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_1883_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1884_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1885_nonzero__mult__div__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1886_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1887_nonzero__mult__div__cancel__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1888_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1889_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1890_nonzero__mult__div__cancel__right,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1891_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1892_nonzero__mult__div__cancel__right,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1893_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1894_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1895_div__self,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ A @ A )
= one_one_Code_integer ) ) ).
% div_self
thf(fact_1896_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_1897_div__self,axiom,
! [A: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ A @ A )
= one_one_Code_natural ) ) ).
% div_self
thf(fact_1898_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_1899_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_1900_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_1901_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1902_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
& ( ( ( numeral_numeral_rat @ W )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1903_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1904_diff__numeral__special_I12_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(12)
thf(fact_1905_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_1906_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1907_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1908_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1909_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1910_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1911_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: code_integer,Z4: code_integer] : ( Y4 = Z4 ) )
= ( ^ [A3: code_integer,B3: code_integer] :
( ( minus_8373710615458151222nteger @ A3 @ B3 )
= zero_z3403309356797280102nteger ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1912_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A3: rat,B3: rat] :
( ( minus_minus_rat @ A3 @ B3 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1913_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1914_left__diff__distrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1915_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1916_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1917_right__diff__distrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1918_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1919_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1920_left__diff__distrib_H,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ C ) @ A )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1921_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
= ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1922_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1923_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1924_right__diff__distrib_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1925_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1926_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1927_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1928_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1929_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1930_minus__diff__commute,axiom,
! [B: rat,A: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1931_not__iszero__1,axiom,
~ ( ring_16219924574208605041nteger @ one_one_Code_integer ) ).
% not_iszero_1
thf(fact_1932_not__iszero__1,axiom,
~ ( ring_1_iszero_rat @ one_one_rat ) ).
% not_iszero_1
thf(fact_1933_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_1934_diff__eq,axiom,
( minus_1356011639430497352at_nat
= ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ X3 @ ( uminus6524753893492686040at_nat @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1935_diff__eq,axiom,
( minus_711738161318947805_int_o
= ( ^ [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ X3 @ ( uminus7117520113953359693_int_o @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1936_diff__eq,axiom,
( minus_6821098565838606101char_o
= ( ^ [X3: list_char > $o,Y3: list_char > $o] : ( inf_inf_list_char_o @ X3 @ ( uminus7154032370948614053char_o @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1937_diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1938_diff__eq,axiom,
( minus_3524152463667985524t_unit
= ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ X3 @ ( uminus2952777764628376836t_unit @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1939_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_1940_not__iszero__neg__1,axiom,
~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_iszero_neg_1
thf(fact_1941_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_iszero_neg_1
thf(fact_1942_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1943_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1944_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ( divide_divide_rat @ B @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1945_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1946_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1947_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1948_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_1949_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_1950_div__minus1__right,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A ) ) ).
% div_minus1_right
thf(fact_1951_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1952_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1953_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_1954_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_1955_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1956_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1957_Diff__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= A2 ) ).
% Diff_empty
thf(fact_1958_Diff__empty,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Diff_empty
thf(fact_1959_Diff__empty,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Diff_empty
thf(fact_1960_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_1961_empty__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
= bot_bo2099793752762293965at_nat ) ).
% empty_Diff
thf(fact_1962_empty__Diff,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_1963_empty__Diff,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_1964_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1965_Diff__cancel,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ A2 )
= bot_bo2099793752762293965at_nat ) ).
% Diff_cancel
thf(fact_1966_Diff__cancel,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ A2 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_1967_Diff__cancel,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ A2 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_1968_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1969_Un__Diff__cancel,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( minus_5060654252129873198at_nat @ B2 @ A2 ) )
= ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1970_Un__Diff__cancel,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ A2 ) )
= ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1971_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1972_Un__Diff__cancel2,axiom,
! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ B2 @ A2 ) @ A2 )
= ( sup_su3035147773818789531at_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1973_Un__Diff__cancel2,axiom,
! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B2 @ A2 ) @ A2 )
= ( sup_su5525570899277871387at_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1974_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1975_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1976_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1977_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1978_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1979_Diff__UNIV,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ top_to4669805908274784177at_nat )
= bot_bo2099793752762293965at_nat ) ).
% Diff_UNIV
thf(fact_1980_Diff__UNIV,axiom,
! [A2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A2 @ top_top_set_list_nat )
= bot_bot_set_list_nat ) ).
% Diff_UNIV
thf(fact_1981_Diff__UNIV,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_1982_Diff__UNIV,axiom,
! [A2: set_rat] :
( ( minus_minus_set_rat @ A2 @ top_top_set_rat )
= bot_bot_set_rat ) ).
% Diff_UNIV
thf(fact_1983_Diff__UNIV,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ top_top_set_int )
= bot_bot_set_int ) ).
% Diff_UNIV
thf(fact_1984_Diff__UNIV,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1985_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_1986_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ).
% div_minus_minus
thf(fact_1987_Diff__disjoint,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) )
= bot_bo2099793752762293965at_nat ) ).
% Diff_disjoint
thf(fact_1988_Diff__disjoint,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ A2 @ ( minus_minus_set_o @ B2 @ A2 ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_1989_Diff__disjoint,axiom,
! [A2: set_int,B2: set_int] :
( ( inf_inf_set_int @ A2 @ ( minus_minus_set_int @ B2 @ A2 ) )
= bot_bot_set_int ) ).
% Diff_disjoint
thf(fact_1990_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1991_Diff__Compl,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( uminus6524753893492686040at_nat @ B2 ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ).
% Diff_Compl
thf(fact_1992_Diff__Compl,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Compl
thf(fact_1993_inter__compl__diff__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A2 @ ( uminus6524753893492686040at_nat @ B2 ) )
= ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ).
% inter_compl_diff_conv
thf(fact_1994_inter__compl__diff__conv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% inter_compl_diff_conv
thf(fact_1995_Compl__Diff__eq,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( uminus2330091110623919550at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1996_Compl__Diff__eq,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1997_Compl__Diff__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_1998_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1999_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2000_div__mult__mult1__if,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ( C = zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= zero_z3403309356797280102nteger ) )
& ( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2001_div__mult__mult1__if,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( ( C = zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
= zero_z2226904508553997617atural ) )
& ( ( C != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
= ( divide5121882707175180666atural @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2002_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2003_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2004_div__mult__mult2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2005_div__mult__mult2,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
= ( divide5121882707175180666atural @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2006_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2007_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2008_div__mult__mult1,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2009_div__mult__mult1,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
= ( divide5121882707175180666atural @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2010_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2011_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2012_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2013_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2014_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2015_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_2016_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_2017_Int__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_2018_Int__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_2019_Diff__Int2,axiom,
! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
= ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_2020_Diff__Int2,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_2021_Diff__Diff__Int,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_2022_Diff__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_2023_Diff__Int__distrib,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
= ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_2024_Diff__Int__distrib,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_2025_Diff__Int__distrib2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ C2 )
= ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_2026_Diff__Int__distrib2,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_2027_Un__Diff,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( minus_5060654252129873198at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ C2 )
= ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A2 @ C2 ) @ ( minus_5060654252129873198at_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_2028_Un__Diff,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C2 )
= ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A2 @ C2 ) @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_2029_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_2030_set__diff__diff__left,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( minus_5060654252129873198at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) @ C2 )
= ( minus_5060654252129873198at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) ) ).
% set_diff_diff_left
thf(fact_2031_set__diff__diff__left,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ C2 )
= ( minus_3314409938677909166at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% set_diff_diff_left
thf(fact_2032_set__diff__diff__left,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% set_diff_diff_left
thf(fact_2033_Diff__triv,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
=> ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_2034_Diff__triv,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_2035_Diff__triv,axiom,
! [A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( minus_minus_set_int @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_2036_Diff__triv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_2037_Int__Diff__disjoint,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
= bot_bo2099793752762293965at_nat ) ).
% Int_Diff_disjoint
thf(fact_2038_Int__Diff__disjoint,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ ( minus_minus_set_o @ A2 @ B2 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_2039_Int__Diff__disjoint,axiom,
! [A2: set_int,B2: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ A2 @ B2 ) )
= bot_bot_set_int ) ).
% Int_Diff_disjoint
thf(fact_2040_Int__Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_2041_disjoint__alt__simp1,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
= A2 )
= ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp1
thf(fact_2042_disjoint__alt__simp1,axiom,
! [A2: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A2 @ B2 )
= A2 )
= ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ).
% disjoint_alt_simp1
thf(fact_2043_disjoint__alt__simp1,axiom,
! [A2: set_int,B2: set_int] :
( ( ( minus_minus_set_int @ A2 @ B2 )
= A2 )
= ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int ) ) ).
% disjoint_alt_simp1
thf(fact_2044_disjoint__alt__simp1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 )
= ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% disjoint_alt_simp1
thf(fact_2045_disjoint__alt__simp2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
!= A2 )
= ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
!= bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp2
thf(fact_2046_disjoint__alt__simp2,axiom,
! [A2: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A2 @ B2 )
!= A2 )
= ( ( inf_inf_set_o @ A2 @ B2 )
!= bot_bot_set_o ) ) ).
% disjoint_alt_simp2
thf(fact_2047_disjoint__alt__simp2,axiom,
! [A2: set_int,B2: set_int] :
( ( ( minus_minus_set_int @ A2 @ B2 )
!= A2 )
= ( ( inf_inf_set_int @ A2 @ B2 )
!= bot_bot_set_int ) ) ).
% disjoint_alt_simp2
thf(fact_2048_disjoint__alt__simp2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
!= A2 )
= ( ( inf_inf_set_nat @ A2 @ B2 )
!= bot_bot_set_nat ) ) ).
% disjoint_alt_simp2
thf(fact_2049_Diff__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
= ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( minus_1356011639430497352at_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_2050_Diff__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( minus_5060654252129873198at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) )
= ( inf_in1697001100524423349at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) @ ( minus_5060654252129873198at_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_2051_Diff__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
= ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( minus_3314409938677909166at_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_2052_Diff__Un,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_2053_Diff__Int,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
= ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( minus_1356011639430497352at_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_2054_Diff__Int,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( minus_5060654252129873198at_nat @ A2 @ ( inf_in1697001100524423349at_nat @ B2 @ C2 ) )
= ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) @ ( minus_5060654252129873198at_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_2055_Diff__Int,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A2 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
= ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( minus_3314409938677909166at_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_2056_Diff__Int,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_2057_Int__Diff__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_2058_Int__Diff__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_2059_Int__Diff__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_2060_Int__Diff__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_2061_Un__Diff__Int,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_2062_Un__Diff__Int,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_2063_Un__Diff__Int,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_2064_Un__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_2065_Compl__eq__Diff__UNIV,axiom,
( uminus3195874150345416415st_nat
= ( minus_7954133019191499631st_nat @ top_top_set_list_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_2066_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_2067_Compl__eq__Diff__UNIV,axiom,
( uminus2201863774496077783et_rat
= ( minus_minus_set_rat @ top_top_set_rat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_2068_Compl__eq__Diff__UNIV,axiom,
( uminus1532241313380277803et_int
= ( minus_minus_set_int @ top_top_set_int ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_2069_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_2070_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_2071_divide__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_2072_times__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_2073_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_2074_div__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% div_minus_right
thf(fact_2075_minus__divide__left,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_2076_minus__divide__divide,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ).
% minus_divide_divide
thf(fact_2077_minus__divide__right,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_divide_right
thf(fact_2078_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( times_times_rat @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2079_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C )
= A )
= ( B
= ( times_times_rat @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2080_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= B )
=> ( A
= ( divide_divide_rat @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2081_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C ) )
=> ( ( divide_divide_rat @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2082_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_2083_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C )
= A )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_2084_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y )
= ( divide_divide_rat @ W @ Z ) )
= ( ( times_times_rat @ X @ Z )
= ( times_times_rat @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2085_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2086_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_2087_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_2088_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_2089_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_2090_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2091_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2092_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
= ( ( times_times_rat @ C @ B )
= ( uminus_uminus_rat @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2093_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= C )
= ( ( uminus_uminus_rat @ A )
= ( times_times_rat @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2094_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
= A )
= ( ( ( C != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B )
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2095_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= ( uminus_uminus_rat @ B ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2096_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B != zero_zero_rat )
& ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_2097_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2098_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2099_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2100_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_2101_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_2102_bits__div__by__1,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
= A ) ).
% bits_div_by_1
thf(fact_2103_bits__div__by__1,axiom,
! [A: code_natural] :
( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
= A ) ).
% bits_div_by_1
thf(fact_2104_inf__period_I1_J,axiom,
! [P: code_integer > $o,D3: code_integer,Q2: code_integer > $o] :
( ! [X2: code_integer,K2: code_integer] :
( ( P @ X2 )
= ( P @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
=> ( ! [X2: code_integer,K2: code_integer] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
=> ! [X5: code_integer,K3: code_integer] :
( ( ( P @ X5 )
& ( Q2 @ X5 ) )
= ( ( P @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2105_inf__period_I1_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X2: rat,K2: rat] :
( ( P @ X2 )
= ( P @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ( ! [X2: rat,K2: rat] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ! [X5: rat,K3: rat] :
( ( ( P @ X5 )
& ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2106_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X5: int,K3: int] :
( ( ( P @ X5 )
& ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2107_inf__period_I2_J,axiom,
! [P: code_integer > $o,D3: code_integer,Q2: code_integer > $o] :
( ! [X2: code_integer,K2: code_integer] :
( ( P @ X2 )
= ( P @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
=> ( ! [X2: code_integer,K2: code_integer] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
=> ! [X5: code_integer,K3: code_integer] :
( ( ( P @ X5 )
| ( Q2 @ X5 ) )
= ( ( P @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2108_inf__period_I2_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X2: rat,K2: rat] :
( ( P @ X2 )
= ( P @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ( ! [X2: rat,K2: rat] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ! [X5: rat,K3: rat] :
( ( ( P @ X5 )
| ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2109_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X5: int,K3: int] :
( ( ( P @ X5 )
| ( Q2 @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2110_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2111_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2112_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2113_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2114_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2115_diff__numeral__special_I6_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% diff_numeral_special(6)
thf(fact_2116_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2117_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2118_left__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,C: int,A: int,B: int] :
( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2119_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2120_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: rat,A: rat,B: rat] :
( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2121_left__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,C: int,A: int,B: int] :
( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2122_left__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2123_left__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2124_left__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: int,A: int,B: int] :
( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
=> ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2125_left__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2126_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2127_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2128_right__diff__distrib__NO__MATCH,axiom,
! [X: nat,Y: nat,A: int,B: int,C: int] :
( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2129_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2130_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: rat,B: rat,C: rat] :
( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2131_right__diff__distrib__NO__MATCH,axiom,
! [X: int,Y: int,A: int,B: int,C: int] :
( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2132_right__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2133_right__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2134_right__diff__distrib__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: int,B: int,C: int] :
( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2135_right__diff__distrib__NO__MATCH,axiom,
! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2136_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2137_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2138_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2139_dual__order_Orefl,axiom,
! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ A @ A ) ).
% dual_order.refl
thf(fact_2140_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_2141_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_2142_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_2143_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_2144_order__refl,axiom,
! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).
% order_refl
thf(fact_2145_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_2146_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_2147_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_2148_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_2149_empty__subsetI,axiom,
! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% empty_subsetI
thf(fact_2150_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_2151_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_2152_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_2153_subset__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% subset_empty
thf(fact_2154_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_2155_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_2156_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_2157_Int__subset__iff,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
= ( ( ord_le3146513528884898305at_nat @ C2 @ A2 )
& ( ord_le3146513528884898305at_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_2158_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_2159_Un__subset__iff,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_le8081472938463900775at_nat @ A2 @ C2 )
& ( ord_le8081472938463900775at_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_2160_Un__subset__iff,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_le1268244103169919719at_nat @ A2 @ C2 )
& ( ord_le1268244103169919719at_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_2161_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_2162_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
= ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2163_Compl__anti__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_2164_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_2165_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_2166_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_2167_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_2168_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_2169_compl__le__compl__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) )
= ( ord_le3221252021190050221t_unit @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_2170_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_2171_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_2172_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_2173_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_2174_compl__less__compl__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( uminus2952777764628376836t_unit @ Y ) )
= ( ord_le361264281704409273t_unit @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_2175_le__inf__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
= ( ( ord_le3146513528884898305at_nat @ X @ Y )
& ( ord_le3146513528884898305at_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2176_le__inf__iff,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
= ( ( ord_le8369615600986905444_int_o @ X @ Y )
& ( ord_le8369615600986905444_int_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2177_le__inf__iff,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) )
= ( ( ord_le4796328588573674190char_o @ X @ Y )
& ( ord_le4796328588573674190char_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2178_le__inf__iff,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) )
= ( ( ord_le2510731241096832064er_nat @ X @ Y )
& ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2179_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2180_le__inf__iff,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) )
= ( ( ord_less_eq_rat @ X @ Y )
& ( ord_less_eq_rat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2181_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2182_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_2183_inf_Obounded__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
= ( ( ord_le3146513528884898305at_nat @ A @ B )
& ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2184_inf_Obounded__iff,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
= ( ( ord_le8369615600986905444_int_o @ A @ B )
& ( ord_le8369615600986905444_int_o @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2185_inf_Obounded__iff,axiom,
! [A: list_char > $o,B: list_char > $o,C: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ ( inf_inf_list_char_o @ B @ C ) )
= ( ( ord_le4796328588573674190char_o @ A @ B )
& ( ord_le4796328588573674190char_o @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2186_inf_Obounded__iff,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ ( inf_inf_filter_nat @ B @ C ) )
= ( ( ord_le2510731241096832064er_nat @ A @ B )
& ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2187_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2188_inf_Obounded__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
= ( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2189_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2190_inf_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_2191_le__sup__iff,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z )
= ( ( ord_le8081472938463900775at_nat @ X @ Z )
& ( ord_le8081472938463900775at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2192_le__sup__iff,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
= ( ( ord_le1268244103169919719at_nat @ X @ Z )
& ( ord_le1268244103169919719at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2193_le__sup__iff,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ X @ Y ) @ Z )
= ( ( ord_le2510731241096832064er_nat @ X @ Z )
& ( ord_le2510731241096832064er_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2194_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2195_le__sup__iff,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z )
= ( ( ord_less_eq_rat @ X @ Z )
& ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2196_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2197_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
& ( ord_less_eq_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_2198_sup_Obounded__iff,axiom,
! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
= ( ( ord_le8081472938463900775at_nat @ B @ A )
& ( ord_le8081472938463900775at_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2199_sup_Obounded__iff,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
= ( ( ord_le1268244103169919719at_nat @ B @ A )
& ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2200_sup_Obounded__iff,axiom,
! [B: filter_nat,C: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B @ C ) @ A )
= ( ( ord_le2510731241096832064er_nat @ B @ A )
& ( ord_le2510731241096832064er_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2201_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2202_sup_Obounded__iff,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
= ( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2203_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2204_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_2205_Diff__eq__empty__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_2206_Diff__eq__empty__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_2207_Diff__eq__empty__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( ( minus_minus_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_2208_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_2209_diff__ge__0__iff__ge,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( ord_le3102999989581377725nteger @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_2210_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_2211_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_2212_diff__gt__0__iff__gt,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( ord_le6747313008572928689nteger @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2213_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_rat @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2214_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2215_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_2216_neg__0__le__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_2217_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_2218_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_le_0_iff_le
thf(fact_2219_neg__le__0__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_le_0_iff_le
thf(fact_2220_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_2221_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_2222_less__eq__neg__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_2223_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_2224_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_2225_neg__less__eq__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_2226_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_2227_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_2228_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_2229_less__neg__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_2230_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_2231_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_2232_neg__less__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_pos
thf(fact_2233_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_2234_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_2235_neg__0__less__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_2236_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_2237_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_0_iff_less
thf(fact_2238_neg__less__0__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_0_iff_less
thf(fact_2239_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_2240_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_2241_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_2242_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_2243_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_2244_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_2245_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_2246_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2247_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_2248_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2249_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2250_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2251_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2252_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2253_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2254_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2255_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2256_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2257_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2258_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2259_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2260_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2261_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2262_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2263_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2264_minus__set__def,axiom,
( minus_2612819937483484256nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ( minus_357216186751819389_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_2265_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_2266_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 )
@ ^ [X3: int] : ( member_int @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_2267_minus__set__def,axiom,
( minus_1801376950450012436_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ( minus_2803741817741520073at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_2268_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_2269_set__diff__eq,axiom,
( minus_2612819937483484256nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ A4 )
& ~ ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2270_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ~ ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2271_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A4: set_int,B4: set_int] :
( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A4 )
& ~ ( member_int @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2272_set__diff__eq,axiom,
( minus_1801376950450012436_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A4 )
& ~ ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2273_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ~ ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2274_order__less__imp__not__less,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ~ ( ord_less_literal @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2275_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2276_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2277_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_2278_order__le__imp__less__or__eq,axiom,
! [X: literal,Y: literal] :
( ( ord_less_eq_literal @ X @ Y )
=> ( ( ord_less_literal @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2279_order__le__imp__less__or__eq,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_less_filter_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2280_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2281_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2282_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2283_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_2284_linorder__le__less__linear,axiom,
! [X: literal,Y: literal] :
( ( ord_less_eq_literal @ X @ Y )
| ( ord_less_literal @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_2285_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_2286_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_2287_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_2288_order__less__imp__not__eq2,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2289_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2290_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2291_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_2292_order__less__imp__not__eq,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2293_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2294_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2295_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_2296_order__less__le__subst2,axiom,
! [A: literal,B: literal,F2: literal > literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_eq_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2297_order__less__le__subst2,axiom,
! [A: rat,B: rat,F2: rat > literal,C: literal] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2298_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > literal,C: literal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2299_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > literal,C: literal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2300_order__less__le__subst2,axiom,
! [A: literal,B: literal,F2: literal > rat,C: rat] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2301_order__less__le__subst2,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2302_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2303_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2304_order__less__le__subst2,axiom,
! [A: literal,B: literal,F2: literal > nat,C: nat] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2305_order__less__le__subst2,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_2306_order__less__le__subst1,axiom,
! [A: literal,F2: rat > literal,B: rat,C: rat] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2307_order__less__le__subst1,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2308_order__less__le__subst1,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2309_order__less__le__subst1,axiom,
! [A: int,F2: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2310_order__less__le__subst1,axiom,
! [A: literal,F2: nat > literal,B: nat,C: nat] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2311_order__less__le__subst1,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2312_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2313_order__less__le__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2314_order__less__le__subst1,axiom,
! [A: literal,F2: int > literal,B: int,C: int] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2315_order__less__le__subst1,axiom,
! [A: rat,F2: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_2316_order__le__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > literal,C: literal] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2317_order__le__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2318_order__le__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2319_order__le__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2320_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > literal,C: literal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2321_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2322_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2323_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2324_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > literal,C: literal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2325_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > rat,C: rat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_2326_order__le__less__subst1,axiom,
! [A: literal,F2: literal > literal,B: literal,C: literal] :
( ( ord_less_eq_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2327_order__le__less__subst1,axiom,
! [A: literal,F2: rat > literal,B: rat,C: rat] :
( ( ord_less_eq_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2328_order__le__less__subst1,axiom,
! [A: literal,F2: nat > literal,B: nat,C: nat] :
( ( ord_less_eq_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2329_order__le__less__subst1,axiom,
! [A: literal,F2: int > literal,B: int,C: int] :
( ( ord_less_eq_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2330_order__le__less__subst1,axiom,
! [A: rat,F2: literal > rat,B: literal,C: literal] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2331_order__le__less__subst1,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2332_order__le__less__subst1,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2333_order__le__less__subst1,axiom,
! [A: rat,F2: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2334_order__le__less__subst1,axiom,
! [A: nat,F2: literal > nat,B: literal,C: literal] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2335_order__le__less__subst1,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_2336_linorder__less__linear,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
| ( X = Y )
| ( ord_less_literal @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2337_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2338_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2339_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_2340_order__less__le__trans,axiom,
! [X: literal,Y: literal,Z: literal] :
( ( ord_less_literal @ X @ Y )
=> ( ( ord_less_eq_literal @ Y @ Z )
=> ( ord_less_literal @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2341_order__less__le__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_less_filter_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ Z )
=> ( ord_less_filter_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2342_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2343_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2344_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2345_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_2346_order__less__imp__triv,axiom,
! [X: literal,Y: literal,P: $o] :
( ( ord_less_literal @ X @ Y )
=> ( ( ord_less_literal @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2347_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2348_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2349_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_2350_order__le__less__trans,axiom,
! [X: literal,Y: literal,Z: literal] :
( ( ord_less_eq_literal @ X @ Y )
=> ( ( ord_less_literal @ Y @ Z )
=> ( ord_less_literal @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2351_order__le__less__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_less_filter_nat @ Y @ Z )
=> ( ord_less_filter_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2352_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2353_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2354_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2355_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_2356_order__neq__le__trans,axiom,
! [A: literal,B: literal] :
( ( A != B )
=> ( ( ord_less_eq_literal @ A @ B )
=> ( ord_less_literal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2357_order__neq__le__trans,axiom,
! [A: filter_nat,B: filter_nat] :
( ( A != B )
=> ( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ord_less_filter_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2358_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2359_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2360_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2361_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_2362_order__less__not__sym,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ~ ( ord_less_literal @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2363_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2364_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2365_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_2366_order__le__neq__trans,axiom,
! [A: literal,B: literal] :
( ( ord_less_eq_literal @ A @ B )
=> ( ( A != B )
=> ( ord_less_literal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2367_order__le__neq__trans,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_filter_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2368_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2369_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2370_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2371_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_2372_order__antisym__conv,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_2373_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_2374_order__antisym__conv,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_2375_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_2376_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_2377_order__less__subst2,axiom,
! [A: literal,B: literal,F2: literal > literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2378_order__less__subst2,axiom,
! [A: literal,B: literal,F2: literal > rat,C: rat] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2379_order__less__subst2,axiom,
! [A: literal,B: literal,F2: literal > nat,C: nat] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2380_order__less__subst2,axiom,
! [A: literal,B: literal,F2: literal > int,C: int] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2381_order__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > literal,C: literal] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2382_order__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2383_order__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2384_order__less__subst2,axiom,
! [A: rat,B: rat,F2: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2385_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > literal,C: literal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_literal @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2386_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_2387_order__less__subst1,axiom,
! [A: literal,F2: literal > literal,B: literal,C: literal] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2388_order__less__subst1,axiom,
! [A: literal,F2: rat > literal,B: rat,C: rat] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2389_order__less__subst1,axiom,
! [A: literal,F2: nat > literal,B: nat,C: nat] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2390_order__less__subst1,axiom,
! [A: literal,F2: int > literal,B: int,C: int] :
( ( ord_less_literal @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2391_order__less__subst1,axiom,
! [A: rat,F2: literal > rat,B: literal,C: literal] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2392_order__less__subst1,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2393_order__less__subst1,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2394_order__less__subst1,axiom,
! [A: rat,F2: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2395_order__less__subst1,axiom,
! [A: nat,F2: literal > nat,B: literal,C: literal] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2396_order__less__subst1,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_2397_order__less__irrefl,axiom,
! [X: literal] :
~ ( ord_less_literal @ X @ X ) ).
% order_less_irrefl
thf(fact_2398_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_2399_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_2400_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_2401_order__less__imp__le,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ( ord_less_eq_literal @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2402_order__less__imp__le,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_less_filter_nat @ X @ Y )
=> ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2403_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2404_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2405_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2406_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_2407_ord__less__eq__subst,axiom,
! [A: literal,B: literal,F2: literal > literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2408_ord__less__eq__subst,axiom,
! [A: literal,B: literal,F2: literal > rat,C: rat] :
( ( ord_less_literal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2409_ord__less__eq__subst,axiom,
! [A: literal,B: literal,F2: literal > nat,C: nat] :
( ( ord_less_literal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2410_ord__less__eq__subst,axiom,
! [A: literal,B: literal,F2: literal > int,C: int] :
( ( ord_less_literal @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2411_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > literal,C: literal] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2412_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2413_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2414_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2415_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > literal,C: literal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2416_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_2417_ord__eq__less__subst,axiom,
! [A: literal,F2: literal > literal,B: literal,C: literal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2418_ord__eq__less__subst,axiom,
! [A: rat,F2: literal > rat,B: literal,C: literal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2419_ord__eq__less__subst,axiom,
! [A: nat,F2: literal > nat,B: literal,C: literal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2420_ord__eq__less__subst,axiom,
! [A: int,F2: literal > int,B: literal,C: literal] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_literal @ B @ C )
=> ( ! [X2: literal,Y2: literal] :
( ( ord_less_literal @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2421_ord__eq__less__subst,axiom,
! [A: literal,F2: rat > literal,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2422_ord__eq__less__subst,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2423_ord__eq__less__subst,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2424_ord__eq__less__subst,axiom,
! [A: int,F2: rat > int,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2425_ord__eq__less__subst,axiom,
! [A: literal,F2: nat > literal,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_literal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_literal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2426_ord__eq__less__subst,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_2427_linorder__not__less,axiom,
! [X: literal,Y: literal] :
( ( ~ ( ord_less_literal @ X @ Y ) )
= ( ord_less_eq_literal @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2428_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2429_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2430_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_2431_linorder__le__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_2432_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_2433_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_2434_order__less__trans,axiom,
! [X: literal,Y: literal,Z: literal] :
( ( ord_less_literal @ X @ Y )
=> ( ( ord_less_literal @ Y @ Z )
=> ( ord_less_literal @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_2435_order__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_2436_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_2437_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_2438_order__less__asym_H,axiom,
! [A: literal,B: literal] :
( ( ord_less_literal @ A @ B )
=> ~ ( ord_less_literal @ B @ A ) ) ).
% order_less_asym'
thf(fact_2439_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_2440_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_2441_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_2442_linorder__neq__iff,axiom,
! [X: literal,Y: literal] :
( ( X != Y )
= ( ( ord_less_literal @ X @ Y )
| ( ord_less_literal @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2443_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2444_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2445_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_2446_order__less__asym,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ~ ( ord_less_literal @ Y @ X ) ) ).
% order_less_asym
thf(fact_2447_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_2448_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_2449_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_2450_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2451_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2452_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F2: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2453_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2454_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2455_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2456_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > rat,C: rat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2457_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2458_ord__le__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2459_ord__le__eq__subst,axiom,
! [A: filter_nat,B: filter_nat,F2: filter_nat > rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_2460_ord__eq__le__subst,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2461_ord__eq__le__subst,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2462_ord__eq__le__subst,axiom,
! [A: int,F2: rat > int,B: rat,C: rat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2463_ord__eq__le__subst,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2464_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2465_ord__eq__le__subst,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2466_ord__eq__le__subst,axiom,
! [A: rat,F2: int > rat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2467_ord__eq__le__subst,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2468_ord__eq__le__subst,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2469_ord__eq__le__subst,axiom,
! [A: rat,F2: filter_nat > rat,B: filter_nat,C: filter_nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ! [X2: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_2470_linorder__not__le,axiom,
! [X: literal,Y: literal] :
( ( ~ ( ord_less_eq_literal @ X @ Y ) )
= ( ord_less_literal @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2471_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y ) )
= ( ord_less_rat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2472_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2473_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_2474_linorder__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_linear
thf(fact_2475_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_2476_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_2477_order__less__le,axiom,
( ord_less_literal
= ( ^ [X3: literal,Y3: literal] :
( ( ord_less_eq_literal @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2478_order__less__le,axiom,
( ord_less_filter_nat
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2479_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2480_order__less__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2481_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2482_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_2483_order__le__less,axiom,
( ord_less_eq_literal
= ( ^ [X3: literal,Y3: literal] :
( ( ord_less_literal @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2484_order__le__less,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( ord_less_filter_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2485_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2486_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2487_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2488_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_2489_order__eq__refl,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( X = Y )
=> ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_2490_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_2491_order__eq__refl,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_2492_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_2493_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_2494_linorder__neqE,axiom,
! [X: literal,Y: literal] :
( ( X != Y )
=> ( ~ ( ord_less_literal @ X @ Y )
=> ( ord_less_literal @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2495_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2496_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2497_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_2498_order__subst2,axiom,
! [A: rat,B: rat,F2: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2499_order__subst2,axiom,
! [A: rat,B: rat,F2: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2500_order__subst2,axiom,
! [A: rat,B: rat,F2: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2501_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2502_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2503_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2504_order__subst2,axiom,
! [A: int,B: int,F2: int > rat,C: rat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2505_order__subst2,axiom,
! [A: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2506_order__subst2,axiom,
! [A: int,B: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2507_order__subst2,axiom,
! [A: filter_nat,B: filter_nat,F2: filter_nat > rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F2 @ B ) @ C )
=> ( ! [X2: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_2508_order__subst1,axiom,
! [A: rat,F2: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2509_order__subst1,axiom,
! [A: rat,F2: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2510_order__subst1,axiom,
! [A: rat,F2: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_rat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_rat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2511_order__subst1,axiom,
! [A: nat,F2: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2512_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2513_order__subst1,axiom,
! [A: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2514_order__subst1,axiom,
! [A: int,F2: rat > int,B: rat,C: rat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2515_order__subst1,axiom,
! [A: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2516_order__subst1,axiom,
! [A: int,F2: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2517_order__subst1,axiom,
! [A: filter_nat,F2: rat > filter_nat,B: rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X2: rat,Y2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_le2510731241096832064er_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_2518_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
& ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_2519_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_2520_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_2521_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_2522_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_2523_antisym,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_2524_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_2525_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_2526_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_2527_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_2528_dual__order_Ostrict__implies__order,axiom,
! [B: literal,A: literal] :
( ( ord_less_literal @ B @ A )
=> ( ord_less_eq_literal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2529_dual__order_Ostrict__implies__order,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_less_filter_nat @ B @ A )
=> ( ord_le2510731241096832064er_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2530_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2531_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2532_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2533_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_2534_dual__order_Ostrict__implies__not__eq,axiom,
! [B: literal,A: literal] :
( ( ord_less_literal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2535_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2536_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2537_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_2538_order_Ostrict__implies__order,axiom,
! [A: literal,B: literal] :
( ( ord_less_literal @ A @ B )
=> ( ord_less_eq_literal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2539_order_Ostrict__implies__order,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_less_filter_nat @ A @ B )
=> ( ord_le2510731241096832064er_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2540_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2541_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2542_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2543_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_2544_dual__order_Ostrict__iff__not,axiom,
( ord_less_literal
= ( ^ [B3: literal,A3: literal] :
( ( ord_less_eq_literal @ B3 @ A3 )
& ~ ( ord_less_eq_literal @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2545_dual__order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
& ~ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2546_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2547_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2548_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2549_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_2550_dual__order_Ostrict__trans2,axiom,
! [B: literal,A: literal,C: literal] :
( ( ord_less_literal @ B @ A )
=> ( ( ord_less_eq_literal @ C @ B )
=> ( ord_less_literal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2551_dual__order_Ostrict__trans2,axiom,
! [B: filter_nat,A: filter_nat,C: filter_nat] :
( ( ord_less_filter_nat @ B @ A )
=> ( ( ord_le2510731241096832064er_nat @ C @ B )
=> ( ord_less_filter_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2552_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2553_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2554_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2555_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_2556_dual__order_Ostrict__trans1,axiom,
! [B: literal,A: literal,C: literal] :
( ( ord_less_eq_literal @ B @ A )
=> ( ( ord_less_literal @ C @ B )
=> ( ord_less_literal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2557_dual__order_Ostrict__trans1,axiom,
! [B: filter_nat,A: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( ord_less_filter_nat @ C @ B )
=> ( ord_less_filter_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2558_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2559_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2560_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2561_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_2562_order_Ostrict__implies__not__eq,axiom,
! [A: literal,B: literal] :
( ( ord_less_literal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2563_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2564_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2565_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_2566_dual__order_Ostrict__iff__order,axiom,
( ord_less_literal
= ( ^ [B3: literal,A3: literal] :
( ( ord_less_eq_literal @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2567_dual__order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2568_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2569_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2570_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2571_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_2572_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_literal
= ( ^ [B3: literal,A3: literal] :
( ( ord_less_literal @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2573_dual__order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( ( ord_less_filter_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2574_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2575_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2576_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2577_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_2578_dual__order_Ostrict__trans,axiom,
! [B: literal,A: literal,C: literal] :
( ( ord_less_literal @ B @ A )
=> ( ( ord_less_literal @ C @ B )
=> ( ord_less_literal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2579_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2580_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2581_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_2582_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ! [W2: rat] :
( ( ord_less_rat @ X @ W2 )
=> ( ( ord_less_rat @ W2 @ Y )
=> ( ord_less_eq_rat @ W2 @ Z ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_2583_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W2: rat] :
( ( ord_less_rat @ Z @ W2 )
=> ( ( ord_less_rat @ W2 @ X )
=> ( ord_less_eq_rat @ Y @ W2 ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_2584_not__less__iff__gr__or__eq,axiom,
! [X: literal,Y: literal] :
( ( ~ ( ord_less_literal @ X @ Y ) )
= ( ( ord_less_literal @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2585_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2586_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2587_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_2588_order_Ostrict__iff__not,axiom,
( ord_less_literal
= ( ^ [A3: literal,B3: literal] :
( ( ord_less_eq_literal @ A3 @ B3 )
& ~ ( ord_less_eq_literal @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2589_order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
& ~ ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2590_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2591_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2592_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2593_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_2594_order_Ostrict__trans2,axiom,
! [A: literal,B: literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_eq_literal @ B @ C )
=> ( ord_less_literal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2595_order_Ostrict__trans2,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_less_filter_nat @ A @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_less_filter_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2596_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2597_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2598_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2599_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_2600_order_Ostrict__trans1,axiom,
! [A: literal,B: literal,C: literal] :
( ( ord_less_eq_literal @ A @ B )
=> ( ( ord_less_literal @ B @ C )
=> ( ord_less_literal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2601_order_Ostrict__trans1,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ord_less_filter_nat @ B @ C )
=> ( ord_less_filter_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2602_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2603_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2604_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2605_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_2606_order_Ostrict__iff__order,axiom,
( ord_less_literal
= ( ^ [A3: literal,B3: literal] :
( ( ord_less_eq_literal @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2607_order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2608_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2609_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2610_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2611_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_2612_order_Oorder__iff__strict,axiom,
( ord_less_eq_literal
= ( ^ [A3: literal,B3: literal] :
( ( ord_less_literal @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2613_order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( ord_less_filter_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2614_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2615_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2616_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2617_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_2618_order_Ostrict__trans,axiom,
! [A: literal,B: literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( ord_less_literal @ B @ C )
=> ( ord_less_literal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2619_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2620_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2621_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_2622_linorder__less__wlog,axiom,
! [P: literal > literal > $o,A: literal,B: literal] :
( ! [A6: literal,B6: literal] :
( ( ord_less_literal @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: literal] : ( P @ A6 @ A6 )
=> ( ! [A6: literal,B6: literal] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2623_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A6: rat,B6: rat] :
( ( ord_less_rat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: rat] : ( P @ A6 @ A6 )
=> ( ! [A6: rat,B6: rat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2624_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2625_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B6: int] :
( ( ord_less_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int] : ( P @ A6 @ A6 )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_2626_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P2 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_2627_dual__order_Oirrefl,axiom,
! [A: literal] :
~ ( ord_less_literal @ A @ A ) ).
% dual_order.irrefl
thf(fact_2628_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_2629_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_2630_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_2631_dual__order_Otrans,axiom,
! [B: filter_nat,A: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( ord_le2510731241096832064er_nat @ C @ B )
=> ( ord_le2510731241096832064er_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_2632_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_2633_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_2634_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_2635_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_2636_dual__order_Oasym,axiom,
! [B: literal,A: literal] :
( ( ord_less_literal @ B @ A )
=> ~ ( ord_less_literal @ A @ B ) ) ).
% dual_order.asym
thf(fact_2637_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_2638_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_2639_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_2640_dual__order_Oantisym,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_2641_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_2642_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_2643_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_2644_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_2645_not__le__imp__less,axiom,
! [Y: literal,X: literal] :
( ~ ( ord_less_eq_literal @ Y @ X )
=> ( ord_less_literal @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2646_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_rat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2647_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2648_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_2649_less__le__not__le,axiom,
( ord_less_literal
= ( ^ [X3: literal,Y3: literal] :
( ( ord_less_eq_literal @ X3 @ Y3 )
& ~ ( ord_less_eq_literal @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2650_less__le__not__le,axiom,
( ord_less_filter_nat
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y3 )
& ~ ( ord_le2510731241096832064er_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2651_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2652_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ~ ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2653_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2654_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_2655_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
& ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_2656_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_2657_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_2658_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_2659_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_2660_linorder__cases,axiom,
! [X: literal,Y: literal] :
( ~ ( ord_less_literal @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_literal @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2661_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2662_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2663_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_2664_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X2: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ X2 @ Z ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_le
thf(fact_2665_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X2: rat] :
( ( ord_less_rat @ Z @ X2 )
=> ( ord_less_eq_rat @ Y @ X2 ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_ge
thf(fact_2666_linorder__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A6: rat,B6: rat] :
( ( ord_less_eq_rat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: rat,B6: rat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_2667_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_2668_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: int,B6: int] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_2669_antisym__conv3,axiom,
! [Y: literal,X: literal] :
( ~ ( ord_less_literal @ Y @ X )
=> ( ( ~ ( ord_less_literal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2670_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2671_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2672_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_2673_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_2674_ord__less__eq__trans,axiom,
! [A: literal,B: literal,C: literal] :
( ( ord_less_literal @ A @ B )
=> ( ( B = C )
=> ( ord_less_literal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2675_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2676_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2677_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_2678_ord__eq__less__trans,axiom,
! [A: literal,B: literal,C: literal] :
( ( A = B )
=> ( ( ord_less_literal @ B @ C )
=> ( ord_less_literal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2679_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2680_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2681_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_2682_order__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ Z )
=> ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_2683_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_2684_order__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_2685_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_2686_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_2687_order_Otrans,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_2688_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_2689_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% order.trans
thf(fact_2690_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_2691_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_2692_order_Oasym,axiom,
! [A: literal,B: literal] :
( ( ord_less_literal @ A @ B )
=> ~ ( ord_less_literal @ B @ A ) ) ).
% order.asym
thf(fact_2693_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_2694_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_2695_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_2696_order__antisym,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_2697_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_2698_order__antisym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_2699_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_2700_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_2701_antisym__conv2,axiom,
! [X: literal,Y: literal] :
( ( ord_less_eq_literal @ X @ Y )
=> ( ( ~ ( ord_less_literal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2702_antisym__conv2,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2703_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2704_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2705_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2706_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_2707_antisym__conv1,axiom,
! [X: literal,Y: literal] :
( ~ ( ord_less_literal @ X @ Y )
=> ( ( ord_less_eq_literal @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2708_antisym__conv1,axiom,
! [X: filter_nat,Y: filter_nat] :
( ~ ( ord_less_filter_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2709_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2710_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2711_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2712_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_2713_ord__le__eq__trans,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_2714_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_2715_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_2716_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_2717_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_2718_ord__eq__le__trans,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( A = B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_2719_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_2720_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_2721_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_2722_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_2723_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y3 )
& ( ord_le2510731241096832064er_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_2724_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_2725_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
& ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_2726_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_2727_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_2728_less__imp__neq,axiom,
! [X: literal,Y: literal] :
( ( ord_less_literal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_2729_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_2730_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_2731_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_2732_le__cases3,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_2733_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_2734_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_2735_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z2: rat] :
( ( ord_less_rat @ X @ Z2 )
& ( ord_less_rat @ Z2 @ Y ) ) ) ).
% dense
thf(fact_2736_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_2737_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_2738_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_2739_nless__le,axiom,
! [A: literal,B: literal] :
( ( ~ ( ord_less_literal @ A @ B ) )
= ( ~ ( ord_less_eq_literal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2740_nless__le,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ~ ( ord_less_filter_nat @ A @ B ) )
= ( ~ ( ord_le2510731241096832064er_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2741_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2742_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2743_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2744_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_2745_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_2746_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_2747_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_2748_lt__ex,axiom,
! [X: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X ) ).
% lt_ex
thf(fact_2749_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_2750_leI,axiom,
! [X: literal,Y: literal] :
( ~ ( ord_less_literal @ X @ Y )
=> ( ord_less_eq_literal @ Y @ X ) ) ).
% leI
thf(fact_2751_leI,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% leI
thf(fact_2752_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_2753_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_2754_leD,axiom,
! [Y: literal,X: literal] :
( ( ord_less_eq_literal @ Y @ X )
=> ~ ( ord_less_literal @ X @ Y ) ) ).
% leD
thf(fact_2755_leD,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ~ ( ord_less_filter_nat @ X @ Y ) ) ).
% leD
thf(fact_2756_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_2757_leD,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_rat @ X @ Y ) ) ).
% leD
thf(fact_2758_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_2759_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_2760_pred__subset__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
( ( ord_le4340812435750786203eger_o
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R )
@ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S ) )
= ( ord_le653643898420964396nteger @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2761_pred__subset__eq2,axiom,
! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
( ( ord_le8000401564054156549_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S ) )
= ( ord_le5966269811547037844et_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2762_pred__subset__eq2,axiom,
! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
( ( ord_le6753239538765779593_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S ) )
= ( ord_le4763372923235995152et_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2763_pred__subset__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ( ord_le5643404153117327598_int_o
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
= ( ord_le8725513860283290265nt_int @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2764_pred__subset__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ( ord_le2124322318746777828_int_o
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
@ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
= ( ord_le135402666524580259nt_int @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_2765_mult__less__le__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2766_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2767_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2768_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2769_mult__le__less__imp__less,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2770_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2771_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2772_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2773_mult__right__le__imp__le,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2774_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2775_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2776_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2777_mult__left__le__imp__le,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2778_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2779_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2780_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2781_mult__le__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2782_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2783_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2784_mult__le__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2785_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2786_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2787_mult__less__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2788_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2789_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2790_mult__strict__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2791_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2792_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2793_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2794_mult__right__less__imp__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2795_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2796_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2797_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2798_mult__less__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2799_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2800_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2801_mult__strict__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2802_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2803_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2804_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2805_mult__left__less__imp__less,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2806_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2807_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2808_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2809_mult__le__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2810_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2811_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2812_mult__le__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2813_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2814_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2815_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_2816_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_2817_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_2818_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_2819_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_2820_less__numeral__extra_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% less_numeral_extra(4)
thf(fact_2821_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_2822_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_2823_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_2824_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_2825_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_2826_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_2827_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_2828_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( ord_less_rat @ A @ B )
= ( ord_less_rat @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_2829_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_2830_diff__strict__mono,axiom,
! [A: rat,B: rat,D2: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ D2 @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_2831_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_2832_bot_Oextremum__strict,axiom,
! [A: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% bot.extremum_strict
thf(fact_2833_bot_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_2834_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_2835_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_2836_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_2837_bot_Onot__eq__extremum,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( A != bot_bo2099793752762293965at_nat )
= ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_2838_bot_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_2839_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_2840_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_2841_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_2842_not__psubset__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% not_psubset_empty
thf(fact_2843_not__psubset__empty,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_2844_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_2845_not__psubset__empty,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_2846_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2847_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2848_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2849_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_2850_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% less_minus_iff
thf(fact_2851_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_2852_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_2853_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_2854_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_2855_compl__less__swap2,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ X )
=> ( ord_le361264281704409273t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_2856_compl__less__swap1,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le361264281704409273t_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) )
=> ( ord_le361264281704409273t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% compl_less_swap1
thf(fact_2857_le__numeral__extra_I4_J,axiom,
ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).
% le_numeral_extra(4)
thf(fact_2858_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_2859_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_2860_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_2861_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( ord_less_eq_rat @ A @ B )
= ( ord_less_eq_rat @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_2862_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_2863_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_2864_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_2865_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_2866_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_2867_diff__mono,axiom,
! [A: rat,B: rat,D2: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ D2 @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_2868_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_2869_subrelI,axiom,
! [R3: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
( ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ R3 )
=> ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ S3 ) )
=> ( ord_le653643898420964396nteger @ R3 @ S3 ) ) ).
% subrelI
thf(fact_2870_subrelI,axiom,
! [R3: set_Pr3286484037609594932et_nat,S3: set_Pr3286484037609594932et_nat] :
( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ R3 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ S3 ) )
=> ( ord_le5966269811547037844et_nat @ R3 @ S3 ) ) ).
% subrelI
thf(fact_2871_subrelI,axiom,
! [R3: set_Pr8536935166611901872et_nat,S3: set_Pr8536935166611901872et_nat] :
( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ R3 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ S3 ) )
=> ( ord_le4763372923235995152et_nat @ R3 @ S3 ) ) ).
% subrelI
thf(fact_2872_subrelI,axiom,
! [R3: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
( ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ R3 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ S3 ) )
=> ( ord_le8725513860283290265nt_int @ R3 @ S3 ) ) ).
% subrelI
thf(fact_2873_subrelI,axiom,
! [R3: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
( ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ R3 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ S3 ) )
=> ( ord_le135402666524580259nt_int @ R3 @ S3 ) ) ).
% subrelI
thf(fact_2874_bot_Oextremum,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% bot.extremum
thf(fact_2875_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_2876_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_2877_bot_Oextremum,axiom,
! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A ) ).
% bot.extremum
thf(fact_2878_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_2879_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_2880_bot_Oextremum__unique,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_unique
thf(fact_2881_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_2882_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_2883_bot_Oextremum__unique,axiom,
! [A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ bot_bot_filter_nat )
= ( A = bot_bot_filter_nat ) ) ).
% bot.extremum_unique
thf(fact_2884_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_2885_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_2886_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_2887_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_2888_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_2889_bot_Oextremum__uniqueI,axiom,
! [A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ bot_bot_filter_nat )
=> ( A = bot_bot_filter_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_2890_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_2891_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_2892_top_Onot__eq__extremum,axiom,
! [A: set_list_nat] :
( ( A != top_top_set_list_nat )
= ( ord_le1190675801316882794st_nat @ A @ top_top_set_list_nat ) ) ).
% top.not_eq_extremum
thf(fact_2893_top_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != top_top_set_o )
= ( ord_less_set_o @ A @ top_top_set_o ) ) ).
% top.not_eq_extremum
thf(fact_2894_top_Onot__eq__extremum,axiom,
! [A: set_rat] :
( ( A != top_top_set_rat )
= ( ord_less_set_rat @ A @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_2895_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_2896_top_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != top_top_set_int )
= ( ord_less_set_int @ A @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_2897_top_Oextremum__strict,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ top_top_set_list_nat @ A ) ).
% top.extremum_strict
thf(fact_2898_top_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ top_top_set_o @ A ) ).
% top.extremum_strict
thf(fact_2899_top_Oextremum__strict,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A ) ).
% top.extremum_strict
thf(fact_2900_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_2901_top_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A ) ).
% top.extremum_strict
thf(fact_2902_less__infI1,axiom,
! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ X )
=> ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2903_less__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2904_less__infI1,axiom,
! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ A @ X )
=> ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2905_less__infI1,axiom,
! [A: list_char > $o,X: list_char > $o,B: list_char > $o] :
( ( ord_less_list_char_o @ A @ X )
=> ( ord_less_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2906_less__infI1,axiom,
! [A: rat,X: rat,B: rat] :
( ( ord_less_rat @ A @ X )
=> ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2907_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2908_less__infI1,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_int @ A @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_2909_less__infI2,axiom,
! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B @ X )
=> ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2910_less__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2911_less__infI2,axiom,
! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ B @ X )
=> ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2912_less__infI2,axiom,
! [B: list_char > $o,X: list_char > $o,A: list_char > $o] :
( ( ord_less_list_char_o @ B @ X )
=> ( ord_less_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2913_less__infI2,axiom,
! [B: rat,X: rat,A: rat] :
( ( ord_less_rat @ B @ X )
=> ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2914_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2915_less__infI2,axiom,
! [B: int,X: int,A: int] :
( ( ord_less_int @ B @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_2916_inf_Oabsorb3,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ B )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2917_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2918_inf_Oabsorb3,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ A @ B )
=> ( ( inf_in3604695632404883862_int_o @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2919_inf_Oabsorb3,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( ord_less_list_char_o @ A @ B )
=> ( ( inf_inf_list_char_o @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2920_inf_Oabsorb3,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( inf_inf_rat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2921_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2922_inf_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_2923_inf_Oabsorb4,axiom,
! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B @ A )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2924_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2925_inf_Oabsorb4,axiom,
! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ B @ A )
=> ( ( inf_in3604695632404883862_int_o @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2926_inf_Oabsorb4,axiom,
! [B: list_char > $o,A: list_char > $o] :
( ( ord_less_list_char_o @ B @ A )
=> ( ( inf_inf_list_char_o @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2927_inf_Oabsorb4,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( inf_inf_rat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2928_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2929_inf_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_2930_inf_Ostrict__boundedE,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
=> ~ ( ( ord_le7866589430770878221at_nat @ A @ B )
=> ~ ( ord_le7866589430770878221at_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2931_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2932_inf_Ostrict__boundedE,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
=> ~ ( ( ord_le8213806771718485336_int_o @ A @ B )
=> ~ ( ord_le8213806771718485336_int_o @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2933_inf_Ostrict__boundedE,axiom,
! [A: list_char > $o,B: list_char > $o,C: list_char > $o] :
( ( ord_less_list_char_o @ A @ ( inf_inf_list_char_o @ B @ C ) )
=> ~ ( ( ord_less_list_char_o @ A @ B )
=> ~ ( ord_less_list_char_o @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2934_inf_Ostrict__boundedE,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( inf_inf_rat @ B @ C ) )
=> ~ ( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2935_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2936_inf_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_2937_inf_Ostrict__order__iff,axiom,
( ord_le7866589430770878221at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( A3
= ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2938_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( A3
= ( inf_inf_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2939_inf_Ostrict__order__iff,axiom,
( ord_le8213806771718485336_int_o
= ( ^ [A3: product_prod_int_int > $o,B3: product_prod_int_int > $o] :
( ( A3
= ( inf_in3604695632404883862_int_o @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2940_inf_Ostrict__order__iff,axiom,
( ord_less_list_char_o
= ( ^ [A3: list_char > $o,B3: list_char > $o] :
( ( A3
= ( inf_inf_list_char_o @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2941_inf_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( A3
= ( inf_inf_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2942_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2943_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( A3
= ( inf_inf_int @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_2944_inf_Ostrict__coboundedI1,axiom,
! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A @ C )
=> ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2945_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2946_inf_Ostrict__coboundedI1,axiom,
! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ A @ C )
=> ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2947_inf_Ostrict__coboundedI1,axiom,
! [A: list_char > $o,C: list_char > $o,B: list_char > $o] :
( ( ord_less_list_char_o @ A @ C )
=> ( ord_less_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2948_inf_Ostrict__coboundedI1,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ A @ C )
=> ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2949_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2950_inf_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_2951_inf_Ostrict__coboundedI2,axiom,
! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B @ C )
=> ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2952_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2953_inf_Ostrict__coboundedI2,axiom,
! [B: product_prod_int_int > $o,C: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8213806771718485336_int_o @ B @ C )
=> ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2954_inf_Ostrict__coboundedI2,axiom,
! [B: list_char > $o,C: list_char > $o,A: list_char > $o] :
( ( ord_less_list_char_o @ B @ C )
=> ( ord_less_list_char_o @ ( inf_inf_list_char_o @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2955_inf_Ostrict__coboundedI2,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2956_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2957_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_2958_less__supI1,axiom,
! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ X @ A )
=> ( ord_le7642048601412989811at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_2959_less__supI1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ X @ A )
=> ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_2960_less__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_2961_less__supI1,axiom,
! [X: rat,A: rat,B: rat] :
( ( ord_less_rat @ X @ A )
=> ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).
% less_supI1
thf(fact_2962_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_2963_less__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_int @ X @ A )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_2964_less__supI2,axiom,
! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ X @ B )
=> ( ord_le7642048601412989811at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_2965_less__supI2,axiom,
! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ X @ B )
=> ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_2966_less__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_2967_less__supI2,axiom,
! [X: rat,B: rat,A: rat] :
( ( ord_less_rat @ X @ B )
=> ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).
% less_supI2
thf(fact_2968_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_2969_less__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_int @ X @ B )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_2970_sup_Oabsorb3,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ B @ A )
=> ( ( sup_su3035147773818789531at_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2971_sup_Oabsorb3,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ B @ A )
=> ( ( sup_su5525570899277871387at_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2972_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2973_sup_Oabsorb3,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( sup_sup_rat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2974_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2975_sup_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_2976_sup_Oabsorb4,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ A @ B )
=> ( ( sup_su3035147773818789531at_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2977_sup_Oabsorb4,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ A @ B )
=> ( ( sup_su5525570899277871387at_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2978_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2979_sup_Oabsorb4,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( sup_sup_rat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2980_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2981_sup_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_2982_sup_Ostrict__boundedE,axiom,
! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
=> ~ ( ( ord_le7642048601412989811at_nat @ B @ A )
=> ~ ( ord_le7642048601412989811at_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2983_sup_Ostrict__boundedE,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
=> ~ ( ( ord_le2604355607129572851at_nat @ B @ A )
=> ~ ( ord_le2604355607129572851at_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2984_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2985_sup_Ostrict__boundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( sup_sup_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2986_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2987_sup_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_2988_sup_Ostrict__order__iff,axiom,
( ord_le7642048601412989811at_nat
= ( ^ [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
( ( A3
= ( sup_su3035147773818789531at_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2989_sup_Ostrict__order__iff,axiom,
( ord_le2604355607129572851at_nat
= ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
( ( A3
= ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2990_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2991_sup_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( A3
= ( sup_sup_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2992_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2993_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( A3
= ( sup_sup_int @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_2994_sup_Ostrict__coboundedI1,axiom,
! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ C @ A )
=> ( ord_le7642048601412989811at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_2995_sup_Ostrict__coboundedI1,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ C @ A )
=> ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_2996_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_2997_sup_Ostrict__coboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ A )
=> ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_2998_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_2999_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_3000_sup_Ostrict__coboundedI2,axiom,
! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le7642048601412989811at_nat @ C @ B )
=> ( ord_le7642048601412989811at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3001_sup_Ostrict__coboundedI2,axiom,
! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ C @ B )
=> ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3002_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3003_sup_Ostrict__coboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3004_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3005_sup_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3006_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_minus_iff
thf(fact_3007_le__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% le_minus_iff
thf(fact_3008_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_3009_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_3010_minus__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_3011_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_3012_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_imp_neg_le
thf(fact_3013_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% le_imp_neg_le
thf(fact_3014_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_3015_compl__le__swap2,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ X )
=> ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_3016_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_3017_compl__le__swap1,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) )
=> ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% compl_le_swap1
thf(fact_3018_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_3019_compl__mono,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ Y ) @ ( uminus2952777764628376836t_unit @ X ) ) ) ).
% compl_mono
thf(fact_3020_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_3021_subset__minus__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
=> ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat ) ) ).
% subset_minus_empty
thf(fact_3022_subset__minus__empty,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( minus_minus_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ).
% subset_minus_empty
thf(fact_3023_subset__minus__empty,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( minus_minus_set_int @ A2 @ B2 )
= bot_bot_set_int ) ) ).
% subset_minus_empty
thf(fact_3024_subset__minus__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% subset_minus_empty
thf(fact_3025_top_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ top_top_set_list_nat @ A )
=> ( A = top_top_set_list_nat ) ) ).
% top.extremum_uniqueI
thf(fact_3026_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_3027_top_Oextremum__uniqueI,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
=> ( A = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_3028_top_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A )
=> ( A = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_3029_top_Oextremum__uniqueI,axiom,
! [A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A )
=> ( A = top_top_filter_nat ) ) ).
% top.extremum_uniqueI
thf(fact_3030_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_3031_top_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ top_top_set_list_nat @ A )
= ( A = top_top_set_list_nat ) ) ).
% top.extremum_unique
thf(fact_3032_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_3033_top_Oextremum__unique,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
= ( A = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_3034_top_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A )
= ( A = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_3035_top_Oextremum__unique,axiom,
! [A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A )
= ( A = top_top_filter_nat ) ) ).
% top.extremum_unique
thf(fact_3036_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_3037_top__greatest,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ top_top_set_list_nat ) ).
% top_greatest
thf(fact_3038_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_3039_top__greatest,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% top_greatest
thf(fact_3040_top__greatest,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% top_greatest
thf(fact_3041_top__greatest,axiom,
! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ A @ top_top_filter_nat ) ).
% top_greatest
thf(fact_3042_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_3043_inf__sup__ord_I2_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3044_inf__sup__ord_I2_J,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3045_inf__sup__ord_I2_J,axiom,
! [X: list_char > $o,Y: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3046_inf__sup__ord_I2_J,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3047_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3048_inf__sup__ord_I2_J,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3049_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3050_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_3051_inf__sup__ord_I1_J,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3052_inf__sup__ord_I1_J,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3053_inf__sup__ord_I1_J,axiom,
! [X: list_char > $o,Y: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3054_inf__sup__ord_I1_J,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3055_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3056_inf__sup__ord_I1_J,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3057_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3058_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_3059_inf__le1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3060_inf__le1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3061_inf__le1,axiom,
! [X: list_char > $o,Y: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3062_inf__le1,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3063_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3064_inf__le1,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3065_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3066_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_3067_inf__le2,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3068_inf__le2,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3069_inf__le2,axiom,
! [X: list_char > $o,Y: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3070_inf__le2,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3071_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3072_inf__le2,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3073_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3074_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_3075_le__infE,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) )
=> ~ ( ( ord_le3146513528884898305at_nat @ X @ A )
=> ~ ( ord_le3146513528884898305at_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_3076_le__infE,axiom,
! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) )
=> ~ ( ( ord_le8369615600986905444_int_o @ X @ A )
=> ~ ( ord_le8369615600986905444_int_o @ X @ B ) ) ) ).
% le_infE
thf(fact_3077_le__infE,axiom,
! [X: list_char > $o,A: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X @ ( inf_inf_list_char_o @ A @ B ) )
=> ~ ( ( ord_le4796328588573674190char_o @ X @ A )
=> ~ ( ord_le4796328588573674190char_o @ X @ B ) ) ) ).
% le_infE
thf(fact_3078_le__infE,axiom,
! [X: filter_nat,A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ A @ B ) )
=> ~ ( ( ord_le2510731241096832064er_nat @ X @ A )
=> ~ ( ord_le2510731241096832064er_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_3079_le__infE,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_3080_le__infE,axiom,
! [X: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) )
=> ~ ( ( ord_less_eq_rat @ X @ A )
=> ~ ( ord_less_eq_rat @ X @ B ) ) ) ).
% le_infE
thf(fact_3081_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_3082_le__infE,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
=> ~ ( ( ord_less_eq_int @ X @ A )
=> ~ ( ord_less_eq_int @ X @ B ) ) ) ).
% le_infE
thf(fact_3083_le__infI,axiom,
! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ A )
=> ( ( ord_le3146513528884898305at_nat @ X @ B )
=> ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3084_le__infI,axiom,
! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X @ A )
=> ( ( ord_le8369615600986905444_int_o @ X @ B )
=> ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3085_le__infI,axiom,
! [X: list_char > $o,A: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X @ A )
=> ( ( ord_le4796328588573674190char_o @ X @ B )
=> ( ord_le4796328588573674190char_o @ X @ ( inf_inf_list_char_o @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3086_le__infI,axiom,
! [X: filter_nat,A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ A )
=> ( ( ord_le2510731241096832064er_nat @ X @ B )
=> ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3087_le__infI,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3088_le__infI,axiom,
! [X: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ X @ B )
=> ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3089_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3090_le__infI,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).
% le_infI
thf(fact_3091_inf__mono,axiom,
! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ C )
=> ( ( ord_le3146513528884898305at_nat @ B @ D2 )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3092_inf__mono,axiom,
! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o,D2: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ C )
=> ( ( ord_le8369615600986905444_int_o @ B @ D2 )
=> ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ ( inf_in3604695632404883862_int_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3093_inf__mono,axiom,
! [A: list_char > $o,C: list_char > $o,B: list_char > $o,D2: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ C )
=> ( ( ord_le4796328588573674190char_o @ B @ D2 )
=> ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ ( inf_inf_list_char_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3094_inf__mono,axiom,
! [A: filter_nat,C: filter_nat,B: filter_nat,D2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ C )
=> ( ( ord_le2510731241096832064er_nat @ B @ D2 )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ ( inf_inf_filter_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3095_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3096_inf__mono,axiom,
! [A: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ C )
=> ( ( ord_less_eq_rat @ B @ D2 )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3097_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3098_inf__mono,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_3099_le__infI1,axiom,
! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ X )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3100_le__infI1,axiom,
! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ X )
=> ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3101_le__infI1,axiom,
! [A: list_char > $o,X: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ X )
=> ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3102_le__infI1,axiom,
! [A: filter_nat,X: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ X )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3103_le__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3104_le__infI1,axiom,
! [A: rat,X: rat,B: rat] :
( ( ord_less_eq_rat @ A @ X )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3105_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3106_le__infI1,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_3107_le__infI2,axiom,
! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ X )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3108_le__infI2,axiom,
! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ B @ X )
=> ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3109_le__infI2,axiom,
! [B: list_char > $o,X: list_char > $o,A: list_char > $o] :
( ( ord_le4796328588573674190char_o @ B @ X )
=> ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3110_le__infI2,axiom,
! [B: filter_nat,X: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ X )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3111_le__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3112_le__infI2,axiom,
! [B: rat,X: rat,A: rat] :
( ( ord_less_eq_rat @ B @ X )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3113_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3114_le__infI2,axiom,
! [B: int,X: int,A: int] :
( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_3115_inf_OorderE,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( A
= ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3116_inf_OorderE,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ B )
=> ( A
= ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3117_inf_OorderE,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ B )
=> ( A
= ( inf_inf_list_char_o @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3118_inf_OorderE,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( A
= ( inf_inf_filter_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3119_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3120_inf_OorderE,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( A
= ( inf_inf_rat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3121_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3122_inf_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( inf_inf_int @ A @ B ) ) ) ).
% inf.orderE
thf(fact_3123_inf_OorderI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( A
= ( inf_in2572325071724192079at_nat @ A @ B ) )
=> ( ord_le3146513528884898305at_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_3124_inf_OorderI,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( A
= ( inf_in3604695632404883862_int_o @ A @ B ) )
=> ( ord_le8369615600986905444_int_o @ A @ B ) ) ).
% inf.orderI
thf(fact_3125_inf_OorderI,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( A
= ( inf_inf_list_char_o @ A @ B ) )
=> ( ord_le4796328588573674190char_o @ A @ B ) ) ).
% inf.orderI
thf(fact_3126_inf_OorderI,axiom,
! [A: filter_nat,B: filter_nat] :
( ( A
= ( inf_inf_filter_nat @ A @ B ) )
=> ( ord_le2510731241096832064er_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_3127_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_3128_inf_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( inf_inf_rat @ A @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% inf.orderI
thf(fact_3129_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_3130_inf_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( inf_inf_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% inf.orderI
thf(fact_3131_inf__unique,axiom,
! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X2 @ Y2 )
=> ( ( ord_le3146513528884898305at_nat @ X2 @ Z2 )
=> ( ord_le3146513528884898305at_nat @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_in2572325071724192079at_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3132_inf__unique,axiom,
! [F2: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o,X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X2 @ Y2 )
=> ( ( ord_le8369615600986905444_int_o @ X2 @ Z2 )
=> ( ord_le8369615600986905444_int_o @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_in3604695632404883862_int_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3133_inf__unique,axiom,
! [F2: ( list_char > $o ) > ( list_char > $o ) > list_char > $o,X: list_char > $o,Y: list_char > $o] :
( ! [X2: list_char > $o,Y2: list_char > $o] : ( ord_le4796328588573674190char_o @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: list_char > $o,Y2: list_char > $o] : ( ord_le4796328588573674190char_o @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: list_char > $o,Y2: list_char > $o,Z2: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X2 @ Y2 )
=> ( ( ord_le4796328588573674190char_o @ X2 @ Z2 )
=> ( ord_le4796328588573674190char_o @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_list_char_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3134_inf__unique,axiom,
! [F2: filter_nat > filter_nat > filter_nat,X: filter_nat,Y: filter_nat] :
( ! [X2: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: filter_nat,Y2: filter_nat,Z2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
=> ( ( ord_le2510731241096832064er_nat @ X2 @ Z2 )
=> ( ord_le2510731241096832064er_nat @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_filter_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3135_inf__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3136_inf__unique,axiom,
! [F2: rat > rat > rat,X: rat,Y: rat] :
( ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: rat,Y2: rat,Z2: rat] :
( ( ord_less_eq_rat @ X2 @ Y2 )
=> ( ( ord_less_eq_rat @ X2 @ Z2 )
=> ( ord_less_eq_rat @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_rat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3137_inf__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3138_inf__unique,axiom,
! [F2: int > int > int,X: int,Y: int] :
( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F2 @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F2 @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Z2 )
=> ( ord_less_eq_int @ X2 @ ( F2 @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_3139_le__iff__inf,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3140_le__iff__inf,axiom,
( ord_le8369615600986905444_int_o
= ( ^ [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3141_le__iff__inf,axiom,
( ord_le4796328588573674190char_o
= ( ^ [X3: list_char > $o,Y3: list_char > $o] :
( ( inf_inf_list_char_o @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3142_le__iff__inf,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( inf_inf_filter_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3143_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3144_le__iff__inf,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y3: rat] :
( ( inf_inf_rat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3145_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3146_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( inf_inf_int @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_3147_inf_Oabsorb1,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3148_inf_Oabsorb1,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ B )
=> ( ( inf_in3604695632404883862_int_o @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3149_inf_Oabsorb1,axiom,
! [A: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ B )
=> ( ( inf_inf_list_char_o @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3150_inf_Oabsorb1,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( inf_inf_filter_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3151_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3152_inf_Oabsorb1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( inf_inf_rat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3153_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3154_inf_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_3155_inf_Oabsorb2,axiom,
! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ A )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3156_inf_Oabsorb2,axiom,
! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ B @ A )
=> ( ( inf_in3604695632404883862_int_o @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3157_inf_Oabsorb2,axiom,
! [B: list_char > $o,A: list_char > $o] :
( ( ord_le4796328588573674190char_o @ B @ A )
=> ( ( inf_inf_list_char_o @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3158_inf_Oabsorb2,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( inf_inf_filter_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3159_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3160_inf_Oabsorb2,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( inf_inf_rat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3161_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3162_inf_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_3163_inf__absorb1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ( inf_in2572325071724192079at_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3164_inf__absorb1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X @ Y )
=> ( ( inf_in3604695632404883862_int_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3165_inf__absorb1,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X @ Y )
=> ( ( inf_inf_list_char_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3166_inf__absorb1,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( inf_inf_filter_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3167_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3168_inf__absorb1,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( inf_inf_rat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3169_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3170_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_3171_inf__absorb2,axiom,
! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ Y @ X )
=> ( ( inf_in2572325071724192079at_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3172_inf__absorb2,axiom,
! [Y: product_prod_int_int > $o,X: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ Y @ X )
=> ( ( inf_in3604695632404883862_int_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3173_inf__absorb2,axiom,
! [Y: list_char > $o,X: list_char > $o] :
( ( ord_le4796328588573674190char_o @ Y @ X )
=> ( ( inf_inf_list_char_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3174_inf__absorb2,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( inf_inf_filter_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3175_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3176_inf__absorb2,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( inf_inf_rat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3177_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3178_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_3179_inf_OboundedE,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
=> ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
=> ~ ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3180_inf_OboundedE,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
=> ~ ( ( ord_le8369615600986905444_int_o @ A @ B )
=> ~ ( ord_le8369615600986905444_int_o @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3181_inf_OboundedE,axiom,
! [A: list_char > $o,B: list_char > $o,C: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ ( inf_inf_list_char_o @ B @ C ) )
=> ~ ( ( ord_le4796328588573674190char_o @ A @ B )
=> ~ ( ord_le4796328588573674190char_o @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3182_inf_OboundedE,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ ( inf_inf_filter_nat @ B @ C ) )
=> ~ ( ( ord_le2510731241096832064er_nat @ A @ B )
=> ~ ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3183_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3184_inf_OboundedE,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
=> ~ ( ( ord_less_eq_rat @ A @ B )
=> ~ ( ord_less_eq_rat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3185_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3186_inf_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_3187_inf_OboundedI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( ord_le3146513528884898305at_nat @ A @ C )
=> ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3188_inf_OboundedI,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ B )
=> ( ( ord_le8369615600986905444_int_o @ A @ C )
=> ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3189_inf_OboundedI,axiom,
! [A: list_char > $o,B: list_char > $o,C: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ B )
=> ( ( ord_le4796328588573674190char_o @ A @ C )
=> ( ord_le4796328588573674190char_o @ A @ ( inf_inf_list_char_o @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3190_inf_OboundedI,axiom,
! [A: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( ord_le2510731241096832064er_nat @ A @ C )
=> ( ord_le2510731241096832064er_nat @ A @ ( inf_inf_filter_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3191_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3192_inf_OboundedI,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ A @ C )
=> ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3193_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3194_inf_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_3195_inf__greatest,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X @ Y )
=> ( ( ord_le3146513528884898305at_nat @ X @ Z )
=> ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3196_inf__greatest,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ X @ Y )
=> ( ( ord_le8369615600986905444_int_o @ X @ Z )
=> ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3197_inf__greatest,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( ord_le4796328588573674190char_o @ X @ Y )
=> ( ( ord_le4796328588573674190char_o @ X @ Z )
=> ( ord_le4796328588573674190char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3198_inf__greatest,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ X @ Z )
=> ( ord_le2510731241096832064er_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3199_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3200_inf__greatest,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Z )
=> ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3201_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3202_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_3203_inf_Oorder__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( A3
= ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3204_inf_Oorder__iff,axiom,
( ord_le8369615600986905444_int_o
= ( ^ [A3: product_prod_int_int > $o,B3: product_prod_int_int > $o] :
( A3
= ( inf_in3604695632404883862_int_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3205_inf_Oorder__iff,axiom,
( ord_le4796328588573674190char_o
= ( ^ [A3: list_char > $o,B3: list_char > $o] :
( A3
= ( inf_inf_list_char_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3206_inf_Oorder__iff,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( A3
= ( inf_inf_filter_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3207_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( A3
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3208_inf_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( A3
= ( inf_inf_rat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3209_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3210_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( A3
= ( inf_inf_int @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_3211_inf_Ocobounded1,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3212_inf_Ocobounded1,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3213_inf_Ocobounded1,axiom,
! [A: list_char > $o,B: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3214_inf_Ocobounded1,axiom,
! [A: filter_nat,B: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3215_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3216_inf_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3217_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3218_inf_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_3219_inf_Ocobounded2,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3220_inf_Ocobounded2,axiom,
! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3221_inf_Ocobounded2,axiom,
! [A: list_char > $o,B: list_char > $o] : ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3222_inf_Ocobounded2,axiom,
! [A: filter_nat,B: filter_nat] : ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3223_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3224_inf_Ocobounded2,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3225_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3226_inf_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_3227_inf_Oabsorb__iff1,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3228_inf_Oabsorb__iff1,axiom,
( ord_le8369615600986905444_int_o
= ( ^ [A3: product_prod_int_int > $o,B3: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3229_inf_Oabsorb__iff1,axiom,
( ord_le4796328588573674190char_o
= ( ^ [A3: list_char > $o,B3: list_char > $o] :
( ( inf_inf_list_char_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3230_inf_Oabsorb__iff1,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( inf_inf_filter_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3231_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3232_inf_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( inf_inf_rat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3233_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3234_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( inf_inf_int @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_3235_inf_Oabsorb__iff2,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3236_inf_Oabsorb__iff2,axiom,
( ord_le8369615600986905444_int_o
= ( ^ [B3: product_prod_int_int > $o,A3: product_prod_int_int > $o] :
( ( inf_in3604695632404883862_int_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3237_inf_Oabsorb__iff2,axiom,
( ord_le4796328588573674190char_o
= ( ^ [B3: list_char > $o,A3: list_char > $o] :
( ( inf_inf_list_char_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3238_inf_Oabsorb__iff2,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( ( inf_inf_filter_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3239_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3240_inf_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( inf_inf_rat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3241_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3242_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( inf_inf_int @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_3243_inf_OcoboundedI1,axiom,
! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ C )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3244_inf_OcoboundedI1,axiom,
! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ A @ C )
=> ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3245_inf_OcoboundedI1,axiom,
! [A: list_char > $o,C: list_char > $o,B: list_char > $o] :
( ( ord_le4796328588573674190char_o @ A @ C )
=> ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3246_inf_OcoboundedI1,axiom,
! [A: filter_nat,C: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ C )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3247_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3248_inf_OcoboundedI1,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ C )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3249_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3250_inf_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_3251_inf_OcoboundedI2,axiom,
! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B @ C )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3252_inf_OcoboundedI2,axiom,
! [B: product_prod_int_int > $o,C: product_prod_int_int > $o,A: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ B @ C )
=> ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3253_inf_OcoboundedI2,axiom,
! [B: list_char > $o,C: list_char > $o,A: list_char > $o] :
( ( ord_le4796328588573674190char_o @ B @ C )
=> ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3254_inf_OcoboundedI2,axiom,
! [B: filter_nat,C: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3255_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3256_inf_OcoboundedI2,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3257_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3258_inf_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_3259_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3260_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3261_inf__sup__ord_I4_J,axiom,
! [Y: filter_nat,X: filter_nat] : ( ord_le2510731241096832064er_nat @ Y @ ( sup_sup_filter_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3262_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3263_inf__sup__ord_I4_J,axiom,
! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3264_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3265_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_3266_inf__sup__ord_I3_J,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3267_inf__sup__ord_I3_J,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3268_inf__sup__ord_I3_J,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3269_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3270_inf__sup__ord_I3_J,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3271_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3272_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_3273_le__supE,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ X )
=> ~ ( ( ord_le8081472938463900775at_nat @ A @ X )
=> ~ ( ord_le8081472938463900775at_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_3274_le__supE,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X )
=> ~ ( ( ord_le1268244103169919719at_nat @ A @ X )
=> ~ ( ord_le1268244103169919719at_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_3275_le__supE,axiom,
! [A: filter_nat,B: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A @ B ) @ X )
=> ~ ( ( ord_le2510731241096832064er_nat @ A @ X )
=> ~ ( ord_le2510731241096832064er_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_3276_le__supE,axiom,
! [A: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_3277_le__supE,axiom,
! [A: rat,B: rat,X: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_rat @ A @ X )
=> ~ ( ord_less_eq_rat @ B @ X ) ) ) ).
% le_supE
thf(fact_3278_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_3279_le__supE,axiom,
! [A: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_3280_le__supI,axiom,
! [A: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A @ X )
=> ( ( ord_le8081472938463900775at_nat @ B @ X )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3281_le__supI,axiom,
! [A: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A @ X )
=> ( ( ord_le1268244103169919719at_nat @ B @ X )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3282_le__supI,axiom,
! [A: filter_nat,X: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ X )
=> ( ( ord_le2510731241096832064er_nat @ B @ X )
=> ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3283_le__supI,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3284_le__supI,axiom,
! [A: rat,X: rat,B: rat] :
( ( ord_less_eq_rat @ A @ X )
=> ( ( ord_less_eq_rat @ B @ X )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3285_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3286_le__supI,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_3287_sup__ge1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3288_sup__ge1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3289_sup__ge1,axiom,
! [X: filter_nat,Y: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3290_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3291_sup__ge1,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3292_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_3293_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_3294_sup__ge2,axiom,
! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3295_sup__ge2,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3296_sup__ge2,axiom,
! [Y: filter_nat,X: filter_nat] : ( ord_le2510731241096832064er_nat @ Y @ ( sup_sup_filter_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3297_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3298_sup__ge2,axiom,
! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3299_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_3300_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_3301_le__supI1,axiom,
! [X: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ X @ A )
=> ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3302_le__supI1,axiom,
! [X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ X @ A )
=> ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3303_le__supI1,axiom,
! [X: filter_nat,A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ A )
=> ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3304_le__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3305_le__supI1,axiom,
! [X: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3306_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_3307_le__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_3308_le__supI2,axiom,
! [X: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ X @ B )
=> ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3309_le__supI2,axiom,
! [X: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ X @ B )
=> ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3310_le__supI2,axiom,
! [X: filter_nat,B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ B )
=> ( ord_le2510731241096832064er_nat @ X @ ( sup_sup_filter_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3311_le__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3312_le__supI2,axiom,
! [X: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ X @ B )
=> ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3313_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_3314_le__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_3315_sup_Omono,axiom,
! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ C @ A )
=> ( ( ord_le8081472938463900775at_nat @ D2 @ B )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ C @ D2 ) @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3316_sup_Omono,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ C @ A )
=> ( ( ord_le1268244103169919719at_nat @ D2 @ B )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D2 ) @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3317_sup_Omono,axiom,
! [C: filter_nat,A: filter_nat,D2: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ C @ A )
=> ( ( ord_le2510731241096832064er_nat @ D2 @ B )
=> ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ C @ D2 ) @ ( sup_sup_filter_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3318_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3319_sup_Omono,axiom,
! [C: rat,A: rat,D2: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ( ord_less_eq_rat @ D2 @ B )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D2 ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3320_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3321_sup_Omono,axiom,
! [C: int,A: int,D2: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D2 @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_3322_sup__mono,axiom,
! [A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A @ C )
=> ( ( ord_le8081472938463900775at_nat @ B @ D2 )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ ( sup_su3035147773818789531at_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3323_sup__mono,axiom,
! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A @ C )
=> ( ( ord_le1268244103169919719at_nat @ B @ D2 )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3324_sup__mono,axiom,
! [A: filter_nat,C: filter_nat,B: filter_nat,D2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ C )
=> ( ( ord_le2510731241096832064er_nat @ B @ D2 )
=> ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ A @ B ) @ ( sup_sup_filter_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3325_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3326_sup__mono,axiom,
! [A: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ C )
=> ( ( ord_less_eq_rat @ B @ D2 )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3327_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3328_sup__mono,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_3329_sup__least,axiom,
! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ Y @ X )
=> ( ( ord_le8081472938463900775at_nat @ Z @ X )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3330_sup__least,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ Y @ X )
=> ( ( ord_le1268244103169919719at_nat @ Z @ X )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3331_sup__least,axiom,
! [Y: filter_nat,X: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( ord_le2510731241096832064er_nat @ Z @ X )
=> ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3332_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3333_sup__least,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_less_eq_rat @ Z @ X )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3334_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3335_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_3336_le__iff__sup,axiom,
( ord_le8081472938463900775at_nat
= ( ^ [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3337_le__iff__sup,axiom,
( ord_le1268244103169919719at_nat
= ( ^ [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3338_le__iff__sup,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [X3: filter_nat,Y3: filter_nat] :
( ( sup_sup_filter_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3339_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3340_le__iff__sup,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y3: rat] :
( ( sup_sup_rat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3341_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3342_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( sup_sup_int @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_3343_sup_OorderE,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ B @ A )
=> ( A
= ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3344_sup_OorderE,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ B @ A )
=> ( A
= ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3345_sup_OorderE,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( A
= ( sup_sup_filter_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3346_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3347_sup_OorderE,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( A
= ( sup_sup_rat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3348_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3349_sup_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( sup_sup_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_3350_sup_OorderI,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( A
= ( sup_su3035147773818789531at_nat @ A @ B ) )
=> ( ord_le8081472938463900775at_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_3351_sup_OorderI,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( A
= ( sup_su5525570899277871387at_nat @ A @ B ) )
=> ( ord_le1268244103169919719at_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_3352_sup_OorderI,axiom,
! [A: filter_nat,B: filter_nat] :
( ( A
= ( sup_sup_filter_nat @ A @ B ) )
=> ( ord_le2510731241096832064er_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_3353_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_3354_sup_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( sup_sup_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% sup.orderI
thf(fact_3355_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_3356_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% sup.orderI
thf(fact_3357_sup__unique,axiom,
! [F2: set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat > set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_Pr8551490117392284871at_nat,Y2: set_Pr8551490117392284871at_nat,Z2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ Y2 @ X2 )
=> ( ( ord_le8081472938463900775at_nat @ Z2 @ X2 )
=> ( ord_le8081472938463900775at_nat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_su3035147773818789531at_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3358_sup__unique,axiom,
! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ Y2 @ X2 )
=> ( ( ord_le1268244103169919719at_nat @ Z2 @ X2 )
=> ( ord_le1268244103169919719at_nat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_su5525570899277871387at_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3359_sup__unique,axiom,
! [F2: filter_nat > filter_nat > filter_nat,X: filter_nat,Y: filter_nat] :
( ! [X2: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: filter_nat,Y2: filter_nat] : ( ord_le2510731241096832064er_nat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: filter_nat,Y2: filter_nat,Z2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y2 @ X2 )
=> ( ( ord_le2510731241096832064er_nat @ Z2 @ X2 )
=> ( ord_le2510731241096832064er_nat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_filter_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3360_sup__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X2 )
=> ( ord_less_eq_set_nat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3361_sup__unique,axiom,
! [F2: rat > rat > rat,X: rat,Y: rat] :
( ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: rat,Y2: rat,Z2: rat] :
( ( ord_less_eq_rat @ Y2 @ X2 )
=> ( ( ord_less_eq_rat @ Z2 @ X2 )
=> ( ord_less_eq_rat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_rat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3362_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3363_sup__unique,axiom,
! [F2: int > int > int,X: int,Y: int] :
( ! [X2: int,Y2: int] : ( ord_less_eq_int @ X2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F2 @ X2 @ Y2 ) )
=> ( ! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ Z2 @ X2 )
=> ( ord_less_eq_int @ ( F2 @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_3364_sup_Oabsorb1,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ B @ A )
=> ( ( sup_su3035147773818789531at_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3365_sup_Oabsorb1,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ B @ A )
=> ( ( sup_su5525570899277871387at_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3366_sup_Oabsorb1,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( sup_sup_filter_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3367_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3368_sup_Oabsorb1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( sup_sup_rat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3369_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3370_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_3371_sup_Oabsorb2,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A @ B )
=> ( ( sup_su3035147773818789531at_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3372_sup_Oabsorb2,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A @ B )
=> ( ( sup_su5525570899277871387at_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3373_sup_Oabsorb2,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A @ B )
=> ( ( sup_sup_filter_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3374_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3375_sup_Oabsorb2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( sup_sup_rat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3376_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3377_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_3378_sup__absorb1,axiom,
! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ Y @ X )
=> ( ( sup_su3035147773818789531at_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3379_sup__absorb1,axiom,
! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ Y @ X )
=> ( ( sup_su5525570899277871387at_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3380_sup__absorb1,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( sup_sup_filter_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3381_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3382_sup__absorb1,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( sup_sup_rat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3383_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3384_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_3385_sup__absorb2,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ X @ Y )
=> ( ( sup_su3035147773818789531at_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3386_sup__absorb2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ X @ Y )
=> ( ( sup_su5525570899277871387at_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3387_sup__absorb2,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( sup_sup_filter_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3388_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3389_sup__absorb2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( sup_sup_rat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3390_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3391_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_3392_sup_OboundedE,axiom,
! [B: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A )
=> ~ ( ( ord_le8081472938463900775at_nat @ B @ A )
=> ~ ( ord_le8081472938463900775at_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3393_sup_OboundedE,axiom,
! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
=> ~ ( ( ord_le1268244103169919719at_nat @ B @ A )
=> ~ ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3394_sup_OboundedE,axiom,
! [B: filter_nat,C: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B @ C ) @ A )
=> ~ ( ( ord_le2510731241096832064er_nat @ B @ A )
=> ~ ( ord_le2510731241096832064er_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3395_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3396_sup_OboundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_rat @ B @ A )
=> ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3397_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3398_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_3399_sup_OboundedI,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ B @ A )
=> ( ( ord_le8081472938463900775at_nat @ C @ A )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3400_sup_OboundedI,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ B @ A )
=> ( ( ord_le1268244103169919719at_nat @ C @ A )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3401_sup_OboundedI,axiom,
! [B: filter_nat,A: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( ord_le2510731241096832064er_nat @ C @ A )
=> ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3402_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3403_sup_OboundedI,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3404_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3405_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_3406_sup_Oorder__iff,axiom,
( ord_le8081472938463900775at_nat
= ( ^ [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
( A3
= ( sup_su3035147773818789531at_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3407_sup_Oorder__iff,axiom,
( ord_le1268244103169919719at_nat
= ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
( A3
= ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3408_sup_Oorder__iff,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( A3
= ( sup_sup_filter_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3409_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3410_sup_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( A3
= ( sup_sup_rat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3411_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3412_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( A3
= ( sup_sup_int @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_3413_sup_Ocobounded1,axiom,
! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3414_sup_Ocobounded1,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3415_sup_Ocobounded1,axiom,
! [A: filter_nat,B: filter_nat] : ( ord_le2510731241096832064er_nat @ A @ ( sup_sup_filter_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3416_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3417_sup_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3418_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3419_sup_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_3420_sup_Ocobounded2,axiom,
! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3421_sup_Ocobounded2,axiom,
! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3422_sup_Ocobounded2,axiom,
! [B: filter_nat,A: filter_nat] : ( ord_le2510731241096832064er_nat @ B @ ( sup_sup_filter_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3423_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3424_sup_Ocobounded2,axiom,
! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3425_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3426_sup_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_3427_sup_Oabsorb__iff1,axiom,
( ord_le8081472938463900775at_nat
= ( ^ [B3: set_Pr8551490117392284871at_nat,A3: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3428_sup_Oabsorb__iff1,axiom,
( ord_le1268244103169919719at_nat
= ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3429_sup_Oabsorb__iff1,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B3: filter_nat,A3: filter_nat] :
( ( sup_sup_filter_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3430_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3431_sup_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( sup_sup_rat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3432_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3433_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( sup_sup_int @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_3434_sup_Oabsorb__iff2,axiom,
( ord_le8081472938463900775at_nat
= ( ^ [A3: set_Pr8551490117392284871at_nat,B3: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3435_sup_Oabsorb__iff2,axiom,
( ord_le1268244103169919719at_nat
= ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3436_sup_Oabsorb__iff2,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A3: filter_nat,B3: filter_nat] :
( ( sup_sup_filter_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3437_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3438_sup_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( sup_sup_rat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3439_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3440_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( sup_sup_int @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_3441_sup_OcoboundedI1,axiom,
! [C: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ C @ A )
=> ( ord_le8081472938463900775at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3442_sup_OcoboundedI1,axiom,
! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ C @ A )
=> ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3443_sup_OcoboundedI1,axiom,
! [C: filter_nat,A: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ C @ A )
=> ( ord_le2510731241096832064er_nat @ C @ ( sup_sup_filter_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3444_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3445_sup_OcoboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3446_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3447_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_3448_sup_OcoboundedI2,axiom,
! [C: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ C @ B )
=> ( ord_le8081472938463900775at_nat @ C @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3449_sup_OcoboundedI2,axiom,
! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ C @ B )
=> ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3450_sup_OcoboundedI2,axiom,
! [C: filter_nat,B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ C @ B )
=> ( ord_le2510731241096832064er_nat @ C @ ( sup_sup_filter_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3451_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3452_sup_OcoboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3453_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3454_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_3455_subset__UNIV,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ top_top_set_list_nat ) ).
% subset_UNIV
thf(fact_3456_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_3457_subset__UNIV,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_3458_subset__UNIV,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% subset_UNIV
thf(fact_3459_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_3460_inter__eq__subsetI,axiom,
! [S: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ S @ S4 )
=> ( ( ( inf_in2572325071724192079at_nat @ A2 @ S4 )
= ( inf_in2572325071724192079at_nat @ B2 @ S4 ) )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ S )
= ( inf_in2572325071724192079at_nat @ B2 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_3461_inter__eq__subsetI,axiom,
! [S: set_nat,S4: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ S @ S4 )
=> ( ( ( inf_inf_set_nat @ A2 @ S4 )
= ( inf_inf_set_nat @ B2 @ S4 ) )
=> ( ( inf_inf_set_nat @ A2 @ S )
= ( inf_inf_set_nat @ B2 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_3462_Int__Collect__mono,axiom,
! [A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( ord_le483042692224249369nt_int @ A2 @ B2 )
=> ( ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le483042692224249369nt_int @ ( inf_in8396524679539076455nt_int @ A2 @ ( collec5210948495886036740nt_int @ P ) ) @ ( inf_in8396524679539076455nt_int @ B2 @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3463_Int__Collect__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3464_Int__Collect__mono,axiom,
! [A2: set_int,B2: set_int,P: int > $o,Q2: int > $o] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B2 @ ( collect_int @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3465_Int__Collect__mono,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ B2 )
=> ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B2 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3466_Int__Collect__mono,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B2 @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3467_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_3468_Int__greatest,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ C2 @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ C2 @ B2 )
=> ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_3469_Int__greatest,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_3470_Int__absorb2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_3471_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_3472_Int__absorb1,axiom,
! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_3473_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_3474_Int__lower2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_3475_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_3476_Int__lower1,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_3477_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_3478_Int__mono,axiom,
! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ C2 )
=> ( ( ord_le3146513528884898305at_nat @ B2 @ D3 )
=> ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_3479_Int__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_3480_subset__Un__eq,axiom,
( ord_le8081472938463900775at_nat
= ( ^ [A4: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_3481_subset__Un__eq,axiom,
( ord_le1268244103169919719at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_3482_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_3483_subset__UnE,axiom,
! [C2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
=> ~ ! [A7: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A7 @ A2 )
=> ! [B7: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ B7 @ B2 )
=> ( C2
!= ( sup_su3035147773818789531at_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_3484_subset__UnE,axiom,
! [C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
=> ~ ! [A7: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A7 @ A2 )
=> ! [B7: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ B7 @ B2 )
=> ( C2
!= ( sup_su5525570899277871387at_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_3485_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A2 )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_3486_Un__absorb2,axiom,
! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ B2 @ A2 )
=> ( ( sup_su3035147773818789531at_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_3487_Un__absorb2,axiom,
! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
=> ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_3488_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_3489_Un__absorb1,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A2 @ B2 )
=> ( ( sup_su3035147773818789531at_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_3490_Un__absorb1,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
=> ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_3491_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_3492_Un__upper2,axiom,
! [B2: set_Pr8551490117392284871at_nat,A2: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_3493_Un__upper2,axiom,
! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_3494_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_3495_Un__upper1,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_3496_Un__upper1,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_3497_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_3498_Un__least,axiom,
! [A2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A2 @ C2 )
=> ( ( ord_le8081472938463900775at_nat @ B2 @ C2 )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_3499_Un__least,axiom,
! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ C2 )
=> ( ( ord_le1268244103169919719at_nat @ B2 @ C2 )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_3500_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_3501_Un__mono,axiom,
! [A2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,D3: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A2 @ C2 )
=> ( ( ord_le8081472938463900775at_nat @ B2 @ D3 )
=> ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) @ ( sup_su3035147773818789531at_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_3502_Un__mono,axiom,
! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D3: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ C2 )
=> ( ( ord_le1268244103169919719at_nat @ B2 @ D3 )
=> ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_3503_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_3504_Diff__partition,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ A2 @ B2 )
=> ( ( sup_su3035147773818789531at_nat @ A2 @ ( minus_5060654252129873198at_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_3505_Diff__partition,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
=> ( ( sup_su5525570899277871387at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_3506_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_3507_Diff__subset__conv,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) @ C2 )
= ( ord_le8081472938463900775at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_3508_Diff__subset__conv,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ C2 )
= ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_3509_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_3510_relH__subset,axiom,
! [Bs: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,As: set_nat] :
( ( relH @ Bs @ H3 @ H4 )
=> ( ( ord_less_eq_set_nat @ As @ Bs )
=> ( relH @ As @ H3 @ H4 ) ) ) ).
% relH_subset
thf(fact_3511_mult__le__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3512_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3513_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3514_mult__le__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3515_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3516_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3517_mult__le__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3518_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3519_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3520_mult__le__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3521_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3522_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3523_mult__less__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3524_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3525_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3526_mult__less__cancel__left2,axiom,
! [C: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3527_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3528_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3529_mult__less__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3530_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3531_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3532_mult__less__cancel__right2,axiom,
! [A: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3533_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3534_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3535_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ Z2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3536_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_3537_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_3538_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_3539_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_3540_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_3541_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_3542_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_3543_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_3544_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_3545_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_3546_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_3547_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_3548_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3549_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3550_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_3551_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_3552_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_3553_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_3554_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_3555_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_3556_less__eq__assn__def,axiom,
( ord_less_eq_assn
= ( ^ [A3: assn,B3: assn] :
( A3
= ( inf_inf_assn @ A3 @ B3 ) ) ) ) ).
% less_eq_assn_def
thf(fact_3557_mult__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_3558_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_3559_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_3560_not__square__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).
% not_square_less_zero
thf(fact_3561_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_3562_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_3563_mult__less__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_3564_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_3565_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_3566_mult__neg__pos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_neg_pos
thf(fact_3567_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_3568_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_3569_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_3570_mult__pos__neg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_pos_neg
thf(fact_3571_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_3572_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_3573_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_3574_mult__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_3575_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_3576_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_3577_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_3578_mult__pos__neg2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_pos_neg2
thf(fact_3579_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_3580_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_3581_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_3582_zero__less__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% zero_less_mult_iff
thf(fact_3583_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_3584_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_3585_zero__less__mult__pos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_3586_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_3587_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_3588_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_3589_zero__less__mult__pos2,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_3590_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_3591_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_3592_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_3593_mult__less__cancel__left__neg,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_3594_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_3595_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_3596_mult__less__cancel__left__pos,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_3597_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_3598_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_3599_mult__strict__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A )
=> ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_3600_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_3601_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_3602_mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_3603_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_3604_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_3605_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_3606_mult__less__cancel__left__disj,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
& ( ord_le6747313008572928689nteger @ A @ B ) )
| ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_3607_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_3608_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_3609_mult__strict__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A )
=> ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_3610_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_3611_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_3612_mult__strict__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_3613_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_3614_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_3615_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_3616_mult__less__cancel__right__disj,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
& ( ord_le6747313008572928689nteger @ A @ B ) )
| ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_3617_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_3618_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_3619_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3620_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3621_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3622_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3623_zero__less__one,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% zero_less_one
thf(fact_3624_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_3625_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_3626_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_3627_not__one__less__zero,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% not_one_less_zero
thf(fact_3628_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_3629_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_3630_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_3631_less__numeral__extra_I1_J,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% less_numeral_extra(1)
thf(fact_3632_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_3633_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_3634_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_3635_less__iff__diff__less__0,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).
% less_iff_diff_less_0
thf(fact_3636_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_3637_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_3638_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_3639_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_3640_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).
% not_numeral_less_one
thf(fact_3641_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_3642_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural ) ).
% not_numeral_less_one
thf(fact_3643_less__1__mult,axiom,
! [M: code_integer,N: code_integer] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3644_less__1__mult,axiom,
! [M: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3645_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3646_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_3647_mult__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_3648_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_3649_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_3650_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_3651_mult__mono_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_3652_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_3653_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_3654_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_3655_zero__le__square,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).
% zero_le_square
thf(fact_3656_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_3657_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_3658_split__mult__pos__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_3659_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_3660_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_3661_mult__left__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_3662_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_3663_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_3664_mult__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_3665_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_3666_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_3667_mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_3668_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_3669_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_3670_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_3671_mult__right__mono__neg,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_3672_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_3673_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_3674_mult__right__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_3675_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_3676_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_3677_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_3678_mult__le__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_3679_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_3680_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_3681_split__mult__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ).
% split_mult_neg_le
thf(fact_3682_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_3683_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_3684_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_3685_mult__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_3686_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_3687_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_3688_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_3689_mult__nonneg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonneg_nonpos
thf(fact_3690_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_3691_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_3692_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_3693_mult__nonpos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonpos_nonneg
thf(fact_3694_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_3695_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_3696_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_3697_mult__nonneg__nonpos2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_3698_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_3699_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_3700_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_3701_zero__le__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% zero_le_mult_iff
thf(fact_3702_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_3703_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_3704_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3705_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3706_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3707_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3708_zero__less__one__class_Ozero__le__one,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% zero_less_one_class.zero_le_one
thf(fact_3709_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_3710_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_3711_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_3712_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3713_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3714_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3715_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3716_not__one__le__zero,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% not_one_le_zero
thf(fact_3717_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_3718_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_3719_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_3720_le__iff__diff__le__0,axiom,
( ord_le3102999989581377725nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).
% le_iff_diff_le_0
thf(fact_3721_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_3722_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_3723_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_3724_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_3725_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_3726_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_3727_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_3728_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_3729_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_3730_less__minus__one__simps_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(4)
thf(fact_3731_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(4)
thf(fact_3732_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_3733_less__minus__one__simps_I2_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% less_minus_one_simps(2)
thf(fact_3734_less__minus__one__simps_I2_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% less_minus_one_simps(2)
thf(fact_3735_one__le__numeral,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).
% one_le_numeral
thf(fact_3736_one__le__numeral,axiom,
! [N: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) ) ).
% one_le_numeral
thf(fact_3737_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_3738_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_3739_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_3740_diff__shunt__var,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ X @ Y )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_3741_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_3742_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_3743_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_3744_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_3745_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_3746_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_3747_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_3748_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_3749_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_3750_le__minus__one__simps_I4_J,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(4)
thf(fact_3751_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_3752_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_3753_le__minus__one__simps_I2_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% le_minus_one_simps(2)
thf(fact_3754_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_3755_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_3756_disjoint__alt__simp3,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ A2 )
= ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
!= bot_bo2099793752762293965at_nat ) ) ).
% disjoint_alt_simp3
thf(fact_3757_disjoint__alt__simp3,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ A2 )
= ( ( inf_inf_set_o @ A2 @ B2 )
!= bot_bot_set_o ) ) ).
% disjoint_alt_simp3
thf(fact_3758_disjoint__alt__simp3,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 )
= ( ( inf_inf_set_int @ A2 @ B2 )
!= bot_bot_set_int ) ) ).
% disjoint_alt_simp3
thf(fact_3759_disjoint__alt__simp3,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 )
= ( ( inf_inf_set_nat @ A2 @ B2 )
!= bot_bot_set_nat ) ) ).
% disjoint_alt_simp3
thf(fact_3760_distrib__inf__le,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3761_distrib__inf__le,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) @ ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3762_distrib__inf__le,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] : ( ord_le4796328588573674190char_o @ ( sup_sup_list_char_o @ ( inf_inf_list_char_o @ X @ Y ) @ ( inf_inf_list_char_o @ X @ Z ) ) @ ( inf_inf_list_char_o @ X @ ( sup_sup_list_char_o @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3763_distrib__inf__le,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z ) ) @ ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3764_distrib__inf__le,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3765_distrib__inf__le,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ ( inf_inf_filter_nat @ X @ Y ) @ ( inf_inf_filter_nat @ X @ Z ) ) @ ( inf_inf_filter_nat @ X @ ( sup_sup_filter_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3766_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3767_distrib__inf__le,axiom,
! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X @ Y ) @ ( inf_inf_rat @ X @ Z ) ) @ ( inf_inf_rat @ X @ ( sup_sup_rat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3768_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3769_distrib__inf__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_3770_distrib__sup__le,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3771_distrib__sup__le,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) @ ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3772_distrib__sup__le,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] : ( ord_le4796328588573674190char_o @ ( sup_sup_list_char_o @ X @ ( inf_inf_list_char_o @ Y @ Z ) ) @ ( inf_inf_list_char_o @ ( sup_sup_list_char_o @ X @ Y ) @ ( sup_sup_list_char_o @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3773_distrib__sup__le,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z ) ) @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3774_distrib__sup__le,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3775_distrib__sup__le,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ( ord_le2510731241096832064er_nat @ ( sup_sup_filter_nat @ X @ ( inf_inf_filter_nat @ Y @ Z ) ) @ ( inf_inf_filter_nat @ ( sup_sup_filter_nat @ X @ Y ) @ ( sup_sup_filter_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3776_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3777_distrib__sup__le,axiom,
! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X @ Y ) @ ( sup_sup_rat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3778_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3779_distrib__sup__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_3780_disjoint__mono,axiom,
! [A: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ A5 )
=> ( ( ord_le3146513528884898305at_nat @ B @ B5 )
=> ( ( ( inf_in2572325071724192079at_nat @ A5 @ B5 )
= bot_bo2099793752762293965at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A @ B )
= bot_bo2099793752762293965at_nat ) ) ) ) ).
% disjoint_mono
thf(fact_3781_disjoint__mono,axiom,
! [A: set_o,A5: set_o,B: set_o,B5: set_o] :
( ( ord_less_eq_set_o @ A @ A5 )
=> ( ( ord_less_eq_set_o @ B @ B5 )
=> ( ( ( inf_inf_set_o @ A5 @ B5 )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ) ) ).
% disjoint_mono
thf(fact_3782_disjoint__mono,axiom,
! [A: set_int,A5: set_int,B: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A @ A5 )
=> ( ( ord_less_eq_set_int @ B @ B5 )
=> ( ( ( inf_inf_set_int @ A5 @ B5 )
= bot_bot_set_int )
=> ( ( inf_inf_set_int @ A @ B )
= bot_bot_set_int ) ) ) ) ).
% disjoint_mono
thf(fact_3783_disjoint__mono,axiom,
! [A: set_nat,A5: set_nat,B: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A @ A5 )
=> ( ( ord_less_eq_set_nat @ B @ B5 )
=> ( ( ( inf_inf_set_nat @ A5 @ B5 )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ) ).
% disjoint_mono
thf(fact_3784_Un__Int__assoc__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) )
= ( ord_le3146513528884898305at_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_3785_Un__Int__assoc__eq,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) @ C2 )
= ( inf_in1697001100524423349at_nat @ A2 @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) )
= ( ord_le8081472938463900775at_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_3786_Un__Int__assoc__eq,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) @ C2 )
= ( inf_in7913087082777306421at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) )
= ( ord_le1268244103169919719at_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_3787_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_3788_subset__Compl__self__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% subset_Compl_self_eq
thf(fact_3789_subset__Compl__self__eq,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_3790_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_3791_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_3792_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_3793_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_3794_in__range__subset,axiom,
! [As: set_nat,As3: set_nat,H3: heap_e7401611519738050253t_unit] :
( ( ord_less_eq_set_nat @ As @ As3 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% in_range_subset
thf(fact_3795_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_3796_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_3797_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_3798_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_3799_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_3800_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_3801_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_3802_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_3803_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_3804_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_3805_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_3806_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_3807_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_3808_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_3809_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_3810_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_3811_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_3812_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_3813_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_3814_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
=> ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_3815_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_3816_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_3817_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_3818_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_3819_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_3820_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_3821_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_less_one
thf(fact_3822_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_3823_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_3824_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_less_numeral
thf(fact_3825_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_3826_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_3827_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_less_neg_one
thf(fact_3828_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_3829_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_3830_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_3831_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_3832_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_3833_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_3834_mult__left__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3835_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3836_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3837_mult__right__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3838_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3839_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3840_mult__le__one,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).
% mult_le_one
thf(fact_3841_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_3842_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_3843_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_3844_mult__left__le,axiom,
! [C: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_3845_mult__left__le,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_3846_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_3847_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_3848_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_3849_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_3850_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_3851_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_3852_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_3853_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_3854_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_3855_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_3856_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_3857_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_3858_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_3859_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_3860_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3861_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3862_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3863_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_3864_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_3865_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_3866_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_3867_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_le_numeral
thf(fact_3868_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_3869_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% neg_numeral_le_neg_one
thf(fact_3870_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_3871_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_3872_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_3873_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_3874_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_3875_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_3876_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_3877_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_3878_inf__shunt,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
( ( ( inf_in3604695632404883862_int_o @ X @ Y )
= bot_bo8147686125503663512_int_o )
= ( ord_le8369615600986905444_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).
% inf_shunt
thf(fact_3879_inf__shunt,axiom,
! [X: list_char > $o,Y: list_char > $o] :
( ( ( inf_inf_list_char_o @ X @ Y )
= bot_bot_list_char_o )
= ( ord_le4796328588573674190char_o @ X @ ( uminus7154032370948614053char_o @ Y ) ) ) ).
% inf_shunt
thf(fact_3880_inf__shunt,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ X @ Y )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_3881_inf__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).
% inf_shunt
thf(fact_3882_inf__shunt,axiom,
! [X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% inf_shunt
thf(fact_3883_inf__shunt,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% inf_shunt
thf(fact_3884_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_3885_sup__shunt,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] :
( ( ( sup_su3035147773818789531at_nat @ X @ Y )
= top_to8764741339697478167at_nat )
= ( ord_le8081472938463900775at_nat @ ( uminus2330091110623919550at_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3886_sup__shunt,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( ( sup_su5525570899277871387at_nat @ X @ Y )
= top_to6833984726390702231at_nat )
= ( ord_le1268244103169919719at_nat @ ( uminus935396558254630718at_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3887_sup__shunt,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ( sup_sup_set_list_nat @ X @ Y )
= top_top_set_list_nat )
= ( ord_le6045566169113846134st_nat @ ( uminus3195874150345416415st_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3888_sup__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= top_top_set_o )
= ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3889_sup__shunt,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( sup_sup_set_rat @ X @ Y )
= top_top_set_rat )
= ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3890_sup__shunt,axiom,
! [X: set_int,Y: set_int] :
( ( ( sup_sup_set_int @ X @ Y )
= top_top_set_int )
= ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3891_sup__shunt,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( sup_sup_Product_unit @ X @ Y )
= top_top_Product_unit )
= ( ord_le3221252021190050221t_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3892_sup__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_3893_shunt1,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
= ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3894_shunt1,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
= ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3895_shunt1,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ Y ) @ Z )
= ( ord_le4796328588573674190char_o @ X @ ( sup_sup_list_char_o @ ( uminus7154032370948614053char_o @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3896_shunt1,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ Z )
= ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3897_shunt1,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ Z )
= ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3898_shunt1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
= ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3899_shunt1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_3900_shunt2,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z )
= ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3901_shunt2,axiom,
! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) @ Z )
= ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3902_shunt2,axiom,
! [X: list_char > $o,Y: list_char > $o,Z: list_char > $o] :
( ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ X @ ( uminus7154032370948614053char_o @ Y ) ) @ Z )
= ( ord_le4796328588573674190char_o @ X @ ( sup_sup_list_char_o @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3903_shunt2,axiom,
! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ ( uminus2330091110623919550at_nat @ Y ) ) @ Z )
= ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3904_shunt2,axiom,
! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ ( uminus935396558254630718at_nat @ Y ) ) @ Z )
= ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3905_shunt2,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) @ Z )
= ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3906_shunt2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z )
= ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% shunt2
thf(fact_3907_sup__neg__inf,axiom,
! [P4: set_Pr1261947904930325089at_nat,Q4: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ P4 @ ( sup_su6327502436637775413at_nat @ Q4 @ R3 ) )
= ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P4 @ ( uminus6524753893492686040at_nat @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3908_sup__neg__inf,axiom,
! [P4: product_prod_int_int > $o,Q4: product_prod_int_int > $o,R3: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ P4 @ ( sup_su8463660629351352368_int_o @ Q4 @ R3 ) )
= ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ P4 @ ( uminus7117520113953359693_int_o @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3909_sup__neg__inf,axiom,
! [P4: list_char > $o,Q4: list_char > $o,R3: list_char > $o] :
( ( ord_le4796328588573674190char_o @ P4 @ ( sup_sup_list_char_o @ Q4 @ R3 ) )
= ( ord_le4796328588573674190char_o @ ( inf_inf_list_char_o @ P4 @ ( uminus7154032370948614053char_o @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3910_sup__neg__inf,axiom,
! [P4: set_Pr8551490117392284871at_nat,Q4: set_Pr8551490117392284871at_nat,R3: set_Pr8551490117392284871at_nat] :
( ( ord_le8081472938463900775at_nat @ P4 @ ( sup_su3035147773818789531at_nat @ Q4 @ R3 ) )
= ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ P4 @ ( uminus2330091110623919550at_nat @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3911_sup__neg__inf,axiom,
! [P4: set_Pr4329608150637261639at_nat,Q4: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ P4 @ ( sup_su5525570899277871387at_nat @ Q4 @ R3 ) )
= ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P4 @ ( uminus935396558254630718at_nat @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3912_sup__neg__inf,axiom,
! [P4: product_unit,Q4: product_unit,R3: product_unit] :
( ( ord_le3221252021190050221t_unit @ P4 @ ( sup_sup_Product_unit @ Q4 @ R3 ) )
= ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ P4 @ ( uminus2952777764628376836t_unit @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3913_sup__neg__inf,axiom,
! [P4: set_nat,Q4: set_nat,R3: set_nat] :
( ( ord_less_eq_set_nat @ P4 @ ( sup_sup_set_nat @ Q4 @ R3 ) )
= ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P4 @ ( uminus5710092332889474511et_nat @ Q4 ) ) @ R3 ) ) ).
% sup_neg_inf
thf(fact_3914_disjoint__eq__subset__Compl,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_3915_disjoint__eq__subset__Compl,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_3916_disjoint__eq__subset__Compl,axiom,
! [A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_3917_disjoint__eq__subset__Compl,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_3918_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_3919_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_3920_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_3921_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_3922_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_3923_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_3924_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_3925_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_3926_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_3927_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_3928_mult__le__cancel__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
= ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3929_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3930_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3931_mult__le__cancel__iff2,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
= ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3932_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3933_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3934_mult__less__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
= ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3935_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3936_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3937_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila5207306265035627125at_nat @ sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat
@ ^ [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y3 @ X3 )
@ ^ [X3: set_Pr8551490117392284871at_nat,Y3: set_Pr8551490117392284871at_nat] : ( ord_le7642048601412989811at_nat @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3938_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila6534579987270727413at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat
@ ^ [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ X3 )
@ ^ [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le2604355607129572851at_nat @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3939_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila4918476307565957903at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y3 @ X3 )
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3940_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2554085542299052326_set_o @ sup_sup_set_o @ bot_bot_set_o
@ ^ [X3: set_o,Y3: set_o] : ( ord_less_eq_set_o @ Y3 @ X3 )
@ ^ [X3: set_o,Y3: set_o] : ( ord_less_set_o @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3941_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila6712789903965657268et_int @ sup_sup_set_int @ bot_bot_set_int
@ ^ [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ Y3 @ X3 )
@ ^ [X3: set_int,Y3: set_int] : ( ord_less_set_int @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3942_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila5926144314033625522er_nat @ sup_sup_filter_nat @ bot_bot_filter_nat
@ ^ [X3: filter_nat,Y3: filter_nat] : ( ord_le2510731241096832064er_nat @ Y3 @ X3 )
@ ^ [X3: filter_nat,Y3: filter_nat] : ( ord_less_filter_nat @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3943_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila1667268886620078168et_nat @ sup_sup_set_nat @ bot_bot_set_nat
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ X3 )
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_set_nat @ Y3 @ X3 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3944_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila4918476307565957903at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ ord_le3146513528884898305at_nat @ ord_le7866589430770878221at_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3945_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila8052717425514686422_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ ord_le8369615600986905444_int_o @ ord_le8213806771718485336_int_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3946_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila3134969884451571676char_o @ inf_inf_list_char_o @ top_top_list_char_o @ ord_le4796328588573674190char_o @ ord_less_list_char_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3947_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2613578647866687720st_nat @ inf_inf_set_list_nat @ top_top_set_list_nat @ ord_le6045566169113846134st_nat @ ord_le1190675801316882794st_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3948_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2554085542299052326_set_o @ inf_inf_set_o @ top_top_set_o @ ord_less_eq_set_o @ ord_less_set_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3949_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila7382412365081457248et_rat @ inf_inf_set_rat @ top_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3950_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila6712789903965657268et_int @ inf_inf_set_int @ top_top_set_int @ ord_less_eq_set_int @ ord_less_set_int ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3951_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila5926144314033625522er_nat @ inf_inf_filter_nat @ top_top_filter_nat @ ord_le2510731241096832064er_nat @ ord_less_filter_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3952_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3953_bot_Oordering__top__axioms,axiom,
( orderi2172309028950807442at_nat
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y3 @ X3 )
@ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y3 @ X3 )
@ bot_bo2099793752762293965at_nat ) ).
% bot.ordering_top_axioms
thf(fact_3954_bot_Oordering__top__axioms,axiom,
( ordering_top_set_o
@ ^ [X3: set_o,Y3: set_o] : ( ord_less_eq_set_o @ Y3 @ X3 )
@ ^ [X3: set_o,Y3: set_o] : ( ord_less_set_o @ Y3 @ X3 )
@ bot_bot_set_o ) ).
% bot.ordering_top_axioms
thf(fact_3955_bot_Oordering__top__axioms,axiom,
( ordering_top_set_int
@ ^ [X3: set_int,Y3: set_int] : ( ord_less_eq_set_int @ Y3 @ X3 )
@ ^ [X3: set_int,Y3: set_int] : ( ord_less_set_int @ Y3 @ X3 )
@ bot_bot_set_int ) ).
% bot.ordering_top_axioms
thf(fact_3956_bot_Oordering__top__axioms,axiom,
( orderi2807725203697798511er_nat
@ ^ [X3: filter_nat,Y3: filter_nat] : ( ord_le2510731241096832064er_nat @ Y3 @ X3 )
@ ^ [X3: filter_nat,Y3: filter_nat] : ( ord_less_filter_nat @ Y3 @ X3 )
@ bot_bot_filter_nat ) ).
% bot.ordering_top_axioms
thf(fact_3957_bot_Oordering__top__axioms,axiom,
( ordering_top_set_nat
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ X3 )
@ ^ [X3: set_nat,Y3: set_nat] : ( ord_less_set_nat @ Y3 @ X3 )
@ bot_bot_set_nat ) ).
% bot.ordering_top_axioms
thf(fact_3958_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_3959_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_3960_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_3961_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_3962_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_3963_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_3964_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_3965_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_3966_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_3967_add__neg__numeral__special_I6_J,axiom,
! [M: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_3968_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_3969_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_3970_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_3971_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_3972_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_3973_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_3974_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_3975_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_3976_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_3977_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_3978_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_3979_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_3980_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_3981_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_3982_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_3983_add_Oright__neutral,axiom,
! [A: literal] :
( ( plus_plus_literal @ A @ zero_zero_literal )
= A ) ).
% add.right_neutral
thf(fact_3984_add_Oright__neutral,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% add.right_neutral
thf(fact_3985_add_Oright__neutral,axiom,
! [A: multis2468970476368604999at_nat] :
( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
= A ) ).
% add.right_neutral
thf(fact_3986_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_3987_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_3988_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_3989_double__zero__sym,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( plus_p5714425477246183910nteger @ A @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% double_zero_sym
thf(fact_3990_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_3991_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_3992_add__cancel__left__left,axiom,
! [B: code_integer,A: code_integer] :
( ( ( plus_p5714425477246183910nteger @ B @ A )
= A )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_left
thf(fact_3993_add__cancel__left__left,axiom,
! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
( ( ( plus_p7104986032573967614at_nat @ B @ A )
= A )
= ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_left_left
thf(fact_3994_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_3995_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_3996_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_3997_add__cancel__left__right,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= A )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_right
thf(fact_3998_add__cancel__left__right,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( ( plus_p7104986032573967614at_nat @ A @ B )
= A )
= ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_left_right
thf(fact_3999_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_4000_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_4001_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_4002_add__cancel__right__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( plus_p5714425477246183910nteger @ B @ A ) )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_left
thf(fact_4003_add__cancel__right__left,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( A
= ( plus_p7104986032573967614at_nat @ B @ A ) )
= ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_right_left
thf(fact_4004_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_4005_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_4006_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_4007_add__cancel__right__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( plus_p5714425477246183910nteger @ A @ B ) )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_right
thf(fact_4008_add__cancel__right__right,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( A
= ( plus_p7104986032573967614at_nat @ A @ B ) )
= ( B = zero_z1048942125864253310at_nat ) ) ).
% add_cancel_right_right
thf(fact_4009_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_4010_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_4011_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_4012_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_4013_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_4014_add__0,axiom,
! [A: literal] :
( ( plus_plus_literal @ zero_zero_literal @ A )
= A ) ).
% add_0
thf(fact_4015_add__0,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
= A ) ).
% add_0
thf(fact_4016_add__0,axiom,
! [A: multis2468970476368604999at_nat] :
( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
= A ) ).
% add_0
thf(fact_4017_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_4018_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_4019_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_4020_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_4021_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_4022_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_4023_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_4024_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_4025_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_4026_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_4027_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_4028_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_4029_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_4030_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4031_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4032_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_4033_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4034_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4035_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_4036_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4037_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4038_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_4039_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4040_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4041_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_4042_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_4043_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_add_distrib
thf(fact_4044_minus__add__distrib,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_4045_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4046_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4047_minus__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4048_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4049_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4050_add__minus__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4051_add__le__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% add_le_same_cancel1
thf(fact_4052_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_4053_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_4054_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_4055_add__le__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% add_le_same_cancel2
thf(fact_4056_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_4057_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_4058_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_4059_le__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% le_add_same_cancel1
thf(fact_4060_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_4061_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_4062_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_4063_le__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% le_add_same_cancel2
thf(fact_4064_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_4065_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_4066_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_4067_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_4068_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_4069_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_4070_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_4071_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_4072_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_4073_add__less__same__cancel1,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% add_less_same_cancel1
thf(fact_4074_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_4075_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_4076_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_4077_add__less__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% add_less_same_cancel2
thf(fact_4078_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_4079_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_4080_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_4081_less__add__same__cancel1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% less_add_same_cancel1
thf(fact_4082_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_4083_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_4084_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_4085_less__add__same__cancel2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% less_add_same_cancel2
thf(fact_4086_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_4087_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_4088_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_4089_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_4090_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_4091_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_4092_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_4093_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_4094_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_4095_diff__add__zero,axiom,
! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
( ( minus_4286766774447292334at_nat @ A @ ( plus_p7104986032573967614at_nat @ A @ B ) )
= zero_z1048942125864253310at_nat ) ).
% diff_add_zero
thf(fact_4096_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_4097_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4098_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4099_distrib__right__numeral,axiom,
! [A: code_integer,B: code_integer,V: num] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4100_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4101_distrib__right__numeral,axiom,
! [A: code_natural,B: code_natural,V: num] :
( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ ( numera5444537566228673987atural @ V ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ V ) ) @ ( times_2397367101498566445atural @ B @ ( numera5444537566228673987atural @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_4102_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4103_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4104_distrib__left__numeral,axiom,
! [V: num,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4105_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4106_distrib__left__numeral,axiom,
! [V: num,B: code_natural,C: code_natural] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B @ C ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_4107_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_4108_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_4109_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_4110_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_4111_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_4112_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_4113_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4114_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4115_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4116_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4117_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4118_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4119_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_4120_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_4121_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6620942414471956472nteger @ N )
= one_one_Code_integer )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_4122_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_4123_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera5444537566228673987atural @ N )
= one_one_Code_natural )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_4124_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4125_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4126_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_Code_integer
= ( numera6620942414471956472nteger @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4127_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4128_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_Code_natural
= ( numera5444537566228673987atural @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_4129_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4130_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4131_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4132_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4133_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4134_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4135_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4136_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4137_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( minus_minus_rat @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4138_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4139_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4140_diff__minus__eq__add,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4141_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_4142_add__neg__numeral__special_I7_J,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(7)
thf(fact_4143_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_4144_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_4145_add__neg__numeral__special_I8_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(8)
thf(fact_4146_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_4147_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_4148_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_4149_div__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_4150_div__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
= ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_4151_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_4152_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_4153_div__mult__self3,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_4154_div__mult__self3,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
= ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_4155_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_4156_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_4157_div__mult__self2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_4158_div__mult__self2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
= ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_4159_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_4160_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_4161_div__mult__self1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
= ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_4162_div__mult__self1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
= ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_4163_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_4164_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_4165_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_4166_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_4167_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
= ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_4168_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_4169_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_4170_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_4171_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_4172_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
= ( numera5444537566228673987atural @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_4173_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4174_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4175_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_4176_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4177_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4178_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_4179_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4180_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4181_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4182_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4183_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4184_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4185_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_4186_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_4187_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_4188_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_4189_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_4190_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_4191_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_4192_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_4193_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_4194_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_4195_pred__subset__eq,axiom,
! [R: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
( ( ord_le8334417538754933252_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ R )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ S ) )
= ( ord_le483042692224249369nt_int @ R @ S ) ) ).
% pred_subset_eq
thf(fact_4196_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ R )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_4197_pred__subset__eq,axiom,
! [R: set_int,S: set_int] :
( ( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ R )
@ ^ [X3: int] : ( member_int @ X3 @ S ) )
= ( ord_less_eq_set_int @ R @ S ) ) ).
% pred_subset_eq
thf(fact_4198_pred__subset__eq,axiom,
! [R: set_Pr4532377907799695533_nat_o,S: set_Pr4532377907799695533_nat_o] :
( ( ord_le1304494283415849680at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ R )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ S ) )
= ( ord_le2965882846123202637_nat_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_4199_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_4200_less__eq__set__def,axiom,
( ord_le483042692224249369nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( ord_le8334417538754933252_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_4201_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_4202_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B4: set_int] :
( ord_less_eq_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 )
@ ^ [X3: int] : ( member_int @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_4203_less__eq__set__def,axiom,
( ord_le2965882846123202637_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ord_le1304494283415849680at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_4204_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_4205_less__set__def,axiom,
( ord_le1924305788584680229nt_int
= ( ^ [A4: set_se6260736226359567993nt_int,B4: set_se6260736226359567993nt_int] :
( ord_le2688692977766382584_int_o
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ A4 )
@ ^ [X3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_4206_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 )
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_4207_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_4208_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B4: set_int] :
( ord_less_int_o
@ ^ [X3: int] : ( member_int @ X3 @ A4 )
@ ^ [X3: int] : ( member_int @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_4209_less__set__def,axiom,
( ord_le2453136405763929_nat_o
= ( ^ [A4: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ord_le5728734702541193924at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A4 )
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_4210_less__assn__def,axiom,
( ord_less_assn
= ( ^ [A3: assn,B3: assn] :
( ( ord_less_eq_assn @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% less_assn_def
thf(fact_4211_Collect__subset,axiom,
! [A2: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ord_le483042692224249369nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_4212_Collect__subset,axiom,
! [A2: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_4213_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_4214_Collect__subset,axiom,
! [A2: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ord_le2965882846123202637_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_4215_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_4216_conj__subset__def,axiom,
! [A2: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( ord_le483042692224249369nt_int @ A2
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le483042692224249369nt_int @ A2 @ ( collec5210948495886036740nt_int @ P ) )
& ( ord_le483042692224249369nt_int @ A2 @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4217_conj__subset__def,axiom,
! [A2: set_set_nat,P: set_nat > $o,Q2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( collect_set_nat @ P ) )
& ( ord_le6893508408891458716et_nat @ A2 @ ( collect_set_nat @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4218_conj__subset__def,axiom,
! [A2: set_int,P: int > $o,Q2: int > $o] :
( ( ord_less_eq_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_less_eq_set_int @ A2 @ ( collect_int @ P ) )
& ( ord_less_eq_set_int @ A2 @ ( collect_int @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4219_conj__subset__def,axiom,
! [A2: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ord_le2965882846123202637_nat_o @ A2
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le2965882846123202637_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
& ( ord_le2965882846123202637_nat_o @ A2 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4220_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_4221_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_4222_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_4223_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_4224_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_4225_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_4226_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [F2: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F2 @ Z @ Less_eq @ Less )
=> ( ( ( F2 @ A @ B )
= Z )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_4227_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [F2: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: nat,B: nat] :
( ( semila1623282765462674594er_nat @ F2 @ Z @ Less_eq @ Less )
=> ( ( Z
= ( F2 @ A @ B ) )
= ( ( A = Z )
& ( B = Z ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_4228_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_4229_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_4230_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_4231_group__cancel_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_4232_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_4233_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_4234_group__cancel_Oadd2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B2 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_4235_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_4236_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_4237_add_Oassoc,axiom,
! [A: literal,B: literal,C: literal] :
( ( plus_plus_literal @ ( plus_plus_literal @ A @ B ) @ C )
= ( plus_plus_literal @ A @ ( plus_plus_literal @ B @ C ) ) ) ).
% add.assoc
thf(fact_4238_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_4239_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_4240_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_4241_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_4242_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_4243_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_4244_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_4245_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_4246_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_4247_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_4248_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_4249_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_4250_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_4251_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_4252_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_4253_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_4254_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_4255_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_4256_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_4257_add_Oright__assoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.right_assoc
thf(fact_4258_add_Oright__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.right_assoc
thf(fact_4259_add_Oright__assoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.right_assoc
thf(fact_4260_add_Oright__commute,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% add.right_commute
thf(fact_4261_add_Oright__commute,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ).
% add.right_commute
thf(fact_4262_add_Oright__commute,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% add.right_commute
thf(fact_4263_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_4264_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_4265_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_4266_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_4267_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_4268_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_4269_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_4270_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_4271_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_4272_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_4273_add__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_4274_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_4275_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_4276_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_4277_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_4278_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_4279_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_4280_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_4281_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_4282_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_4283_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_4284_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_4285_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_4286_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_4287_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_4288_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_4289_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_4290_comm__monoid__add__class_Oadd__0,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_4291_comm__monoid__add__class_Oadd__0,axiom,
! [A: multis2468970476368604999at_nat] :
( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_4292_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_4293_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_4294_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_4295_add_Ocomm__neutral,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% add.comm_neutral
thf(fact_4296_add_Ocomm__neutral,axiom,
! [A: multis2468970476368604999at_nat] :
( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
= A ) ).
% add.comm_neutral
thf(fact_4297_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_4298_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_4299_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_4300_add_Ogroup__left__neutral,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
= A ) ).
% add.group_left_neutral
thf(fact_4301_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_4302_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_4303_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_4304_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_4305_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_4306_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_4307_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_4308_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_4309_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_4310_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_4311_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_4312_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_4313_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_4314_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_4315_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_4316_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_4317_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_4318_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_4319_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_4320_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_4321_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_4322_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_4323_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_4324_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_4325_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_4326_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_4327_combine__common__factor,axiom,
! [A: code_integer,E2: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4328_combine__common__factor,axiom,
! [A: rat,E2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4329_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4330_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_4331_distrib__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% distrib_right
thf(fact_4332_distrib__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% distrib_right
thf(fact_4333_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_4334_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_4335_distrib__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).
% distrib_left
thf(fact_4336_distrib__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% distrib_left
thf(fact_4337_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_4338_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_4339_comm__semiring__class_Odistrib,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4340_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4341_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4342_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_4343_ring__class_Oring__distribs_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4344_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4345_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_4346_ring__class_Oring__distribs_I2_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4347_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4348_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_4349_group__cancel_Osub1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( minus_minus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4350_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4351_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( minus_minus_rat @ A @ B )
= C )
= ( A
= ( plus_plus_rat @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_4352_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_4353_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A
= ( minus_minus_rat @ C @ B ) )
= ( ( plus_plus_rat @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_4354_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_4355_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_4356_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_4357_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4358_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4359_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_4360_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_4361_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4362_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4363_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( ( plus_plus_rat @ C @ B )
= A )
=> ( C
= ( minus_minus_rat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4364_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4365_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4366_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_4367_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_4368_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_4369_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4370_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4371_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4372_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4373_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4374_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4375_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4376_group__cancel_Oneg1,axiom,
! [A2: code_integer,K: code_integer,A: code_integer] :
( ( A2
= ( plus_p5714425477246183910nteger @ K @ A ) )
=> ( ( uminus1351360451143612070nteger @ A2 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4377_group__cancel_Oneg1,axiom,
! [A2: rat,K: rat,A: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( uminus_uminus_rat @ A2 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4378_add_Osafe__commute,axiom,
! [X: rat,Y: rat,A: rat,B: rat] :
( ( syntax3730441303064801268at_rat @ ( plus_plus_rat @ X @ Y ) @ A )
=> ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ B @ A ) ) ) ).
% add.safe_commute
thf(fact_4379_add_Osafe__commute,axiom,
! [X: nat,Y: nat,A: nat,B: nat] :
( ( syntax4682126007086162916at_nat @ ( plus_plus_nat @ X @ Y ) @ A )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ B @ A ) ) ) ).
% add.safe_commute
thf(fact_4380_add_Osafe__commute,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( syntax5678989248478167196nt_int @ ( plus_plus_int @ X @ Y ) @ A )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ B @ A ) ) ) ).
% add.safe_commute
thf(fact_4381_add__nonpos__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
=> ( ( ( plus_p5714425477246183910nteger @ X @ Y )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_4382_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_4383_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_4384_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_4385_add__nonneg__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ( plus_p5714425477246183910nteger @ X @ Y )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_4386_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_4387_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_4388_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_4389_add__nonpos__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_nonpos_nonpos
thf(fact_4390_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_4391_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_4392_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_4393_add__nonneg__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_4394_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_4395_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_4396_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_4397_add__increasing2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_4398_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_4399_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_4400_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_4401_add__decreasing2,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_4402_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_4403_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_4404_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_4405_add__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ B @ C )
=> ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_4406_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_4407_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_4408_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_4409_add__decreasing,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ C @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_4410_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_4411_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_4412_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_4413_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_4414_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_4415_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_4416_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_4417_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_4418_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_4419_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_4420_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_4421_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_4422_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_4423_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_4424_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_4425_pos__add__strict,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_4426_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_4427_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_4428_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_4429_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_4430_add__pos__pos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_4431_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_4432_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_4433_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_4434_add__neg__neg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_neg_neg
thf(fact_4435_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_4436_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_4437_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_4438_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_4439_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_4440_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_4441_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_4442_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_4443_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_4444_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4445_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4446_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4447_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4448_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4449_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4450_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_4451_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4452_add__mono1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).
% add_mono1
thf(fact_4453_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_4454_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_4455_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_4456_less__add__one,axiom,
! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).
% less_add_one
thf(fact_4457_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_4458_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_4459_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_4460_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_4461_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_4462_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_4463_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_4464_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4465_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4466_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_4467_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4468_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4469_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_4470_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4471_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4472_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_4473_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4474_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4475_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_4476_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4477_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4478_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% add_eq_0_iff
thf(fact_4479_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_4480_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_4481_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).
% one_plus_numeral_commute
thf(fact_4482_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_4483_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
= ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).
% one_plus_numeral_commute
thf(fact_4484_square__diff__square__factored,axiom,
! [X: code_integer,Y: code_integer] :
( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4485_square__diff__square__factored,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4486_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4487_eq__add__iff2,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_4488_eq__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_4489_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_4490_eq__add__iff1,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_4491_eq__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_4492_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_4493_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4494_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4495_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4496_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4497_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4498_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_4499_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4500_group__cancel_Osub2,axiom,
! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B2
= ( plus_p5714425477246183910nteger @ K @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B2 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4501_group__cancel_Osub2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( minus_minus_rat @ A @ B2 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_4502_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4503_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
= one_one_Code_integer )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4504_eq__numeral__iff__iszero_I7_J,axiom,
! [X: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
= one_one_rat )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4505_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_4506_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_Code_integer
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_4507_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_one_rat
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_4508_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_4509_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_4510_mult__numeral__1__right,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_4511_mult__numeral__1__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_4512_mult__numeral__1__right,axiom,
! [A: code_natural] :
( ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_4513_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_4514_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_4515_mult__numeral__1,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_4516_mult__numeral__1,axiom,
! [A: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_4517_mult__numeral__1,axiom,
! [A: code_natural] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_4518_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_4519_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_4520_numeral__One,axiom,
( ( numera6620942414471956472nteger @ one )
= one_one_Code_integer ) ).
% numeral_One
thf(fact_4521_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_4522_numeral__One,axiom,
( ( numera5444537566228673987atural @ one )
= one_one_Code_natural ) ).
% numeral_One
thf(fact_4523_add_Ocomm__monoid__axioms,axiom,
comm_m1654649897431166824nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).
% add.comm_monoid_axioms
thf(fact_4524_add_Ocomm__monoid__axioms,axiom,
comm_m1862543879040115196at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).
% add.comm_monoid_axioms
thf(fact_4525_add_Ocomm__monoid__axioms,axiom,
comm_monoid_rat @ plus_plus_rat @ zero_zero_rat ).
% add.comm_monoid_axioms
thf(fact_4526_add_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ plus_plus_nat @ zero_zero_nat ).
% add.comm_monoid_axioms
thf(fact_4527_add_Ocomm__monoid__axioms,axiom,
comm_monoid_int @ plus_plus_int @ zero_zero_int ).
% add.comm_monoid_axioms
thf(fact_4528_add_Omonoid__axioms,axiom,
monoid_literal @ plus_plus_literal @ zero_zero_literal ).
% add.monoid_axioms
thf(fact_4529_add_Omonoid__axioms,axiom,
monoid_Code_integer @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).
% add.monoid_axioms
thf(fact_4530_add_Omonoid__axioms,axiom,
monoid66256789251010760at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).
% add.monoid_axioms
thf(fact_4531_add_Omonoid__axioms,axiom,
monoid_rat @ plus_plus_rat @ zero_zero_rat ).
% add.monoid_axioms
thf(fact_4532_add_Omonoid__axioms,axiom,
monoid_nat @ plus_plus_nat @ zero_zero_nat ).
% add.monoid_axioms
thf(fact_4533_add_Omonoid__axioms,axiom,
monoid_int @ plus_plus_int @ zero_zero_int ).
% add.monoid_axioms
thf(fact_4534_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [F2: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o] :
( ( semila1623282765462674594er_nat @ F2 @ Z @ Less_eq @ Less )
=> ( semila9081495762789891438tr_nat @ F2 @ Z ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_4535_dbl__inc__def,axiom,
( neg_nu5831290666863070958nteger
= ( ^ [X3: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X3 @ X3 ) @ one_one_Code_integer ) ) ) ).
% dbl_inc_def
thf(fact_4536_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_4537_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_4538_add__strict__increasing2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_4539_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_4540_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_4541_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_4542_add__strict__increasing,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_4543_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_4544_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_4545_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_4546_add__pos__nonneg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_4547_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_4548_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_4549_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_4550_add__nonpos__neg,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_nonpos_neg
thf(fact_4551_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_4552_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_4553_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_4554_add__nonneg__pos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_4555_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_4556_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_4557_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_4558_add__neg__nonpos,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_neg_nonpos
thf(fact_4559_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_4560_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_4561_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_4562_sum__squares__ge__zero,axiom,
! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4563_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4564_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_4565_not__sum__squares__lt__zero,axiom,
! [X: code_integer,Y: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).
% not_sum_squares_lt_zero
thf(fact_4566_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_4567_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_4568_discrete,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% discrete
thf(fact_4569_discrete,axiom,
( ord_less_nat
= ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_4570_discrete,axiom,
( ord_less_int
= ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_4571_zero__less__two,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% zero_less_two
thf(fact_4572_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_4573_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_4574_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_4575_le__add__iff1,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_4576_le__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_4577_le__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_4578_le__add__iff2,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_4579_le__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_4580_le__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_4581_less__add__iff1,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_4582_less__add__iff1,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_4583_less__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_4584_less__add__iff2,axiom,
! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_4585_less__add__iff2,axiom,
! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_4586_less__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_4587_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_4588_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_4589_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_4590_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_4591_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_4592_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_4593_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_4594_square__diff__one__factored,axiom,
! [X: code_integer] :
( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).
% square_diff_one_factored
thf(fact_4595_square__diff__one__factored,axiom,
! [X: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_4596_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_4597_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_4598_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_4599_div__add__self2,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% div_add_self2
thf(fact_4600_div__add__self2,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
= ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).
% div_add_self2
thf(fact_4601_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_4602_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_4603_div__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% div_add_self1
thf(fact_4604_div__add__self1,axiom,
! [B: code_natural,A: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
= ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).
% div_add_self1
thf(fact_4605_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_4606_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_4607_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_4608_mult__1s__ring__1_I1_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_4609_mult__1s__ring__1_I1_J,axiom,
! [B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_4610_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_4611_mult__1s__ring__1_I2_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_4612_mult__1s__ring__1_I2_J,axiom,
! [B: rat] :
( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_4613_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_4614_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_4615_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_4616_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_nat @ ( inc @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_4617_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_int @ ( inc @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_4618_numeral__inc,axiom,
! [X: num] :
( ( numera6620942414471956472nteger @ ( inc @ X ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).
% numeral_inc
thf(fact_4619_numeral__inc,axiom,
! [X: num] :
( ( numeral_numeral_rat @ ( inc @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% numeral_inc
thf(fact_4620_numeral__inc,axiom,
! [X: num] :
( ( numera5444537566228673987atural @ ( inc @ X ) )
= ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).
% numeral_inc
thf(fact_4621_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4622_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
= ( numera6620942414471956472nteger @ Y ) )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4623_eq__numeral__iff__iszero_I3_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ Y ) )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4624_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_int @ X )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4625_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numera6620942414471956472nteger @ X )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
= ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4626_eq__numeral__iff__iszero_I2_J,axiom,
! [X: num,Y: num] :
( ( ( numeral_numeral_rat @ X )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
= ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4627_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4628_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4629_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4630_distrib__left__NO__MATCH,axiom,
! [X: nat,Y: nat,A: int,B: int,C: int] :
( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4631_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4632_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: rat,B: rat,C: rat] :
( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4633_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: nat,B: nat,C: nat] :
( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4634_distrib__left__NO__MATCH,axiom,
! [X: int,Y: int,A: int,B: int,C: int] :
( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
=> ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4635_distrib__left__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer,C: code_integer] :
( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
=> ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4636_distrib__left__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
=> ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4637_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4638_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4639_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4640_distrib__right__NO__MATCH,axiom,
! [X: nat,Y: nat,C: int,A: int,B: int] :
( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
=> ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4641_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4642_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: rat,A: rat,B: rat] :
( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4643_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: nat,A: nat,B: nat] :
( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4644_distrib__right__NO__MATCH,axiom,
! [X: int,Y: int,C: int,A: int,B: int] :
( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
=> ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4645_distrib__right__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: code_integer,A: code_integer,B: code_integer] :
( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
=> ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4646_distrib__right__NO__MATCH,axiom,
! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
=> ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4647_dbl__dec__def,axiom,
( neg_nu7757733837767384882nteger
= ( ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X3 @ X3 ) @ one_one_Code_integer ) ) ) ).
% dbl_dec_def
thf(fact_4648_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_4649_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_4650_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_4651_not__iszero__neg__Numeral1,axiom,
~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_4652_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_4653_convex__bound__le,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ A )
=> ( ( ord_le3102999989581377725nteger @ Y @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
=> ( ( ( plus_p5714425477246183910nteger @ U @ V )
= one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4654_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4655_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_4656_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_4657_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_4658_convex__bound__lt,axiom,
! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ A )
=> ( ( ord_le6747313008572928689nteger @ Y @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
=> ( ( ( plus_p5714425477246183910nteger @ U @ V )
= one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4659_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A )
=> ( ( ord_less_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4660_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_4661_scaling__mono,axiom,
! [U: rat,V: rat,R3: rat,S3: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
=> ( ( ord_less_eq_rat @ R3 @ S3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_4662_top_Oordering__top__axioms,axiom,
orderi7613023277850353061st_nat @ ord_le6045566169113846134st_nat @ ord_le1190675801316882794st_nat @ top_top_set_list_nat ).
% top.ordering_top_axioms
thf(fact_4663_top_Oordering__top__axioms,axiom,
ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).
% top.ordering_top_axioms
thf(fact_4664_top_Oordering__top__axioms,axiom,
ordering_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat @ top_top_set_rat ).
% top.ordering_top_axioms
thf(fact_4665_top_Oordering__top__axioms,axiom,
ordering_top_set_int @ ord_less_eq_set_int @ ord_less_set_int @ top_top_set_int ).
% top.ordering_top_axioms
thf(fact_4666_top_Oordering__top__axioms,axiom,
orderi2807725203697798511er_nat @ ord_le2510731241096832064er_nat @ ord_less_filter_nat @ top_top_filter_nat ).
% top.ordering_top_axioms
thf(fact_4667_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_4668_sum__squares__gt__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
= ( ( X != zero_z3403309356797280102nteger )
| ( Y != zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4669_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4670_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_4671_sum__squares__le__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4672_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4673_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_4674_add__scale__eq__noteq,axiom,
! [R3: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( R3 != zero_z3403309356797280102nteger )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R3 @ C ) )
!= ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4675_add__scale__eq__noteq,axiom,
! [R3: rat,A: rat,B: rat,C: rat,D2: rat] :
( ( R3 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R3 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4676_add__scale__eq__noteq,axiom,
! [R3: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R3 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4677_add__scale__eq__noteq,axiom,
! [R3: int,A: int,B: int,C: int,D2: int] :
( ( R3 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R3 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4678_sum__squares__eq__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4679_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4680_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_4681_less__by__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( A2 = bot_bo2099793752762293965at_nat )
=> ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% less_by_empty
thf(fact_4682_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ M @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(2)
thf(fact_4683_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5055182867167087721od_nat @ M @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(2)
thf(fact_4684_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_4685_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_4686_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M @ one ) ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_4687_minus__sub__one__diff__one,axiom,
! [M: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M @ one ) ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% minus_sub_one_diff_one
thf(fact_4688_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_4689_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ V @ W ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_4690_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
= ( plus_plus_rat @ ( neg_numeral_sub_rat @ V @ W ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_4691_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_4692_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ W @ V ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_4693_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
= ( plus_plus_rat @ ( neg_numeral_sub_rat @ W @ V ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_4694_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4695_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( neg_nu5755505904847501662nteger @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4696_add__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( neg_numeral_sub_rat @ M @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4697_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4698_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4699_add__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( neg_numeral_sub_rat @ N @ M ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4700_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_4701_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( neg_nu5755505904847501662nteger @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_4702_diff__numeral__simps_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( neg_numeral_sub_rat @ N @ M ) ) ).
% diff_numeral_simps(4)
thf(fact_4703_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_4704_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ one_one_Code_integer )
= ( neg_nu5755505904847501662nteger @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_4705_diff__numeral__special_I2_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ one_one_rat )
= ( neg_numeral_sub_rat @ M @ one ) ) ).
% diff_numeral_special(2)
thf(fact_4706_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_4707_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( neg_nu5755505904847501662nteger @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_4708_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( neg_numeral_sub_rat @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_4709_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_4710_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
= ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_4711_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
= ( neg_numeral_sub_rat @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_4712_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_4713_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( neg_nu5755505904847501662nteger @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_4714_add__neg__numeral__special_I3_J,axiom,
! [M: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( neg_numeral_sub_rat @ M @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_4715_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_4716_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
= ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_4717_add__neg__numeral__special_I2_J,axiom,
! [M: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
= ( neg_numeral_sub_rat @ one @ M ) ) ).
% add_neg_numeral_special(2)
thf(fact_4718_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_4719_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) )
= ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_4720_add__neg__numeral__special_I1_J,axiom,
! [M: num] :
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) )
= ( neg_numeral_sub_rat @ one @ M ) ) ).
% add_neg_numeral_special(1)
thf(fact_4721_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_4722_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( neg_nu5755505904847501662nteger @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_4723_diff__numeral__special_I8_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( neg_numeral_sub_rat @ one @ M ) ) ).
% diff_numeral_special(8)
thf(fact_4724_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_4725_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_4726_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( neg_numeral_sub_rat @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_4727_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_4728_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_4729_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ X )
= ( X != zero_zero_nat ) ) ).
% nat_geq_1_eq_neqz
thf(fact_4730_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4731_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
= ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4732_eq__numeral__iff__iszero_I4_J,axiom,
! [X: num,Y: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
= ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ Y @ X ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4733_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_4734_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_Code_integer
= ( numera6620942414471956472nteger @ Y ) )
= ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_4735_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ Y ) )
= ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_4736_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numeral_numeral_int @ X )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4737_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numera6620942414471956472nteger @ X )
= one_one_Code_integer )
= ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4738_eq__numeral__iff__iszero_I5_J,axiom,
! [X: num] :
( ( ( numeral_numeral_rat @ X )
= one_one_rat )
= ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ X @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4739_crossproduct__eq,axiom,
! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_4740_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
= ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_4741_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_4742_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_4743_crossproduct__noteq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4744_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) )
!= ( plus_plus_rat @ ( times_times_rat @ A @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4745_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4746_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_4747_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_4748_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_4749_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_4750_subset__emptyI,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ! [X2: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( ord_le483042692224249369nt_int @ A2 @ bot_bo1488462491386950373nt_int ) ) ).
% subset_emptyI
thf(fact_4751_subset__emptyI,axiom,
! [A2: set_set_nat] :
( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_4752_subset__emptyI,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ! [X2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).
% subset_emptyI
thf(fact_4753_subset__emptyI,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_4754_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X2: $o] :
~ ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_4755_subset__emptyI,axiom,
! [A2: set_int] :
( ! [X2: int] :
~ ( member_int @ X2 @ A2 )
=> ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_4756_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_4757_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_4758_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_4759_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_4760_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4761_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4762_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_4763_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_4764_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_4765_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_4766_divmod__step__eq,axiom,
! [L: num,R3: int,Q4: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R3 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R3 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R3 ) ) ) ) ).
% divmod_step_eq
thf(fact_4767_divmod__step__eq,axiom,
! [L: num,R3: nat,Q4: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R3 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R3 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R3 ) ) ) ) ).
% divmod_step_eq
thf(fact_4768_divmod__step__eq,axiom,
! [L: num,R3: code_integer,Q4: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R3 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R3 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R3 ) ) ) ) ).
% divmod_step_eq
thf(fact_4769_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_4770_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_4771_one__add__one,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_4772_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_4773_one__add__one,axiom,
( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
= ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_4774_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_4775_dbl__simps_I3_J,axiom,
( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_4776_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_4777_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_4778_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_4779_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_4780_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_4781_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_4782_one__div__two__eq__zero,axiom,
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% one_div_two_eq_zero
thf(fact_4783_one__div__two__eq__zero,axiom,
( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) ).
% one_div_two_eq_zero
thf(fact_4784_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_4785_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_4786_bits__1__div__2,axiom,
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% bits_1_div_2
thf(fact_4787_bits__1__div__2,axiom,
( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) ).
% bits_1_div_2
thf(fact_4788_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4789_add__neg__numeral__special_I9_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4790_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4791_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_4792_diff__numeral__special_I10_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_4793_diff__numeral__special_I10_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_4794_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_4795_diff__numeral__special_I11_J,axiom,
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_4796_diff__numeral__special_I11_J,axiom,
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_4797_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_4798_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_4799_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_4800_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_4801_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_nu5755505904847501662nteger @ one @ ( bit1 @ L ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_4802_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_rat @ one @ ( bit1 @ L ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_4803_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_4804_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_4805_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_4806_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_4807_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_4808_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_4809_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_4810_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_4811_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
=> ( ! [M3: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M3: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_4812_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_4813_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_4814_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_4815_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_4816_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_4817_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_code(2)
thf(fact_4818_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_4819_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera5444537566228673987atural @ ( bit0 @ N ) )
= ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) ) ).
% numeral_code(2)
thf(fact_4820_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_4821_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_4822_numeral__Bit1,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).
% numeral_Bit1
thf(fact_4823_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_4824_numeral__Bit1,axiom,
! [N: num] :
( ( numera5444537566228673987atural @ ( bit1 @ N ) )
= ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).
% numeral_Bit1
thf(fact_4825_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_4826_ssubst__Pair__rhs,axiom,
! [R3: produc6241069584506657477e_term > option6357759511663192854e_term,S3: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S5: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4827_ssubst__Pair__rhs,axiom,
! [R3: produc3658429121746597890et_nat > $o,S3: produc3658429121746597890et_nat,R: set_Pr3286484037609594932et_nat,S5: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4828_ssubst__Pair__rhs,axiom,
! [R3: produc3658429121746597890et_nat > $o,S3: produc3925858234332021118et_nat,R: set_Pr8536935166611901872et_nat,S5: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4829_ssubst__Pair__rhs,axiom,
! [R3: produc8551481072490612790e_term > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S5: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4830_ssubst__Pair__rhs,axiom,
! [R3: int > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S5: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R3 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R3 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_4831_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_4832_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_4833_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).
% numeral_code(3)
thf(fact_4834_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_4835_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera5444537566228673987atural @ ( bit1 @ N ) )
= ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).
% numeral_code(3)
thf(fact_4836_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_4837_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_4838_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_4839_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_4840_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_4841_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_4842_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_4843_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_4844_mult__2,axiom,
! [Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2
thf(fact_4845_mult__2,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
= ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).
% mult_2
thf(fact_4846_mult__2,axiom,
! [Z: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2
thf(fact_4847_mult__2,axiom,
! [Z: code_natural] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z )
= ( plus_p4538020629002901425atural @ Z @ Z ) ) ).
% mult_2
thf(fact_4848_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4849_mult__2__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4850_mult__2__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4851_mult__2__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4852_mult__2__right,axiom,
! [Z: code_natural] :
( ( times_2397367101498566445atural @ Z @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= ( plus_p4538020629002901425atural @ Z @ Z ) ) ).
% mult_2_right
thf(fact_4853_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_4854_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_4855_left__add__twice,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_4856_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_4857_left__add__twice,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_4858_prop__restrict,axiom,
! [X: set_Pr958786334691620121nt_int,Z5: set_se6260736226359567993nt_int,X7: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( member2340774599025711042nt_int @ X @ Z5 )
=> ( ( ord_le483042692224249369nt_int @ Z5
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_4859_prop__restrict,axiom,
! [X: set_nat,Z5: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ Z5 )
=> ( ( ord_le6893508408891458716et_nat @ Z5
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_4860_prop__restrict,axiom,
! [X: int,Z5: set_int,X7: set_int,P: int > $o] :
( ( member_int @ X @ Z5 )
=> ( ( ord_less_eq_set_int @ Z5
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_4861_prop__restrict,axiom,
! [X: produc3658429121746597890et_nat > $o,Z5: set_Pr4532377907799695533_nat_o,X7: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ X @ Z5 )
=> ( ( ord_le2965882846123202637_nat_o @ Z5
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_4862_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_4863_Collect__restrict,axiom,
! [X7: set_se6260736226359567993nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ord_le483042692224249369nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_4864_Collect__restrict,axiom,
! [X7: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_4865_Collect__restrict,axiom,
! [X7: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_4866_Collect__restrict,axiom,
! [X7: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ord_le2965882846123202637_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_4867_Collect__restrict,axiom,
! [X7: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_4868_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_4869_minus__1__div__2__eq,axiom,
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% minus_1_div_2_eq
thf(fact_4870_set__bit__0,axiom,
! [A: code_integer] :
( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_4871_set__bit__0,axiom,
! [A: code_natural] :
( ( bit_se1617098188084679374atural @ zero_zero_nat @ A )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_4872_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_4873_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_4874_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_4875_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_4876_unset__bit__0,axiom,
! [A: code_integer] :
( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_4877_unset__bit__0,axiom,
! [A: code_natural] :
( ( bit_se7083795435491715335atural @ zero_zero_nat @ A )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_4878_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_4879_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q5: int,R4: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4880_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q5: nat,R4: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) @ one_one_nat ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4881_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R4: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R4 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4882_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q5: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4883_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4884_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4885_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_4886_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
= ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_4887_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
= ( divide_divide_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_4888_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_4889_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_4890_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_4891_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_4892_mod__minus__minus,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_4893_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_4894_case__prod__conv,axiom,
! [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_4895_case__prod__conv,axiom,
! [F2: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_4896_case__prod__conv,axiom,
! [F2: int > int > product_prod_int_int,A: int,B: int] :
( ( produc4245557441103728435nt_int @ F2 @ ( product_Pair_int_int @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_4897_case__prod__conv,axiom,
! [F2: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F2 @ ( product_Pair_int_int @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_4898_case__prod__conv,axiom,
! [F2: int > int > int,A: int,B: int] :
( ( produc8211389475949308722nt_int @ F2 @ ( product_Pair_int_int @ A @ B ) )
= ( F2 @ A @ B ) ) ).
% case_prod_conv
thf(fact_4899_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_4900_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_4901_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self1_is_0
thf(fact_4902_mod__mult__self1__is__0,axiom,
! [B: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
= zero_z2226904508553997617atural ) ).
% mod_mult_self1_is_0
thf(fact_4903_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_4904_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_4905_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self2_is_0
thf(fact_4906_mod__mult__self2__is__0,axiom,
! [A: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
= zero_z2226904508553997617atural ) ).
% mod_mult_self2_is_0
thf(fact_4907_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_4908_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_4909_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_4910_bits__mod__by__1,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% bits_mod_by_1
thf(fact_4911_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_4912_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_4913_mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_4914_mod__by__1,axiom,
! [A: code_natural] :
( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% mod_by_1
thf(fact_4915_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_4916_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_4917_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self4
thf(fact_4918_mod__mult__self4,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self4
thf(fact_4919_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_4920_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_4921_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self3
thf(fact_4922_mod__mult__self3,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self3
thf(fact_4923_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_4924_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_4925_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self2
thf(fact_4926_mod__mult__self2,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self2
thf(fact_4927_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_4928_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_4929_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self1
thf(fact_4930_mod__mult__self1,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% mod_mult_self1
thf(fact_4931_minus__mod__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_4932_minus__mod__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_4933_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_4934_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_4935_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_4936_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_4937_bits__one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_one_mod_two_eq_one
thf(fact_4938_bits__one__mod__two__eq__one,axiom,
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ).
% bits_one_mod_two_eq_one
thf(fact_4939_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_4940_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_4941_one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% one_mod_two_eq_one
thf(fact_4942_one__mod__two__eq__one,axiom,
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ).
% one_mod_two_eq_one
thf(fact_4943_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_4944_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_4945_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_4946_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_4947_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_4948_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
= ( numera5444537566228673987atural @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_4949_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_4950_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_4951_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
= ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_4952_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
= ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_4953_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_4954_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_4955_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_4956_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= ( numera5444537566228673987atural @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_4957_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_4958_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_4959_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_4960_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_4961_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_4962_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_4963_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_4964_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_4965_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_4966_not__mod__2__eq__0__eq__1,axiom,
! [A: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= zero_z2226904508553997617atural )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_4967_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_4968_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_4969_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_4970_not__mod__2__eq__1__eq__0,axiom,
! [A: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= one_one_Code_natural )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_4971_minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% minus_1_mod_2_eq
thf(fact_4972_minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% minus_1_mod_2_eq
thf(fact_4973_bits__minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_minus_1_mod_2_eq
thf(fact_4974_bits__minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_minus_1_mod_2_eq
thf(fact_4975_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_4976_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_4977_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_4978_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_4979_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_4980_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_4981_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q5: int,R4: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4982_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q5: nat,R4: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4983_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R4: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R4 ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4984_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_4985_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_4986_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_4987_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_4988_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_4989_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
= ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_4990_zdiv__zminus1__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_4991_zdiv__zminus2__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_4992_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_4993_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_4994_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_4995_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_4996_verit__le__mono__div__int,axiom,
! [A2: int,B2: int,N: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B2 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_4997_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_4998_divmod__int__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_4999_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_5000_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_5001_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_5002_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_5003_prod_Ocase__distrib,axiom,
! [H3: $o > $o,F2: int > int > $o,Prod: product_prod_int_int] :
( ( H3 @ ( produc4947309494688390418_int_o @ F2 @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5004_prod_Ocase__distrib,axiom,
! [H3: $o > int,F2: int > int > $o,Prod: product_prod_int_int] :
( ( H3 @ ( produc4947309494688390418_int_o @ F2 @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5005_prod_Ocase__distrib,axiom,
! [H3: int > $o,F2: int > int > int,Prod: product_prod_int_int] :
( ( H3 @ ( produc8211389475949308722nt_int @ F2 @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5006_prod_Ocase__distrib,axiom,
! [H3: int > int,F2: int > int > int,Prod: product_prod_int_int] :
( ( H3 @ ( produc8211389475949308722nt_int @ F2 @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5007_prod_Ocase__distrib,axiom,
! [H3: product_prod_int_int > $o,F2: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H3 @ ( produc4245557441103728435nt_int @ F2 @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5008_prod_Ocase__distrib,axiom,
! [H3: product_prod_int_int > int,F2: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H3 @ ( produc4245557441103728435nt_int @ F2 @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5009_prod_Ocase__distrib,axiom,
! [H3: $o > product_prod_int_int,F2: int > int > $o,Prod: product_prod_int_int] :
( ( H3 @ ( produc4947309494688390418_int_o @ F2 @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5010_prod_Ocase__distrib,axiom,
! [H3: int > product_prod_int_int,F2: int > int > int,Prod: product_prod_int_int] :
( ( H3 @ ( produc8211389475949308722nt_int @ F2 @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5011_prod_Ocase__distrib,axiom,
! [H3: product_prod_int_int > product_prod_int_int,F2: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H3 @ ( produc4245557441103728435nt_int @ F2 @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5012_prod_Ocase__distrib,axiom,
! [H3: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F2: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H3 @ ( produc8739625826339149834_nat_o @ F2 @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X12: nat,X23: nat] : ( H3 @ ( F2 @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5013_case__prod__app,axiom,
( produc27273713700761075at_nat
= ( ^ [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( produc2626176000494625587at_nat
@ ^ [L2: nat,R4: nat] : ( F @ L2 @ R4 @ Y3 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_5014_case__prod__app,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( produc6081775807080527818_nat_o
@ ^ [L2: nat,R4: nat] : ( F @ L2 @ R4 @ Y3 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_5015_nested__case__prod__simp,axiom,
( produc27273713700761075at_nat
= ( ^ [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( produc2626176000494625587at_nat
@ ^ [A3: nat,B3: nat] : ( F @ A3 @ B3 @ Y3 )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_5016_nested__case__prod__simp,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( produc6081775807080527818_nat_o
@ ^ [A3: nat,B3: nat] : ( F @ A3 @ B3 @ Y3 )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_5017_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_5018_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_5019_mod__mult__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_5020_mod__mult__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
= ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_5021_mod__mult__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B5 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5022_mod__mult__cong,axiom,
! [A: int,C: int,A5: int,B: int,B5: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B5 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5023_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B5 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5024_mod__mult__cong,axiom,
! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B5: code_natural] :
( ( ( modulo8411746178871703098atural @ A @ C )
= ( modulo8411746178871703098atural @ A5 @ C ) )
=> ( ( ( modulo8411746178871703098atural @ B @ C )
= ( modulo8411746178871703098atural @ B5 @ C ) )
=> ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
= ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A5 @ B5 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_5025_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5026_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5027_mod__mult__mult2,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5028_mod__mult__mult2,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
= ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_5029_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_5030_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_5031_mult__mod__right,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_5032_mult__mod__right,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
= ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_5033_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5034_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5035_mod__mult__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5036_mod__mult__left__eq,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
= ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_5037_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5038_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5039_mod__mult__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5040_mod__mult__right__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
= ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_5041_mod__minus__eq,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5042_mod__minus__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5043_mod__minus__cong,axiom,
! [A: int,B: int,A5: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A5 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5044_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= ( modulo364778990260209775nteger @ A5 @ B ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5045_mod__minus__right,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5046_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5047_split__cong,axiom,
! [Q4: product_prod_nat_nat,F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
( ! [X2: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( produc27273713700761075at_nat @ F2 @ P4 )
= ( produc27273713700761075at_nat @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_5048_split__cong,axiom,
! [Q4: product_prod_nat_nat,F2: nat > nat > product_prod_nat_nat > $o,G2: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [X2: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( produc8739625826339149834_nat_o @ F2 @ P4 )
= ( produc8739625826339149834_nat_o @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_5049_split__cong,axiom,
! [Q4: product_prod_int_int,F2: int > int > product_prod_int_int,G2: int > int > product_prod_int_int,P4: product_prod_int_int] :
( ! [X2: int,Y2: int] :
( ( ( product_Pair_int_int @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( produc4245557441103728435nt_int @ F2 @ P4 )
= ( produc4245557441103728435nt_int @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_5050_split__cong,axiom,
! [Q4: product_prod_int_int,F2: int > int > $o,G2: int > int > $o,P4: product_prod_int_int] :
( ! [X2: int,Y2: int] :
( ( ( product_Pair_int_int @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( produc4947309494688390418_int_o @ F2 @ P4 )
= ( produc4947309494688390418_int_o @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_5051_split__cong,axiom,
! [Q4: product_prod_int_int,F2: int > int > int,G2: int > int > int,P4: product_prod_int_int] :
( ! [X2: int,Y2: int] :
( ( ( product_Pair_int_int @ X2 @ Y2 )
= Q4 )
=> ( ( F2 @ X2 @ Y2 )
= ( G2 @ X2 @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( produc8211389475949308722nt_int @ F2 @ P4 )
= ( produc8211389475949308722nt_int @ G2 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_5052_old_Oprod_Ocase,axiom,
! [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc27273713700761075at_nat @ F2 @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5053_old_Oprod_Ocase,axiom,
! [F2: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
( ( produc8739625826339149834_nat_o @ F2 @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5054_old_Oprod_Ocase,axiom,
! [F2: int > int > product_prod_int_int,X1: int,X22: int] :
( ( produc4245557441103728435nt_int @ F2 @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5055_old_Oprod_Ocase,axiom,
! [F2: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F2 @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5056_old_Oprod_Ocase,axiom,
! [F2: int > int > int,X1: int,X22: int] :
( ( produc8211389475949308722nt_int @ F2 @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F2 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_5057_case__prod__Pair__iden,axiom,
! [P4: produc1908205239877642774nteger] :
( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5058_case__prod__Pair__iden,axiom,
! [P4: produc3925858234332021118et_nat] :
( ( produc4058941399401191971et_nat @ produc5001842942810119800et_nat @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5059_case__prod__Pair__iden,axiom,
! [P4: produc2732055786443039994et_nat] :
( ( produc2377985495875741467et_nat @ produc2245416461498447860et_nat @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5060_case__prod__Pair__iden,axiom,
! [P4: produc2285326912895808259nt_int] :
( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5061_case__prod__Pair__iden,axiom,
! [P4: produc7773217078559923341nt_int] :
( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5062_case__prod__Pair__iden,axiom,
! [P4: product_prod_int_int] :
( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_5063_case__prodE2,axiom,
! [Q2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q2 @ ( produc27273713700761075at_nat @ P @ Z ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( Z
= ( product_Pair_nat_nat @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_5064_case__prodE2,axiom,
! [Q2: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
( ( Q2 @ ( produc8739625826339149834_nat_o @ P @ Z ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( Z
= ( product_Pair_nat_nat @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_5065_case__prodE2,axiom,
! [Q2: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
( ( Q2 @ ( produc4245557441103728435nt_int @ P @ Z ) )
=> ~ ! [X2: int,Y2: int] :
( ( Z
= ( product_Pair_int_int @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_5066_case__prodE2,axiom,
! [Q2: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
( ( Q2 @ ( produc4947309494688390418_int_o @ P @ Z ) )
=> ~ ! [X2: int,Y2: int] :
( ( Z
= ( product_Pair_int_int @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_5067_case__prodE2,axiom,
! [Q2: int > $o,P: int > int > int,Z: product_prod_int_int] :
( ( Q2 @ ( produc8211389475949308722nt_int @ P @ Z ) )
=> ~ ! [X2: int,Y2: int] :
( ( Z
= ( product_Pair_int_int @ X2 @ Y2 ) )
=> ~ ( Q2 @ ( P @ X2 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_5068_case__prod__eta,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] : ( F2 @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5069_case__prod__eta,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] : ( F2 @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5070_case__prod__eta,axiom,
! [F2: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [X3: int,Y3: int] : ( F2 @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5071_case__prod__eta,axiom,
! [F2: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X3: int,Y3: int] : ( F2 @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5072_case__prod__eta,axiom,
! [F2: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int
@ ^ [X3: int,Y3: int] : ( F2 @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5073_cond__case__prod__eta,axiom,
! [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X2: nat,Y2: nat] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
=> ( ( produc27273713700761075at_nat @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5074_cond__case__prod__eta,axiom,
! [F2: nat > nat > product_prod_nat_nat > $o,G2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X2: nat,Y2: nat] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5075_cond__case__prod__eta,axiom,
! [F2: int > int > product_prod_int_int,G2: product_prod_int_int > product_prod_int_int] :
( ! [X2: int,Y2: int] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
=> ( ( produc4245557441103728435nt_int @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5076_cond__case__prod__eta,axiom,
! [F2: int > int > $o,G2: product_prod_int_int > $o] :
( ! [X2: int,Y2: int] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
=> ( ( produc4947309494688390418_int_o @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5077_cond__case__prod__eta,axiom,
! [F2: int > int > int,G2: product_prod_int_int > int] :
( ! [X2: int,Y2: int] :
( ( F2 @ X2 @ Y2 )
= ( G2 @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
=> ( ( produc8211389475949308722nt_int @ F2 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_5078_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D ) ) ) ) ).
% mod_eqE
thf(fact_5079_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
=> ~ ! [D: code_integer] :
( B
!= ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% mod_eqE
thf(fact_5080_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_5081_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
thf(fact_5082_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5083_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q5: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5084_uncurry__def,axiom,
uncurr8011562610307062878at_nat = produc27273713700761075at_nat ).
% uncurry_def
thf(fact_5085_uncurry__def,axiom,
uncurr7511940902602773877_nat_o = produc8739625826339149834_nat_o ).
% uncurry_def
thf(fact_5086_uncurry__def,axiom,
uncurr7650761721940715016nt_int = produc4245557441103728435nt_int ).
% uncurry_def
thf(fact_5087_uncurry__def,axiom,
uncurry_int_int_o = produc4947309494688390418_int_o ).
% uncurry_def
thf(fact_5088_uncurry__def,axiom,
uncurry_int_int_int = produc8211389475949308722nt_int ).
% uncurry_def
thf(fact_5089_xor_Ocomm__monoid__axioms,axiom,
comm_m1654649897431166824nteger @ bit_se3222712562003087583nteger @ zero_z3403309356797280102nteger ).
% xor.comm_monoid_axioms
thf(fact_5090_xor_Ocomm__monoid__axioms,axiom,
comm_monoid_int @ bit_se6526347334894502574or_int @ zero_zero_int ).
% xor.comm_monoid_axioms
thf(fact_5091_xor_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ bit_se6528837805403552850or_nat @ zero_zero_nat ).
% xor.comm_monoid_axioms
thf(fact_5092_xor_Omonoid__axioms,axiom,
monoid_Code_integer @ bit_se3222712562003087583nteger @ zero_z3403309356797280102nteger ).
% xor.monoid_axioms
thf(fact_5093_xor_Omonoid__axioms,axiom,
monoid_int @ bit_se6526347334894502574or_int @ zero_zero_int ).
% xor.monoid_axioms
thf(fact_5094_xor_Omonoid__axioms,axiom,
monoid_nat @ bit_se6528837805403552850or_nat @ zero_zero_nat ).
% xor.monoid_axioms
thf(fact_5095_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_5096_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_5097_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_5098_mult__div__mod__eq,axiom,
! [B: code_natural,A: code_natural] :
( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_5099_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_5100_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_5101_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_5102_mod__mult__div__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_5103_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_5104_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_5105_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_5106_mod__div__mult__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_5107_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_5108_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_5109_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_5110_div__mult__mod__eq,axiom,
! [A: code_natural,B: code_natural] :
( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_5111_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_5112_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_5113_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] :
( A
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_5114_mod__div__decomp,axiom,
! [A: code_natural,B: code_natural] :
( A
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_5115_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5116_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5117_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5118_cancel__div__mod__rules_I1_J,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
= ( plus_p4538020629002901425atural @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_5119_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5120_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5121_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5122_cancel__div__mod__rules_I2_J,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
= ( plus_p4538020629002901425atural @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_5123_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5124_div__mult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5125_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5126_div__mult1__eq,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_5127_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_5128_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_5129_minus__mult__div__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_5130_minus__mult__div__eq__mod,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_5131_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5132_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5133_minus__mod__eq__mult__div,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5134_minus__mod__eq__mult__div,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
= ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_5135_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_5136_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_5137_minus__mod__eq__div__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_5138_minus__mod__eq__div__mult,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
= ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_5139_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_5140_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_5141_minus__div__mult__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_5142_minus__div__mult__eq__mod,axiom,
! [A: code_natural,B: code_natural] :
( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
= ( modulo8411746178871703098atural @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_5143_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_5144_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_5145_divmod__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_5146_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5147_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5148_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5149_verit__le__mono__div,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_5150_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5151_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5152_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5153_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_5154_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5155_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5156_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_5157_mod__double__modulus,axiom,
! [M: code_integer,X: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( modulo364778990260209775nteger @ X @ M ) )
| ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_5158_mod__double__modulus,axiom,
! [M: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_nat @ X @ M ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_5159_mod__double__modulus,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_int @ X @ M ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_5160_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5161_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5162_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_5163_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_5164_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_5165_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_5166_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_5167_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_5168_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_5169_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_5170_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_5171_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_5172_case__prodI2,axiom,
! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5173_case__prodI2,axiom,
! [P4: produc3925858234332021118et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc1437786849005270451_nat_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5174_case__prodI2,axiom,
! [P4: produc2732055786443039994et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( ( P4
= ( produc2245416461498447860et_nat @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc838355143741117751_nat_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5175_case__prodI2,axiom,
! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5176_case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5177_case__prodI2,axiom,
! [P4: product_prod_int_int,C: int > int > $o] :
( ! [A6: int,B6: int] :
( ( P4
= ( product_Pair_int_int @ A6 @ B6 ) )
=> ( C @ A6 @ B6 ) )
=> ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% case_prodI2
thf(fact_5178_case__prodI,axiom,
! [F2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F2 @ A @ B )
=> ( produc6253627499356882019eger_o @ F2 @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_5179_case__prodI,axiom,
! [F2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( F2 @ A @ B )
=> ( produc1437786849005270451_nat_o @ F2 @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5180_case__prodI,axiom,
! [F2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
( ( F2 @ A @ B )
=> ( produc838355143741117751_nat_o @ F2 @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_5181_case__prodI,axiom,
! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F2 @ A @ B )
=> ( produc1573362020775583542_int_o @ F2 @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5182_case__prodI,axiom,
! [F2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F2 @ A @ B )
=> ( produc2558449545302689196_int_o @ F2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5183_case__prodI,axiom,
! [F2: int > int > $o,A: int,B: int] :
( ( F2 @ A @ B )
=> ( produc4947309494688390418_int_o @ F2 @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5184_mem__case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( member_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5185_mem__case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( member_int @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5186_mem__case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( member_set_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5187_mem__case__prodI2,axiom,
! [P4: produc3925858234332021118et_nat,Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ A6 @ B6 ) )
=> ( member_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5188_mem__case__prodI2,axiom,
! [P4: produc3925858234332021118et_nat,Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ A6 @ B6 ) )
=> ( member_int @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5189_mem__case__prodI2,axiom,
! [P4: produc3925858234332021118et_nat,Z: set_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_set_nat] :
( ! [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ A6 @ B6 ) )
=> ( member_set_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_set_nat @ Z @ ( produc2333231157791014625et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5190_mem__case__prodI2,axiom,
! [P4: produc7773217078559923341nt_int,Z: set_Pr958786334691620121nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_se6260736226359567993nt_int] :
( ! [A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ A6 @ B6 ) )
=> ( member2340774599025711042nt_int @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member2340774599025711042nt_int @ Z @ ( produc575502675617541261nt_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5191_mem__case__prodI2,axiom,
! [P4: produc1908205239877642774nteger,Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
( ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ A6 @ B6 ) )
=> ( member_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5192_mem__case__prodI2,axiom,
! [P4: produc1908205239877642774nteger,Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
( ! [A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ A6 @ B6 ) )
=> ( member_int @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5193_mem__case__prodI2,axiom,
! [P4: produc2285326912895808259nt_int,Z: nat,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
( ! [A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ A6 @ B6 ) )
=> ( member_nat @ Z @ ( C @ A6 @ B6 ) ) )
=> ( member_nat @ Z @ ( produc3619988815162578792et_nat @ C @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_5194_mem__case__prodI,axiom,
! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5195_mem__case__prodI,axiom,
! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5196_mem__case__prodI,axiom,
! [Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_set_nat @ Z @ ( C @ A @ B ) )
=> ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5197_mem__case__prodI,axiom,
! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5198_mem__case__prodI,axiom,
! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5199_mem__case__prodI,axiom,
! [Z: set_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_set_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( member_set_nat @ Z @ ( C @ A @ B ) )
=> ( member_set_nat @ Z @ ( produc2333231157791014625et_nat @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5200_mem__case__prodI,axiom,
! [Z: set_Pr958786334691620121nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_se6260736226359567993nt_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member2340774599025711042nt_int @ Z @ ( C @ A @ B ) )
=> ( member2340774599025711042nt_int @ Z @ ( produc575502675617541261nt_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5201_mem__case__prodI,axiom,
! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5202_mem__case__prodI,axiom,
! [Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5203_mem__case__prodI,axiom,
! [Z: nat,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc3619988815162578792et_nat @ C @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5204_case__prodI2_H,axiom,
! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [A6: nat,B6: nat] :
( ( ( product_Pair_nat_nat @ A6 @ B6 )
= P4 )
=> ( C @ A6 @ B6 @ X ) )
=> ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_5205_Collect__const__case__prod,axiom,
! [P: $o] :
( ( P
=> ( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A3: nat,B3: nat] : P ) )
= top_to4669805908274784177at_nat ) )
& ( ~ P
=> ( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A3: nat,B3: nat] : P ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_const_case_prod
thf(fact_5206_Collect__const__case__prod,axiom,
! [P: $o] :
( ( P
=> ( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A3: int,B3: int] : P ) )
= top_to4366644338036079209nt_int ) )
& ( ~ P
=> ( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A3: int,B3: int] : P ) )
= bot_bo1796632182523588997nt_int ) ) ) ).
% Collect_const_case_prod
thf(fact_5207_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5208_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5209_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri6519982836138164636nteger @ ( numeral_numeral_nat @ K ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% signed_take_bit_numeral_of_1
thf(fact_5210_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_numeral_of_1
thf(fact_5211_signed__take__bit__0,axiom,
! [A: code_integer] :
( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5212_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5213_mem__case__prodE,axiom,
! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
=> ~ ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
=> ~ ( member_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5214_mem__case__prodE,axiom,
! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,P4: produc7773217078559923341nt_int] :
( ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) )
=> ~ ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5215_mem__case__prodE,axiom,
! [Z: set_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_set_nat,P4: produc7773217078559923341nt_int] :
( ( member_set_nat @ Z @ ( produc3577156405726439208et_nat @ C @ P4 ) )
=> ~ ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
=> ~ ( member_set_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5216_mem__case__prodE,axiom,
! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,P4: produc3925858234332021118et_nat] :
( ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) )
=> ~ ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
=> ~ ( member_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5217_mem__case__prodE,axiom,
! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,P4: produc3925858234332021118et_nat] :
( ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) )
=> ~ ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5218_mem__case__prodE,axiom,
! [Z: set_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_set_nat,P4: produc3925858234332021118et_nat] :
( ( member_set_nat @ Z @ ( produc2333231157791014625et_nat @ C @ P4 ) )
=> ~ ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
=> ~ ( member_set_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5219_mem__case__prodE,axiom,
! [Z: set_Pr958786334691620121nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_se6260736226359567993nt_int,P4: produc7773217078559923341nt_int] :
( ( member2340774599025711042nt_int @ Z @ ( produc575502675617541261nt_int @ C @ P4 ) )
=> ~ ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
=> ~ ( member2340774599025711042nt_int @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5220_mem__case__prodE,axiom,
! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc1908205239877642774nteger] :
( ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) )
=> ~ ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ X2 @ Y2 ) )
=> ~ ( member_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5221_mem__case__prodE,axiom,
! [Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc1908205239877642774nteger] :
( ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ P4 ) )
=> ~ ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ X2 @ Y2 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5222_mem__case__prodE,axiom,
! [Z: nat,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc2285326912895808259nt_int] :
( ( member_nat @ Z @ ( produc3619988815162578792et_nat @ C @ P4 ) )
=> ~ ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ X2 @ Y2 ) )
=> ~ ( member_nat @ Z @ ( C @ X2 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_5223_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_5224_case__prodE,axiom,
! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
( ( produc6253627499356882019eger_o @ C @ P4 )
=> ~ ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( P4
= ( produc8603105652947943368nteger @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5225_case__prodE,axiom,
! [C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P4: produc3925858234332021118et_nat] :
( ( produc1437786849005270451_nat_o @ C @ P4 )
=> ~ ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( P4
= ( produc5001842942810119800et_nat @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5226_case__prodE,axiom,
! [C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,P4: produc2732055786443039994et_nat] :
( ( produc838355143741117751_nat_o @ C @ P4 )
=> ~ ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( ( P4
= ( produc2245416461498447860et_nat @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5227_case__prodE,axiom,
! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
( ( produc1573362020775583542_int_o @ C @ P4 )
=> ~ ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc5700946648718959541nt_int @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5228_case__prodE,axiom,
! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
( ( produc2558449545302689196_int_o @ C @ P4 )
=> ~ ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( P4
= ( produc4305682042979456191nt_int @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5229_case__prodE,axiom,
! [C: int > int > $o,P4: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P4 )
=> ~ ! [X2: int,Y2: int] :
( ( P4
= ( product_Pair_int_int @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 ) ) ) ).
% case_prodE
thf(fact_5230_case__prodD,axiom,
! [F2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc6253627499356882019eger_o @ F2 @ ( produc8603105652947943368nteger @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5231_case__prodD,axiom,
! [F2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( produc1437786849005270451_nat_o @ F2 @ ( produc5001842942810119800et_nat @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5232_case__prodD,axiom,
! [F2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
( ( produc838355143741117751_nat_o @ F2 @ ( produc2245416461498447860et_nat @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5233_case__prodD,axiom,
! [F2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc1573362020775583542_int_o @ F2 @ ( produc5700946648718959541nt_int @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5234_case__prodD,axiom,
! [F2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc2558449545302689196_int_o @ F2 @ ( produc4305682042979456191nt_int @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5235_case__prodD,axiom,
! [F2: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F2 @ ( product_Pair_int_int @ A @ B ) )
=> ( F2 @ A @ B ) ) ).
% case_prodD
thf(fact_5236_case__prodE_H,axiom,
! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
=> ~ ! [X2: nat,Y2: nat] :
( ( P4
= ( product_Pair_nat_nat @ X2 @ Y2 ) )
=> ~ ( C @ X2 @ Y2 @ Z ) ) ) ).
% case_prodE'
thf(fact_5237_case__prodD_H,axiom,
! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
=> ( R @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_5238_Id__on__def_H,axiom,
! [A2: set_Pr958786334691620121nt_int > $o] :
( ( id_on_6410342593070439734nt_int @ ( collec5210948495886036740nt_int @ A2 ) )
= ( collec2889756465124990930nt_int
@ ( produc1547782455048856492_int_o
@ ^ [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( X3 = Y3 )
& ( A2 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_5239_Id__on__def_H,axiom,
! [A2: set_nat > $o] :
( ( id_on_set_nat @ ( collect_set_nat @ A2 ) )
= ( collec6662362479098859352et_nat
@ ( produc6247414631856714078_nat_o
@ ^ [X3: set_nat,Y3: set_nat] :
( ( X3 = Y3 )
& ( A2 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_5240_Id__on__def_H,axiom,
! [A2: nat > $o] :
( ( id_on_nat @ ( collect_nat @ A2 ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( X3 = Y3 )
& ( A2 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_5241_Id__on__def_H,axiom,
! [A2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( id_on_7438695062414216554_nat_o @ ( collec939566748876313656_nat_o @ A2 ) )
= ( collec7086185654880424402_nat_o
@ ( produc5150382289251079596at_o_o
@ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o] :
( ( X3 = Y3 )
& ( A2 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_5242_Id__on__def_H,axiom,
! [A2: int > $o] :
( ( id_on_int @ ( collect_int @ A2 ) )
= ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X3: int,Y3: int] :
( ( X3 = Y3 )
& ( A2 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_5243_rel__restrict__def,axiom,
( rel_re4349003428253852667nt_int
= ( ^ [R2: set_Pr8057934254988874055nt_int,A4: set_se6260736226359567993nt_int] :
( collec2889756465124990930nt_int
@ ( produc1547782455048856492_int_o
@ ^ [V2: set_Pr958786334691620121nt_int,W3: set_Pr958786334691620121nt_int] :
( ( member5325734588016868240nt_int @ ( produc782388162306416855nt_int @ V2 @ W3 ) @ R2 )
& ~ ( member2340774599025711042nt_int @ V2 @ A4 )
& ~ ( member2340774599025711042nt_int @ W3 @ A4 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5244_rel__restrict__def,axiom,
( rel_restrict_set_nat
= ( ^ [R2: set_Pr5488025237498180813et_nat,A4: set_set_nat] :
( collec6662362479098859352et_nat
@ ( produc6247414631856714078_nat_o
@ ^ [V2: set_nat,W3: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ V2 @ W3 ) @ R2 )
& ~ ( member_set_nat @ V2 @ A4 )
& ~ ( member_set_nat @ W3 @ A4 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5245_rel__restrict__def,axiom,
( rel_restrict_nat
= ( ^ [R2: set_Pr1261947904930325089at_nat,A4: set_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [V2: nat,W3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V2 @ W3 ) @ R2 )
& ~ ( member_nat @ V2 @ A4 )
& ~ ( member_nat @ W3 @ A4 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5246_rel__restrict__def,axiom,
( rel_re3331361160397388719_nat_o
= ( ^ [R2: set_Pr2161125870931222855_nat_o,A4: set_Pr4532377907799695533_nat_o] :
( collec7086185654880424402_nat_o
@ ( produc5150382289251079596at_o_o
@ ^ [V2: produc3658429121746597890et_nat > $o,W3: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V2 @ W3 ) @ R2 )
& ~ ( member6576561426505652726_nat_o @ V2 @ A4 )
& ~ ( member6576561426505652726_nat_o @ W3 @ A4 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5247_rel__restrict__def,axiom,
( rel_restrict_int
= ( ^ [R2: set_Pr958786334691620121nt_int,A4: set_int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [V2: int,W3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V2 @ W3 ) @ R2 )
& ~ ( member_int @ V2 @ A4 )
& ~ ( member_int @ W3 @ A4 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5248_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_5249_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_5250_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_5251_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_5252_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_5253_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_5254_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_5255_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_5256_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_5257_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_5258_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_5259_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_5260_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_5261_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_5262_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_5263_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_5264_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R3: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5265_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_5266_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q4: int,R3: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5267_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_5268_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_5269_split__part,axiom,
! [P: $o,Q2: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A3: int,B3: int] :
( P
& ( Q2 @ A3 @ B3 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P
& ( produc4947309494688390418_int_o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_5270_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_5271_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% signed_take_bit_Suc_1
thf(fact_5272_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_5273_Suc__diff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( suc @ ( minus_minus_nat @ N @ M ) )
= ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).
% Suc_diff
thf(fact_5274_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_5275_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_5276_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_5277_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_5278_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_nu5755505904847501662nteger @ one @ ( bit0 @ L ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_5279_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_rat @ one @ ( bit0 @ L ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_5280_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_5281_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_5282_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_5283_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_5284_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] :
( produc4947309494688390418_int_o
@ ^ [Uu: int,Uv: int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5285_unique__remainder,axiom,
! [A: int,B: int,Q4: int,R3: int,Q6: int,R5: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R5 ) )
=> ( R3 = R5 ) ) ) ).
% unique_remainder
thf(fact_5286_unique__quotient,axiom,
! [A: int,B: int,Q4: int,R3: int,Q6: int,R5: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R5 ) )
=> ( Q4 = Q6 ) ) ) ).
% unique_quotient
thf(fact_5287_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_5288_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_5289_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_5290_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_5291_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_5292_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% eucl_rel_int_by0
thf(fact_5293_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_5294_div__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( ( divide_divide_int @ K @ L )
= Q4 ) ) ).
% div_int_unique
thf(fact_5295_mod__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( ( modulo_modulo_int @ K @ L )
= R3 ) ) ).
% mod_int_unique
thf(fact_5296_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K4: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K4 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_5297_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D: int,I2: int] :
( X
!= ( product_Pair_int_int @ D @ I2 ) ) ).
% small_lazy'.cases
thf(fact_5298_exhaustive__int_H_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F3: int > option6357759511663192854e_term,D: int,I2: int] :
( X
!= ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_5299_full__exhaustive__int_H_Ocases,axiom,
! [X: produc2285326912895808259nt_int] :
~ ! [F3: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I2: int] :
( X
!= ( produc5700946648718959541nt_int @ F3 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_5300_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_5301_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_5302_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_5303_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ one_one_nat @ M )
=> ( ( suc @ ( minus_minus_nat @ N @ M ) )
= ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_5304_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L != zero_zero_int )
=> ( ( K
= ( times_times_int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_5305_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_5306_numeral__BitM,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ ( bitM @ N ) )
= ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ one_one_Code_integer ) ) ).
% numeral_BitM
thf(fact_5307_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bitM @ N ) )
= ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% numeral_BitM
thf(fact_5308_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_5309_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_5310_zminus1__lemma,axiom,
! [A: int,B: int,Q4: int,R3: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
=> ( ( B != zero_zero_int )
=> ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5311_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_5312_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_5313_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5314_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5315_unset__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5316_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_5317_set__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5318_set__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5319_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5320_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_5321_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R3 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R3 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R3 )
& ( ord_less_int @ R3 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R3 )
& ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5322_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M2 @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
@ ( produc2626176000494625587at_nat
@ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5323_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5324_flip__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5325_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5326_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_5327_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_5328_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z3: int] :
( ( ord_less_eq_int @ D4 @ Z3 )
& ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5329_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z3: int] :
( ( ord_less_eq_int @ D4 @ Z6 )
& ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5330_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% take_bit_Suc_1
thf(fact_5331_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_Suc_1
thf(fact_5332_take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% take_bit_Suc_1
thf(fact_5333_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% take_bit_numeral_1
thf(fact_5334_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
= one_one_nat ) ).
% take_bit_numeral_1
thf(fact_5335_take__bit__numeral__1,axiom,
! [L: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
= one_one_int ) ).
% take_bit_numeral_1
thf(fact_5336_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
= zero_z3403309356797280102nteger )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_5337_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_5338_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_5339_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% take_bit_minus
thf(fact_5340_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_5341_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_5342_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
= ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_5343_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
= ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_5344_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_5345_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5346_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
= ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5347_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
= ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5348_take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_5349_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_5350_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_5351_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_Suc_bit1
thf(fact_5352_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_bit1
thf(fact_5353_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).
% take_bit_Suc_bit1
thf(fact_5354_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_bit1
thf(fact_5355_take__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5356_take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5357_take__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5358_take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% take_bit_Suc
thf(fact_5359_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% take_bit_numeral_bit1
thf(fact_5360_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).
% take_bit_numeral_bit1
thf(fact_5361_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).
% take_bit_numeral_bit1
thf(fact_5362_take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_bit1
thf(fact_5363_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_5364_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_5365_take__bit__rec,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_5366_take__bit__rec,axiom,
( bit_se1745604003318907178nteger
= ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_5367_take__bit__rec,axiom,
( bit_se569199155075624693atural
= ( ^ [N2: nat,A3: code_natural] : ( if_Code_natural @ ( N2 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_5368_take__bit__rec,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_5369_mask__numeral,axiom,
! [N: num] :
( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5370_mask__numeral,axiom,
! [N: num] :
( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5371_mask__numeral,axiom,
! [N: num] :
( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5372_mask__numeral,axiom,
! [N: num] :
( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).
% mask_numeral
thf(fact_5373_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K4: int,L2: int] :
( if_int
@ ( ( K4 = zero_zero_int )
| ( L2 = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K4
= ( uminus_uminus_int @ one_one_int ) )
@ L2
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ K4
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_5374_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_5375_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_5376_normalize__negative,axiom,
! [Q4: int,P4: int] :
( ( ord_less_int @ Q4 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_5377_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_5378_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_5379_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_5380_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_5381_odd__two__times__div__two__succ,axiom,
! [A: code_natural] :
( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ one_one_Code_natural )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_5382_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( dvd_dvd_int @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_5383_dvd__minus__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_5384_dvd__minus__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_5385_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( dvd_dvd_int @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_5386_minus__dvd__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
= ( dvd_dvd_Code_integer @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_5387_minus__dvd__iff,axiom,
! [X: rat,Y: rat] :
( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
= ( dvd_dvd_rat @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_5388_power__one,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
= one_one_Code_integer ) ).
% power_one
thf(fact_5389_power__one,axiom,
! [N: nat] :
( ( power_power_assn @ one_one_assn @ N )
= one_one_assn ) ).
% power_one
thf(fact_5390_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_5391_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_5392_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_5393_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_5394_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_5395_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_5396_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_5397_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_5398_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_5399_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_5400_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_5401_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_5402_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_5403_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_5404_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_5405_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_5406_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_5407_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_5408_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_5409_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_5410_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_5411_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_5412_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_5413_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_5414_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_5415_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_5416_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_prod
thf(fact_5417_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_5418_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_5419_power__inject__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ( ( power_8256067586552552935nteger @ A @ M )
= ( power_8256067586552552935nteger @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_5420_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M )
= ( power_power_rat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_5421_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_5422_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_5423_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_5424_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_5425_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_5426_dvd__div__mult__self,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ B )
=> ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_5427_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_5428_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_5429_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_5430_dvd__mult__div__cancel,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ B )
=> ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_5431_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_5432_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_5433_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_div
thf(fact_5434_unit__div,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).
% unit_div
thf(fact_5435_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_5436_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_5437_unit__div__1__unit,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% unit_div_1_unit
thf(fact_5438_unit__div__1__unit,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) @ one_one_Code_natural ) ) ).
% unit_div_1_unit
thf(fact_5439_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_5440_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_5441_unit__div__1__div__1,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_5442_unit__div__1__div__1,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_5443_and_Oleft__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
= A ) ).
% and.left_neutral
thf(fact_5444_and_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
= A ) ).
% and.left_neutral
thf(fact_5445_and_Oright__neutral,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= A ) ).
% and.right_neutral
thf(fact_5446_and_Oright__neutral,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= A ) ).
% and.right_neutral
thf(fact_5447_bit_Oconj__one__right,axiom,
! [X: code_integer] :
( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= X ) ).
% bit.conj_one_right
thf(fact_5448_bit_Oconj__one__right,axiom,
! [X: int] :
( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= X ) ).
% bit.conj_one_right
thf(fact_5449_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_5450_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_5451_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_5452_unit__mult__div__div,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
= ( divide5121882707175180666atural @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_5453_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_5454_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_5455_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_5456_unit__div__mult__self,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_5457_power__strict__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_5458_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_5459_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_5460_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_5461_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_5462_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_5463_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_5464_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5465_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5466_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5467_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
= one_one_Code_integer ) ).
% and_numerals(8)
thf(fact_5468_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
= one_one_Code_natural ) ).
% and_numerals(8)
thf(fact_5469_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
= one_one_int ) ).
% and_numerals(8)
thf(fact_5470_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= one_one_nat ) ).
% and_numerals(8)
thf(fact_5471_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= one_one_Code_integer ) ).
% and_numerals(2)
thf(fact_5472_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= one_one_Code_natural ) ).
% and_numerals(2)
thf(fact_5473_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= one_one_int ) ).
% and_numerals(2)
thf(fact_5474_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_numerals(2)
thf(fact_5475_mask__Suc__0,axiom,
( ( bit_se2119862282449309892nteger @ ( suc @ zero_zero_nat ) )
= one_one_Code_integer ) ).
% mask_Suc_0
thf(fact_5476_mask__Suc__0,axiom,
( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% mask_Suc_0
thf(fact_5477_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_5478_power__add__numeral,axiom,
! [A: code_integer,M: num,N: num] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_5479_power__add__numeral,axiom,
! [A: assn,M: num,N: num] :
( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_5480_power__add__numeral,axiom,
! [A: rat,M: num,N: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_5481_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_5482_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_5483_power__add__numeral2,axiom,
! [A: code_integer,M: num,N: num,B: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_5484_power__add__numeral2,axiom,
! [A: assn,M: num,N: num,B: assn] :
( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_5485_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_5486_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_5487_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_5488_take__bit__minus__one__eq__mask,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_se2119862282449309892nteger @ N ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_5489_take__bit__minus__one__eq__mask,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_5490_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_5491_power__strict__decreasing__iff,axiom,
! [B: code_integer,M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_5492_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_5493_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_5494_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_5495_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_5496_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_5497_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_5498_even__mult__iff,axiom,
! [A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( times_2397367101498566445atural @ A @ B ) )
= ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_5499_power__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_5500_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_5501_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_5502_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_5503_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% and_numerals(5)
thf(fact_5504_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% and_numerals(5)
thf(fact_5505_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
= zero_zero_int ) ).
% and_numerals(5)
thf(fact_5506_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_5507_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= zero_z3403309356797280102nteger ) ).
% and_numerals(1)
thf(fact_5508_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= zero_z2226904508553997617atural ) ).
% and_numerals(1)
thf(fact_5509_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= zero_zero_int ) ).
% and_numerals(1)
thf(fact_5510_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_5511_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_5512_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_5513_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_5514_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_5515_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5516_power2__minus,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5517_power2__minus,axiom,
! [A: rat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5518_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_5519_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_5520_power__decreasing__iff,axiom,
! [B: code_integer,M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_5521_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_5522_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_5523_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_5524_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_5525_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_5526_even__plus__one__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_5527_even__plus__one__iff,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) )
= ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_5528_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_5529_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_5530_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_5531_and__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(4)
thf(fact_5532_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_5533_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_5534_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_5535_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(6)
thf(fact_5536_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5537_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5538_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5539_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5540_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5541_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5542_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5543_power__minus__odd,axiom,
! [N: nat,A: code_integer] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5544_power__minus__odd,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5545_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_5546_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_5547_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_5548_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_5549_even__succ__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_5550_even__succ__div__two,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_5551_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_5552_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_5553_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% odd_succ_div_two
thf(fact_5554_odd__succ__div__two,axiom,
! [A: code_natural] :
( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ) ).
% odd_succ_div_two
thf(fact_5555_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_5556_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_5557_even__succ__div__2,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_5558_even__succ__div__2,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_5559_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_5560_power__minus1__even,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_Code_integer ) ).
% power_minus1_even
thf(fact_5561_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_rat ) ).
% power_minus1_even
thf(fact_5562_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_5563_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% neg_one_odd_power
thf(fact_5564_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% neg_one_odd_power
thf(fact_5565_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_5566_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) ) ).
% neg_one_even_power
thf(fact_5567_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) ) ).
% neg_one_even_power
thf(fact_5568_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_5569_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_5570_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_5571_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_5572_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5573_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5574_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5575_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_5576_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_5577_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_5578_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_5579_even__succ__div__exp,axiom,
! [A: code_natural,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_5580_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5581_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5582_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5583_even__succ__mod__exp,axiom,
! [A: code_natural,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_5584_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_5585_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_5586_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_5587_power__commuting__commutes,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( ( ( times_3573771949741848930nteger @ X @ Y )
= ( times_3573771949741848930nteger @ Y @ X ) )
=> ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_5588_power__commuting__commutes,axiom,
! [X: assn,Y: assn,N: nat] :
( ( ( times_times_assn @ X @ Y )
= ( times_times_assn @ Y @ X ) )
=> ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ Y )
= ( times_times_assn @ Y @ ( power_power_assn @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_5589_power__commuting__commutes,axiom,
! [X: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X @ Y )
= ( times_times_rat @ Y @ X ) )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
= ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_5590_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_5591_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_5592_power__mult__distrib,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_5593_power__mult__distrib,axiom,
! [A: assn,B: assn,N: nat] :
( ( power_power_assn @ ( times_times_assn @ A @ B ) @ N )
= ( times_times_assn @ ( power_power_assn @ A @ N ) @ ( power_power_assn @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_5594_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_5595_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_5596_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_5597_power__commutes,axiom,
! [A: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A )
= ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_commutes
thf(fact_5598_power__commutes,axiom,
! [A: assn,N: nat] :
( ( times_times_assn @ ( power_power_assn @ A @ N ) @ A )
= ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).
% power_commutes
thf(fact_5599_power__commutes,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_commutes
thf(fact_5600_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_5601_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_5602_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ~ ! [K2: code_integer] :
( A
!= ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% dvdE
thf(fact_5603_dvdE,axiom,
! [B: assn,A: assn] :
( ( dvd_dvd_assn @ B @ A )
=> ~ ! [K2: assn] :
( A
!= ( times_times_assn @ B @ K2 ) ) ) ).
% dvdE
thf(fact_5604_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K2: rat] :
( A
!= ( times_times_rat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_5605_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_5606_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_5607_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A
= ( times_3573771949741848930nteger @ B @ K ) )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% dvdI
thf(fact_5608_dvdI,axiom,
! [A: assn,B: assn,K: assn] :
( ( A
= ( times_times_assn @ B @ K ) )
=> ( dvd_dvd_assn @ B @ A ) ) ).
% dvdI
thf(fact_5609_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A
= ( times_times_rat @ B @ K ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_5610_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_5611_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_5612_dvd__def,axiom,
( dvd_dvd_Code_integer
= ( ^ [B3: code_integer,A3: code_integer] :
? [K4: code_integer] :
( A3
= ( times_3573771949741848930nteger @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_5613_dvd__def,axiom,
( dvd_dvd_assn
= ( ^ [B3: assn,A3: assn] :
? [K4: assn] :
( A3
= ( times_times_assn @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_5614_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B3: rat,A3: rat] :
? [K4: rat] :
( A3
= ( times_times_rat @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_5615_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
? [K4: nat] :
( A3
= ( times_times_nat @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_5616_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A3: int] :
? [K4: int] :
( A3
= ( times_times_int @ B3 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_5617_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult
thf(fact_5618_dvd__mult,axiom,
! [A: assn,C: assn,B: assn] :
( ( dvd_dvd_assn @ A @ C )
=> ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).
% dvd_mult
thf(fact_5619_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_5620_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_5621_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_5622_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_5623_dvd__mult2,axiom,
! [A: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ A @ B )
=> ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_5624_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_5625_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_5626_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_5627_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ A @ C ) ) ).
% dvd_mult_left
thf(fact_5628_dvd__mult__left,axiom,
! [A: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
=> ( dvd_dvd_assn @ A @ C ) ) ).
% dvd_mult_left
thf(fact_5629_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_5630_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_5631_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_5632_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% dvd_triv_left
thf(fact_5633_dvd__triv__left,axiom,
! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ A @ B ) ) ).
% dvd_triv_left
thf(fact_5634_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_5635_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_5636_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_5637_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D2 )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_5638_mult__dvd__mono,axiom,
! [A: assn,B: assn,C: assn,D2: assn] :
( ( dvd_dvd_assn @ A @ B )
=> ( ( dvd_dvd_assn @ C @ D2 )
=> ( dvd_dvd_assn @ ( times_times_assn @ A @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_5639_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ C @ D2 )
=> ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_5640_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D2 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_5641_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D2 )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_5642_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ B @ C ) ) ).
% dvd_mult_right
thf(fact_5643_dvd__mult__right,axiom,
! [A: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
=> ( dvd_dvd_assn @ B @ C ) ) ).
% dvd_mult_right
thf(fact_5644_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_5645_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_5646_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_5647_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% dvd_triv_right
thf(fact_5648_dvd__triv__right,axiom,
! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ A ) ) ).
% dvd_triv_right
thf(fact_5649_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_5650_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_5651_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_5652_one__dvd,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% one_dvd
thf(fact_5653_one__dvd,axiom,
! [A: assn] : ( dvd_dvd_assn @ one_one_assn @ A ) ).
% one_dvd
thf(fact_5654_one__dvd,axiom,
! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% one_dvd
thf(fact_5655_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_5656_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_5657_unit__imp__dvd,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_5658_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_5659_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_5660_dvd__unit__imp__unit,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% dvd_unit_imp_unit
thf(fact_5661_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_5662_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_5663_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D4: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D4 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_5664_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D4: int,T2: int] : ( dvd_dvd_int @ D4 @ ( uminus_uminus_int @ T2 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_5665_dvd__power__iff,axiom,
! [X: code_integer,M: nat,N: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
= ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_5666_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_5667_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_5668_dvd__power,axiom,
! [N: nat,X: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% dvd_power
thf(fact_5669_dvd__power,axiom,
! [N: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% dvd_power
thf(fact_5670_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_5671_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_5672_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_5673_subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le7084787975880047091nteger
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_5674_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_5675_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_set_nat
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
@ ( collect_nat
@ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5676_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_set_int
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
@ ( collect_int
@ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
= ( ( dvd_dvd_int @ A @ B )
& ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5677_strict__subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le1307284697595431911nteger
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
@ ( collect_Code_integer
@ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
= ( ( dvd_dvd_Code_integer @ A @ B )
& ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_5678_and__eq__minus__1__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( bit_se3949692690581998587nteger @ A @ B )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( ( A
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
& ( B
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_5679_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se725231765392027082nd_int @ A @ B )
= ( uminus_uminus_int @ one_one_int ) )
= ( ( A
= ( uminus_uminus_int @ one_one_int ) )
& ( B
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_5680_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5681_uminus__power__if,axiom,
! [N: nat,A: code_integer] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5682_uminus__power__if,axiom,
! [N: nat,A: rat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5683_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_5684_one__le__power,axiom,
! [A: code_integer,N: nat] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% one_le_power
thf(fact_5685_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_5686_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_5687_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_5688_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_5689_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_5690_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_5691_left__right__inverse__power,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( ( ( times_3573771949741848930nteger @ X @ Y )
= one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
= one_one_Code_integer ) ) ).
% left_right_inverse_power
thf(fact_5692_left__right__inverse__power,axiom,
! [X: assn,Y: assn,N: nat] :
( ( ( times_times_assn @ X @ Y )
= one_one_assn )
=> ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
= one_one_assn ) ) ).
% left_right_inverse_power
thf(fact_5693_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( ( times_times_rat @ X @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_5694_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_5695_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_5696_power__Suc,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_Suc
thf(fact_5697_power__Suc,axiom,
! [A: assn,N: nat] :
( ( power_power_assn @ A @ ( suc @ N ) )
= ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).
% power_Suc
thf(fact_5698_power__Suc,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_Suc
thf(fact_5699_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_5700_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_5701_power__Suc2,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_5702_power__Suc2,axiom,
! [A: assn,N: nat] :
( ( power_power_assn @ A @ ( suc @ N ) )
= ( times_times_assn @ ( power_power_assn @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_5703_power__Suc2,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_5704_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_5705_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_5706_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
& ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% is_unit_mult_iff
thf(fact_5707_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_5708_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_5709_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_5710_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_5711_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_5712_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_5713_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_5714_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_5715_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_5716_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_5717_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_5718_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_5719_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_5720_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_5721_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ A @ B )
= ( times_3573771949741848930nteger @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_5722_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_5723_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_5724_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ B @ A )
= ( times_3573771949741848930nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_5725_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_5726_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_5727_power__0,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
= one_one_Code_integer ) ).
% power_0
thf(fact_5728_power__0,axiom,
! [A: assn] :
( ( power_power_assn @ A @ zero_zero_nat )
= one_one_assn ) ).
% power_0
thf(fact_5729_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_5730_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_5731_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_5732_power__one__over,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_one_over
thf(fact_5733_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_5734_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_5735_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_5736_dvd__div__mult,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ C @ B )
=> ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ C ) @ A )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_5737_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_5738_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_5739_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_5740_div__mult__swap,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ C @ B )
=> ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_5741_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_5742_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_5743_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_5744_div__div__eq__right,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ C @ B )
=> ( ( dvd_dvd_Code_natural @ B @ A )
=> ( ( divide5121882707175180666atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
= ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_5745_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_5746_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_5747_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_5748_dvd__div__mult2__eq,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ C ) @ A )
=> ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
= ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_5749_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_5750_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_5751_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
=> ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_5752_dvd__mult__imp__div,axiom,
! [A: code_natural,C: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B )
=> ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_5753_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ D2 @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D2 ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_5754_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D2: int,C: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ D2 @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D2 ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_5755_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ D2 @ C )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D2 ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_5756_div__mult__div__if__dvd,axiom,
! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ A )
=> ( ( dvd_dvd_Code_natural @ D2 @ C )
=> ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ ( divide5121882707175180666atural @ C @ D2 ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_5757_power__add,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_add
thf(fact_5758_power__add,axiom,
! [A: assn,M: nat,N: nat] :
( ( power_power_assn @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_assn @ ( power_power_assn @ A @ M ) @ ( power_power_assn @ A @ N ) ) ) ).
% power_add
thf(fact_5759_power__add,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_5760_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_5761_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_5762_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_5763_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_5764_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_5765_unit__div__cancel,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ( ( ( divide5121882707175180666atural @ B @ A )
= ( divide5121882707175180666atural @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_5766_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_5767_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_5768_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_5769_div__unit__dvd__iff,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
= ( dvd_dvd_Code_natural @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_5770_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_5771_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_5772_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_5773_dvd__div__unit__iff,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ C @ B ) )
= ( dvd_dvd_Code_natural @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_5774_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5775_dvd__div__neg,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5776_dvd__div__neg,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5777_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5778_dvd__neg__div,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5779_dvd__neg__div,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5780_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_5781_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% minus_one_power_iff
thf(fact_5782_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% minus_one_power_iff
thf(fact_5783_mask__int__def,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% mask_int_def
thf(fact_5784_mask__eq__exp__minus__1,axiom,
( bit_se2119862282449309892nteger
= ( ^ [N2: nat] : ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5785_mask__eq__exp__minus__1,axiom,
( bit_se943457434206027407atural
= ( ^ [N2: nat] : ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5786_mask__eq__exp__minus__1,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5787_mask__eq__exp__minus__1,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_5788_power__le__one,axiom,
! [A: code_integer,N: nat] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer ) ) ) ).
% power_le_one
thf(fact_5789_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_5790_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_5791_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_5792_power__gt1__lemma,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_5793_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_5794_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_5795_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_5796_power__less__power__Suc,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_5797_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_5798_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_5799_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_5800_unity__coeff__ex,axiom,
! [P: code_integer > $o,L: code_integer] :
( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
= ( ? [X3: code_integer] :
( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_5801_unity__coeff__ex,axiom,
! [P: rat > $o,L: rat] :
( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
= ( ? [X3: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_5802_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
= ( ? [X3: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_5803_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
= ( ? [X3: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
& ( P @ X3 ) ) ) ) ).
% unity_coeff_ex
thf(fact_5804_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
= one_one_Code_integer ) )
& ( ( N != zero_zero_nat )
=> ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
= zero_z3403309356797280102nteger ) ) ) ).
% power_0_left
thf(fact_5805_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_5806_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_5807_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_5808_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C3: code_integer] :
( B
!= ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_5809_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C3: nat] :
( B
!= ( times_times_nat @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_5810_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C3: int] :
( B
!= ( times_times_int @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_5811_power__gt1,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_5812_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_5813_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_5814_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_5815_and_Ocomm__monoid__axioms,axiom,
comm_m1654649897431166824nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).
% and.comm_monoid_axioms
thf(fact_5816_and_Ocomm__monoid__axioms,axiom,
comm_monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).
% and.comm_monoid_axioms
thf(fact_5817_power__less__imp__less__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_5818_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_5819_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_5820_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_5821_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_5822_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_5823_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_5824_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_5825_power__increasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_5826_power__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_5827_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_5828_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_5829_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_5830_power__minus,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_minus
thf(fact_5831_power__minus,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_minus
thf(fact_5832_inf__period_I4_J,axiom,
! [D2: code_integer,D3: code_integer,T3: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ D3 )
=> ! [X5: code_integer,K3: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X5 @ T3 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) @ T3 ) ) ) ) ) ).
% inf_period(4)
thf(fact_5833_inf__period_I4_J,axiom,
! [D2: rat,D3: rat,T3: rat] :
( ( dvd_dvd_rat @ D2 @ D3 )
=> ! [X5: rat,K3: rat] :
( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ T3 ) ) )
= ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) @ T3 ) ) ) ) ) ).
% inf_period(4)
thf(fact_5834_inf__period_I4_J,axiom,
! [D2: int,D3: int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int,K3: int] :
( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) ) )
= ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) @ T3 ) ) ) ) ) ).
% inf_period(4)
thf(fact_5835_inf__period_I3_J,axiom,
! [D2: code_integer,D3: code_integer,T3: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ D3 )
=> ! [X5: code_integer,K3: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X5 @ T3 ) )
= ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) @ T3 ) ) ) ) ).
% inf_period(3)
thf(fact_5836_inf__period_I3_J,axiom,
! [D2: rat,D3: rat,T3: rat] :
( ( dvd_dvd_rat @ D2 @ D3 )
=> ! [X5: rat,K3: rat] :
( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ T3 ) )
= ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D3 ) ) @ T3 ) ) ) ) ).
% inf_period(3)
thf(fact_5837_inf__period_I3_J,axiom,
! [D2: int,D3: int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int,K3: int] :
( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) )
= ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) @ T3 ) ) ) ) ).
% inf_period(3)
thf(fact_5838_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D2 )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D2 @ C ) )
= ( ( times_times_nat @ B @ C )
= ( times_times_nat @ A @ D2 ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_5839_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D2 )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D2 @ C ) )
= ( ( times_times_int @ B @ C )
= ( times_times_int @ A @ D2 ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_5840_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D2 )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ D2 @ C ) )
= ( ( times_3573771949741848930nteger @ B @ C )
= ( times_3573771949741848930nteger @ A @ D2 ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_5841_dvd__div__div__eq__mult,axiom,
! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( C != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ A @ B )
=> ( ( dvd_dvd_Code_natural @ C @ D2 )
=> ( ( ( divide5121882707175180666atural @ B @ A )
= ( divide5121882707175180666atural @ D2 @ C ) )
= ( ( times_2397367101498566445atural @ B @ C )
= ( times_2397367101498566445atural @ A @ D2 ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_5842_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_5843_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_5844_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_5845_dvd__div__iff__mult,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( C != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ C @ B )
=> ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) )
= ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_5846_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_5847_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_5848_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_5849_div__dvd__iff__mult,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( B != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ B @ A )
=> ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
= ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_5850_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C )
= ( B
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_5851_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C )
= ( B
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_5852_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= C )
= ( B
= ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_5853_dvd__div__eq__mult,axiom,
! [A: code_natural,B: code_natural,C: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ A @ B )
=> ( ( ( divide5121882707175180666atural @ B @ A )
= C )
= ( B
= ( times_2397367101498566445atural @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_5854_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_5855_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_5856_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
thf(fact_5857_unit__div__eq__0__iff,axiom,
! [B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( ( divide5121882707175180666atural @ A @ B )
= zero_z2226904508553997617atural )
= ( A = zero_z2226904508553997617atural ) ) ) ).
% unit_div_eq_0_iff
thf(fact_5858_and_Omonoid__axioms,axiom,
monoid_Code_integer @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).
% and.monoid_axioms
thf(fact_5859_and_Omonoid__axioms,axiom,
monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).
% and.monoid_axioms
thf(fact_5860_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= C )
= ( A
= ( times_times_nat @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_5861_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= C )
= ( A
= ( times_times_int @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_5862_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= C )
= ( A
= ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_5863_unit__eq__div1,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( ( divide5121882707175180666atural @ A @ B )
= C )
= ( A
= ( times_2397367101498566445atural @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_5864_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( A
= ( divide_divide_nat @ C @ B ) )
= ( ( times_times_nat @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_5865_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( A
= ( divide_divide_int @ C @ B ) )
= ( ( times_times_int @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_5866_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( A
= ( divide6298287555418463151nteger @ C @ B ) )
= ( ( times_3573771949741848930nteger @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_5867_unit__eq__div2,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( A
= ( divide5121882707175180666atural @ C @ B ) )
= ( ( times_2397367101498566445atural @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_5868_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_5869_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_5870_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_5871_div__mult__unit2,axiom,
! [C: code_natural,B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ B @ A )
=> ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
= ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_5872_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_5873_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_5874_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_5875_unit__div__commute,axiom,
! [B: code_natural,A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_5876_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_5877_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_5878_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_5879_unit__div__mult__swap,axiom,
! [C: code_natural,A: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
=> ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_5880_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_5881_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_5882_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_5883_is__unit__div__mult2__eq,axiom,
! [B: code_natural,C: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
=> ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
= ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_5884_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_5885_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_5886_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
thf(fact_5887_unit__imp__mod__eq__0,axiom,
! [B: code_natural,A: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( modulo8411746178871703098atural @ A @ B )
= zero_z2226904508553997617atural ) ) ).
% unit_imp_mod_eq_0
thf(fact_5888_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5889_power__minus__Bit0,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5890_power__minus__Bit0,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5891_and_Osemilattice__neutr__axioms,axiom,
semila6106414119144251899nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).
% and.semilattice_neutr_axioms
thf(fact_5892_and_Osemilattice__neutr__axioms,axiom,
semila9079005292280841162tr_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).
% and.semilattice_neutr_axioms
thf(fact_5893_power__minus__Bit1,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_5894_power__minus__Bit1,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_5895_power__minus__Bit1,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_5896_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_5897_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_5898_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_5899_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_5900_power__numeral__even,axiom,
! [Z: code_integer,W: num] :
( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_5901_power__numeral__even,axiom,
! [Z: assn,W: num] :
( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_assn @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_5902_power__numeral__even,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_5903_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_5904_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_5905_power__numeral__odd,axiom,
! [Z: code_integer,W: num] :
( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_5906_power__numeral__odd,axiom,
! [Z: assn,W: num] :
( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_assn @ ( times_times_assn @ Z @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_5907_power__numeral__odd,axiom,
! [Z: rat,W: num] :
( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_5908_power__numeral__odd,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_5909_power__numeral__odd,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
= ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_odd
thf(fact_5910_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_5911_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_5912_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_5913_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
= zero_z2226904508553997617atural )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_5914_stable__imp__take__bit__eq,axiom,
! [A: nat,N: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2925701944663578781it_nat @ N @ A )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5915_stable__imp__take__bit__eq,axiom,
! [A: code_integer,N: nat] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1745604003318907178nteger @ N @ A )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se1745604003318907178nteger @ N @ A )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5916_stable__imp__take__bit__eq,axiom,
! [A: code_natural,N: nat] :
( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se569199155075624693atural @ N @ A )
= zero_z2226904508553997617atural ) )
& ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se569199155075624693atural @ N @ A )
= ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5917_stable__imp__take__bit__eq,axiom,
! [A: int,N: nat] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( bit_se2923211474154528505it_int @ N @ A )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_5918_power__Suc__less,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_5919_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_5920_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_5921_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_5922_power__Suc__le__self,axiom,
! [A: code_integer,N: nat] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_5923_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_5924_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_5925_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_5926_power__Suc__less__one,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).
% power_Suc_less_one
thf(fact_5927_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_5928_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_5929_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_5930_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N4 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_5931_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_5932_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_5933_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_5934_power__decreasing,axiom,
! [N: nat,N4: nat,A: code_integer] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N4 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_5935_power__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_5936_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_5937_power__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_5938_power__le__imp__le__exp,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_5939_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_5940_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_5941_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_5942_self__le__power,axiom,
! [A: code_integer,N: nat] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_5943_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_5944_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_5945_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_5946_one__less__power,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_5947_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_5948_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_5949_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_5950_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_5951_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_5952_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_5953_is__unit__div__mult__cancel__right,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ A ) )
= ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_5954_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_5955_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_5956_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_5957_is__unit__div__mult__cancel__left,axiom,
! [A: code_natural,B: code_natural] :
( ( A != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ A @ B ) )
= ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_5958_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B6: nat] :
( ( B6 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B6 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B6 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B6 )
= A )
=> ( ( ( times_times_nat @ A @ B6 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_5959_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B6: int] :
( ( B6 != zero_zero_int )
=> ( ( dvd_dvd_int @ B6 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B6 )
=> ( ( ( divide_divide_int @ one_one_int @ B6 )
= A )
=> ( ( ( times_times_int @ A @ B6 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_5960_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B6: code_integer] :
( ( B6 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B6 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
= B6 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B6 )
= A )
=> ( ( ( times_3573771949741848930nteger @ A @ B6 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C @ A )
!= ( times_3573771949741848930nteger @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_5961_is__unitE,axiom,
! [A: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
=> ~ ( ( A != zero_z2226904508553997617atural )
=> ! [B6: code_natural] :
( ( B6 != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ B6 @ one_one_Code_natural )
=> ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A )
= B6 )
=> ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B6 )
= A )
=> ( ( ( times_2397367101498566445atural @ A @ B6 )
= one_one_Code_natural )
=> ( ( divide5121882707175180666atural @ C @ A )
!= ( times_2397367101498566445atural @ C @ B6 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_5962_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_5963_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_5964_evenE,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: code_integer] :
( A
!= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_5965_evenE,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: code_natural] :
( A
!= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B6 ) ) ) ).
% evenE
thf(fact_5966_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_5967_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_5968_odd__one,axiom,
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% odd_one
thf(fact_5969_odd__one,axiom,
~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ one_one_Code_natural ) ).
% odd_one
thf(fact_5970_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5971_even__minus,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5972_power4__eq__xxxx,axiom,
! [X: code_integer] :
( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_5973_power4__eq__xxxx,axiom,
! [X: assn] :
( ( power_power_assn @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_5974_power4__eq__xxxx,axiom,
! [X: rat] :
( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_5975_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_5976_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_5977_power2__eq__square,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_3573771949741848930nteger @ A @ A ) ) ).
% power2_eq_square
thf(fact_5978_power2__eq__square,axiom,
! [A: assn] :
( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_assn @ A @ A ) ) ).
% power2_eq_square
thf(fact_5979_power2__eq__square,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ A @ A ) ) ).
% power2_eq_square
thf(fact_5980_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_5981_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_5982_one__power2,axiom,
( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% one_power2
thf(fact_5983_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_5984_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_5985_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_5986_power2__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_5987_power2__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_5988_power2__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_rat @ Y ) ) ) ) ).
% power2_eq_iff
thf(fact_5989_power3__eq__cube,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_5990_power3__eq__cube,axiom,
! [A: assn] :
( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_assn @ ( times_times_assn @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_5991_power3__eq__cube,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_5992_power3__eq__cube,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_5993_power3__eq__cube,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_5994_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_5995_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_5996_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5997_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5998_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_5999_even__mult__exp__div__exp__iff,axiom,
! [A: code_natural,M: nat,N: nat] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
= zero_z2226904508553997617atural )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_6000_power__minus_H,axiom,
! [X: int,N: nat] :
( ( nO_MATCH_int_int @ one_one_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ X @ N ) ) ) ) ).
% power_minus'
thf(fact_6001_power__minus_H,axiom,
! [X: code_integer,N: nat] :
( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).
% power_minus'
thf(fact_6002_power__minus_H,axiom,
! [X: rat,N: nat] :
( ( nO_MATCH_rat_rat @ one_one_rat @ X )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ X @ N ) ) ) ) ).
% power_minus'
thf(fact_6003_and__one__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6004_and__one__eq,axiom,
! [A: code_natural] :
( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
= ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6005_and__one__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ A @ one_one_int )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6006_and__one__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_one_eq
thf(fact_6007_one__and__eq,axiom,
! [A: code_integer] :
( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6008_one__and__eq,axiom,
! [A: code_natural] :
( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
= ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6009_one__and__eq,axiom,
! [A: int] :
( ( bit_se725231765392027082nd_int @ one_one_int @ A )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6010_one__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% one_and_eq
thf(fact_6011_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_6012_even__two__times__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_6013_even__two__times__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_6014_even__two__times__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_6015_even__two__times__div__two,axiom,
! [A: code_natural] :
( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_6016_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_6017_power2__eq__1__iff,axiom,
! [A: code_integer] :
( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( A = one_one_Code_integer )
| ( A
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% power2_eq_1_iff
thf(fact_6018_power2__eq__1__iff,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( A = one_one_rat )
| ( A
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% power2_eq_1_iff
thf(fact_6019_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_6020_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_6021_power__diff__power__eq,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
= ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_6022_power__diff__power__eq,axiom,
! [A: code_natural,N: nat,M: nat] :
( ( A != zero_z2226904508553997617atural )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
= ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M ) @ ( power_7079662738309270450atural @ A @ N ) )
= ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_6023_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_6024_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_6025_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_6026_odd__iff__mod__2__eq__one,axiom,
! [A: code_natural] :
( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_6027_normalize__denom__pos,axiom,
! [R3: product_prod_int_int,P4: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_6028_power__eq__if,axiom,
( power_8256067586552552935nteger
= ( ^ [P6: code_integer,M2: nat] : ( if_Code_integer @ ( M2 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P6 @ ( power_8256067586552552935nteger @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6029_power__eq__if,axiom,
( power_power_assn
= ( ^ [P6: assn,M2: nat] : ( if_assn @ ( M2 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P6 @ ( power_power_assn @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6030_power__eq__if,axiom,
( power_power_rat
= ( ^ [P6: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6031_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6032_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6033_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6034_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6035_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6036_normalize__crossproduct,axiom,
! [Q4: int,S3: int,P4: int,R3: int] :
( ( Q4 != zero_zero_int )
=> ( ( S3 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ R3 @ S3 ) ) )
=> ( ( times_times_int @ P4 @ S3 )
= ( times_times_int @ R3 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_6037_power__minus__mult,axiom,
! [N: nat,A: code_integer] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_6038_power__minus__mult,axiom,
! [N: nat,A: assn] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_assn @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_6039_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_6040_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_6041_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_6042_power2__sum,axiom,
! [X: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_6043_power2__sum,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_6044_power2__sum,axiom,
! [X: code_integer,Y: code_integer] :
( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_6045_power2__sum,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_6046_power2__sum,axiom,
! [X: code_natural,Y: code_natural] :
( ( power_7079662738309270450atural @ ( plus_p4538020629002901425atural @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_sum
thf(fact_6047_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_6048_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_6049_oddE,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: code_integer] :
( A
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_6050_oddE,axiom,
! [A: code_natural] :
( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ~ ! [B6: code_natural] :
( A
!= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_natural ) ) ) ).
% oddE
thf(fact_6051_square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
=> ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% square_le_1
thf(fact_6052_square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% square_le_1
thf(fact_6053_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_6054_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_6055_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_6056_parity__cases,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger ) )
=> ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer ) ) ) ).
% parity_cases
thf(fact_6057_parity__cases,axiom,
! [A: code_natural] :
( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= zero_z2226904508553997617atural ) )
=> ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
!= one_one_Code_natural ) ) ) ).
% parity_cases
thf(fact_6058_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_6059_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_6060_mod2__eq__if,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ) ).
% mod2_eq_if
thf(fact_6061_mod2__eq__if,axiom,
! [A: code_natural] :
( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= zero_z2226904508553997617atural ) )
& ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
=> ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= one_one_Code_natural ) ) ) ).
% mod2_eq_if
thf(fact_6062_minus__power__mult__self,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6063_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6064_minus__power__mult__self,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_6065_power__odd__eq,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6066_power__odd__eq,axiom,
! [A: assn,N: nat] :
( ( power_power_assn @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_assn @ A @ ( power_power_assn @ ( power_power_assn @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6067_power__odd__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6068_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6069_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_6070_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_6071_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6072_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_6073_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_6074_power2__diff,axiom,
! [X: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_6075_power2__diff,axiom,
! [X: code_integer,Y: code_integer] :
( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_6076_power2__diff,axiom,
! [X: rat,Y: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% power2_diff
thf(fact_6077_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% mask_half_int
thf(fact_6078_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6079_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6080_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6081_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_natural] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_6082_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_6083_power__minus1__odd,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_6084_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_6085_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_6086_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_6087_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_6088_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_6089_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_6090_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_6091_take__bit__numeral__minus__1__eq,axiom,
! [K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_6092_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top_nat @ dvd_dvd_nat
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
& ( M2 != N2 ) )
@ zero_zero_nat ) ).
% gcd_nat.ordering_top_axioms
thf(fact_6093_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F2: nat > nat > nat,A: nat,B: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
=> ( ( ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F2 @ A @ B @ Acc )
= Acc ) )
& ( ~ ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F2 @ A @ B @ Acc )
= ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F2 @ A @ Acc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_6094_fold__atLeastAtMost__nat_Opelims,axiom,
! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_6095_flip__bit__0,axiom,
! [A: code_integer] :
( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
= ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6096_flip__bit__0,axiom,
! [A: code_natural] :
( ( bit_se168947363167071951atural @ zero_zero_nat @ A )
= ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6097_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6098_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_6099_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6100_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6101_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6102_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_6103_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6104_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6105_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6106_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6107_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6108_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6109_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6110_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6111_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6112_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n356916108424825756nteger @ P )
= one_one_Code_integer )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6113_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= one_one_rat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6114_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6115_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_6116_of__bool__eq_I2_J,axiom,
( ( zero_n356916108424825756nteger @ $true )
= one_one_Code_integer ) ).
% of_bool_eq(2)
thf(fact_6117_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_6118_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_6119_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_6120_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6121_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6122_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6123_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_6124_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n356916108424825756nteger @ ~ P )
= ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6125_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2052037380579107095ol_rat @ ~ P )
= ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6126_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2684676970156552555ol_int @ ~ P )
= ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% of_bool_not_iff
thf(fact_6127_of__int__1,axiom,
( ( ring_18347121197199848620nteger @ one_one_int )
= one_one_Code_integer ) ).
% of_int_1
thf(fact_6128_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_6129_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_6130_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_18347121197199848620nteger @ Z )
= one_one_Code_integer )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6131_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6132_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_rat @ Z )
= one_one_rat )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6133_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_18347121197199848620nteger @ ( times_times_int @ W @ Z ) )
= ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% of_int_mult
thf(fact_6134_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_mult
thf(fact_6135_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_6136_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_6137_of__int__minus,axiom,
! [Z: int] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
= ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).
% of_int_minus
thf(fact_6138_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% of_int_minus
thf(fact_6139_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R3: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R3 ) )
= ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_6140_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_6141_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_6142_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6143_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6144_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6145_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6146_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6147_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_6148_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6149_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6150_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6151_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6152_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6153_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_6154_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6155_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6156_take__bit__of__1,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_1
thf(fact_6157_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6158_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
= ( ring_18347121197199848620nteger @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6159_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6160_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6161_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6162_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6163_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6164_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6165_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6166_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_6167_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6168_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6169_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6170_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_6171_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6172_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se569199155075624693atural @ M @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ N @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6173_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6174_take__bit__of__exp,axiom,
! [M: nat,N: nat] :
( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_of_exp
thf(fact_6175_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6176_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6177_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6178_take__bit__of__2,axiom,
! [N: nat] :
( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_of_2
thf(fact_6179_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6180_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6181_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6182_mult__of__int__commute,axiom,
! [X: int,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6183_mult__of__int__commute,axiom,
! [X: int,Y: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6184_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6185_of__bool__conj,axiom,
! [P: $o,Q2: $o] :
( ( zero_n356916108424825756nteger
@ ( P
& Q2 ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_6186_of__bool__conj,axiom,
! [P: $o,Q2: $o] :
( ( zero_n2052037380579107095ol_rat
@ ( P
& Q2 ) )
= ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_6187_of__bool__conj,axiom,
! [P: $o,Q2: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P
& Q2 ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_6188_of__bool__conj,axiom,
! [P: $o,Q2: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P
& Q2 ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) ) ) ).
% of_bool_conj
thf(fact_6189_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% of_bool_less_eq_one
thf(fact_6190_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% of_bool_less_eq_one
thf(fact_6191_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_6192_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_6193_split__of__bool__asm,axiom,
! [P: code_integer > $o,P4: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_Code_integer ) )
| ( ~ P4
& ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6194_split__of__bool__asm,axiom,
! [P: rat > $o,P4: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_rat ) )
| ( ~ P4
& ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6195_split__of__bool__asm,axiom,
! [P: int > $o,P4: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_int ) )
| ( ~ P4
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6196_split__of__bool__asm,axiom,
! [P: nat > $o,P4: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ one_one_nat ) )
| ( ~ P4
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_6197_split__of__bool,axiom,
! [P: code_integer > $o,P4: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
= ( ( P4
=> ( P @ one_one_Code_integer ) )
& ( ~ P4
=> ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% split_of_bool
thf(fact_6198_split__of__bool,axiom,
! [P: rat > $o,P4: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
= ( ( P4
=> ( P @ one_one_rat ) )
& ( ~ P4
=> ( P @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_6199_split__of__bool,axiom,
! [P: int > $o,P4: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
= ( ( P4
=> ( P @ one_one_int ) )
& ( ~ P4
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_6200_split__of__bool,axiom,
! [P: nat > $o,P4: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
= ( ( P4
=> ( P @ one_one_nat ) )
& ( ~ P4
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_6201_of__bool__def,axiom,
( zero_n356916108424825756nteger
= ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% of_bool_def
thf(fact_6202_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_6203_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_6204_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_6205_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q5: int,R4: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R4 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_6206_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6207_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6208_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6209_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_6210_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A3 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F @ A3 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_6211_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_6212_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_6213_bits__induct,axiom,
! [P: code_integer > $o,A: code_integer] :
( ! [A6: code_integer] :
( ( ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: code_integer,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6214_bits__induct,axiom,
! [P: code_natural > $o,A: code_natural] :
( ! [A6: code_natural] :
( ( ( divide5121882707175180666atural @ A6 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: code_natural,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B6 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A6 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B6 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6215_bits__induct,axiom,
! [P: int > $o,A: int] :
( ! [A6: int] :
( ( ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: int,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6216_bits__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [A6: nat] :
( ( ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: nat,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A6 )
=> ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_6217_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6218_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6219_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6220_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_6221_dvd__productE,axiom,
! [P4: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( P4
= ( times_times_nat @ X2 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X2 @ A )
=> ~ ( dvd_dvd_nat @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6222_dvd__productE,axiom,
! [P4: int,A: int,B: int] :
( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
=> ~ ! [X2: int,Y2: int] :
( ( P4
= ( times_times_int @ X2 @ Y2 ) )
=> ( ( dvd_dvd_int @ X2 @ A )
=> ~ ( dvd_dvd_int @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6223_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B8: nat,C5: nat] :
( ( A
= ( times_times_nat @ B8 @ C5 ) )
& ( dvd_dvd_nat @ B8 @ B )
& ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_6224_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B8: int,C5: int] :
( ( A
= ( times_times_int @ B8 @ C5 ) )
& ( dvd_dvd_int @ B8 @ B )
& ( dvd_dvd_int @ C5 @ C ) ) ) ).
% division_decomp
thf(fact_6225_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_6226_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_6227_xor__one__eq,axiom,
! [A: code_integer] :
( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_6228_xor__one__eq,axiom,
! [A: code_natural] :
( ( bit_se2046307713759805098atural @ A @ one_one_Code_natural )
= ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n8403883297036319079atural
@ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_6229_xor__one__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ A @ one_one_int )
= ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_6230_xor__one__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
= ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% xor_one_eq
thf(fact_6231_one__xor__eq,axiom,
! [A: code_integer] :
( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_6232_one__xor__eq,axiom,
! [A: code_natural] :
( ( bit_se2046307713759805098atural @ one_one_Code_natural @ A )
= ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n8403883297036319079atural
@ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_6233_one__xor__eq,axiom,
! [A: int] :
( ( bit_se6526347334894502574or_int @ one_one_int @ A )
= ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_6234_one__xor__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
= ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_xor_eq
thf(fact_6235_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger
@ ( zero_n356916108424825756nteger
@ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
!= zero_z3403309356797280102nteger )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6236_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
= ( times_2397367101498566445atural
@ ( zero_n8403883297036319079atural
@ ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M )
!= zero_z2226904508553997617atural )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6237_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6238_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_6239_floor__exists1,axiom,
! [X: rat] :
? [X2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X2 @ one_one_int ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
=> ( Y5 = X2 ) ) ) ).
% floor_exists1
thf(fact_6240_floor__exists,axiom,
! [X: rat] :
? [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_6241_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K4: int,L2: int] :
( if_int
@ ( ( member_int @ K4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_6242_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_6243_round__unique,axiom,
! [X: rat,Y: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y ) ) ) ).
% round_unique
thf(fact_6244_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_6245_push__bit__numeral__minus__1,axiom,
! [N: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_6246_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [Q5: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_6247_singletonI,axiom,
! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singletonI
thf(fact_6248_singletonI,axiom,
! [A: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).
% singletonI
thf(fact_6249_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_6250_singletonI,axiom,
! [A: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% singletonI
thf(fact_6251_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_6252_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_6253_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_6254_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_6255_Int__insert__right__if1,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6256_Int__insert__right__if1,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6257_Int__insert__right__if1,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( inf_in8396524679539076455nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ B2 ) )
= ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6258_Int__insert__right__if1,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6259_Int__insert__right__if1,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6260_Int__insert__right__if1,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ B2 ) )
= ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6261_Int__insert__right__if1,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6262_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_6263_Int__insert__right__if0,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ~ ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6264_Int__insert__right__if0,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6265_Int__insert__right__if0,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ~ ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( inf_in8396524679539076455nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ B2 ) )
= ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6266_Int__insert__right__if0,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6267_Int__insert__right__if0,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( inf_inf_set_int @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6268_Int__insert__right__if0,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ B2 ) )
= ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6269_Int__insert__right__if0,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6270_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_6271_insert__inter__insert,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_6272_insert__inter__insert,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_6273_insert__inter__insert,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ ( insert_int @ A @ B2 ) )
= ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_6274_insert__inter__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_6275_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_6276_Int__insert__left__if1,axiom,
! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ C2 )
=> ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B2 ) @ C2 )
= ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6277_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B2: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6278_Int__insert__left__if1,axiom,
! [A: set_Pr958786334691620121nt_int,C2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ A @ C2 )
=> ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B2 ) @ C2 )
= ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6279_Int__insert__left__if1,axiom,
! [A: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C2 )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6280_Int__insert__left__if1,axiom,
! [A: int,C2: set_int,B2: set_int] :
( ( member_int @ A @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C2 )
= ( insert_int @ A @ ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6281_Int__insert__left__if1,axiom,
! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A @ C2 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B2 ) @ C2 )
= ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6282_Int__insert__left__if1,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B2 ) @ C2 )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6283_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_6284_Int__insert__left__if0,axiom,
! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ~ ( member8757157785044589968at_nat @ A @ C2 )
=> ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B2 ) @ C2 )
= ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6285_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B2: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C2 )
= ( inf_inf_set_o @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6286_Int__insert__left__if0,axiom,
! [A: set_Pr958786334691620121nt_int,C2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ~ ( member2340774599025711042nt_int @ A @ C2 )
=> ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B2 ) @ C2 )
= ( inf_in8396524679539076455nt_int @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6287_Int__insert__left__if0,axiom,
! [A: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6288_Int__insert__left__if0,axiom,
! [A: int,C2: set_int,B2: set_int] :
( ~ ( member_int @ A @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C2 )
= ( inf_inf_set_int @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6289_Int__insert__left__if0,axiom,
! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A @ C2 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B2 ) @ C2 )
= ( inf_in1906310914598751387_nat_o @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6290_Int__insert__left__if0,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6291_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_6292_Un__insert__right,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6293_Un__insert__right,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6294_Un__insert__right,axiom,
! [A2: set_int,A: int,B2: set_int] :
( ( sup_sup_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( insert_int @ A @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6295_Un__insert__right,axiom,
! [A2: set_Pr8551490117392284871at_nat,A: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ A2 @ ( insert6337962749363155127at_nat @ A @ B2 ) )
= ( insert6337962749363155127at_nat @ A @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6296_Un__insert__right,axiom,
! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6297_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_6298_Un__insert__left,axiom,
! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ B2 ) @ C2 )
= ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6299_Un__insert__left,axiom,
! [A: $o,B2: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B2 ) @ C2 )
= ( insert_o @ A @ ( sup_sup_set_o @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6300_Un__insert__left,axiom,
! [A: int,B2: set_int,C2: set_int] :
( ( sup_sup_set_int @ ( insert_int @ A @ B2 ) @ C2 )
= ( insert_int @ A @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6301_Un__insert__left,axiom,
! [A: produc4166570645942440679at_nat,B2: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
( ( sup_su3035147773818789531at_nat @ ( insert6337962749363155127at_nat @ A @ B2 ) @ C2 )
= ( insert6337962749363155127at_nat @ A @ ( sup_su3035147773818789531at_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6302_Un__insert__left,axiom,
! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ B2 ) @ C2 )
= ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6303_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_6304_singleton__conv2,axiom,
! [A: produc3843707927480180839at_nat] :
( ( collec6321179662152712658at_nat
@ ( ^ [Y4: produc3843707927480180839at_nat,Z4: produc3843707927480180839at_nat] : ( Y4 = Z4 )
@ A ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singleton_conv2
thf(fact_6305_singleton__conv2,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( collec5210948495886036740nt_int
@ ( ^ [Y4: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] : ( Y4 = Z4 )
@ A ) )
= ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).
% singleton_conv2
thf(fact_6306_singleton__conv2,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 )
@ A ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_6307_singleton__conv2,axiom,
! [A: produc3658429121746597890et_nat > $o] :
( ( collec939566748876313656_nat_o
@ ( ^ [Y4: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat > $o] : ( Y4 = Z4 )
@ A ) )
= ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% singleton_conv2
thf(fact_6308_singleton__conv2,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ( ^ [Y4: product_prod_nat_nat,Z4: product_prod_nat_nat] : ( Y4 = Z4 )
@ A ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv2
thf(fact_6309_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_6310_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_6311_singleton__conv2,axiom,
! [A: int] :
( ( collect_int
@ ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ A ) )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% singleton_conv2
thf(fact_6312_singleton__conv,axiom,
! [A: produc3843707927480180839at_nat] :
( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] : ( X3 = A ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% singleton_conv
thf(fact_6313_singleton__conv,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( X3 = A ) )
= ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).
% singleton_conv
thf(fact_6314_singleton__conv,axiom,
! [A: set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( X3 = A ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_6315_singleton__conv,axiom,
! [A: produc3658429121746597890et_nat > $o] :
( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( X3 = A ) )
= ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% singleton_conv
thf(fact_6316_singleton__conv,axiom,
! [A: product_prod_nat_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( X3 = A ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singleton_conv
thf(fact_6317_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X3: $o] : ( X3 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_6318_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_6319_singleton__conv,axiom,
! [A: int] :
( ( collect_int
@ ^ [X3: int] : ( X3 = A ) )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% singleton_conv
thf(fact_6320_singleton__insert__inj__eq,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
= ( insert9069300056098147895at_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6321_singleton__insert__inj__eq,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6322_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6323_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6324_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6325_singleton__insert__inj__eq_H,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ A2 )
= ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
= ( ( A = B )
& ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_6326_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ A2 )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( ( A = B )
& ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_6327_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_6328_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_6329_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_6330_insert__disjoint_I1_J,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) @ B2 )
= bot_bo228742789529271731at_nat )
= ( ~ ( member8757157785044589968at_nat @ A @ B2 )
& ( ( inf_in7913087082777306421at_nat @ A2 @ B2 )
= bot_bo228742789529271731at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_6331_insert__disjoint_I1_J,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ A2 ) @ B2 )
= bot_bo1488462491386950373nt_int )
= ( ~ ( member2340774599025711042nt_int @ A @ B2 )
& ( ( inf_in8396524679539076455nt_int @ A2 @ B2 )
= bot_bo1488462491386950373nt_int ) ) ) ).
% insert_disjoint(1)
thf(fact_6332_insert__disjoint_I1_J,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B2 )
& ( ( inf_inf_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_6333_insert__disjoint_I1_J,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A2 ) @ B2 )
= bot_bo7824918357723582233_nat_o )
= ( ~ ( member6576561426505652726_nat_o @ A @ B2 )
& ( ( inf_in1906310914598751387_nat_o @ A2 @ B2 )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% insert_disjoint(1)
thf(fact_6334_insert__disjoint_I1_J,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B2 )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B2 )
& ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_6335_insert__disjoint_I1_J,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B2 )
& ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_6336_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_6337_insert__disjoint_I1_J,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ B2 )
= bot_bot_set_int )
= ( ~ ( member_int @ A @ B2 )
& ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_6338_insert__disjoint_I2_J,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat
= ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member8757157785044589968at_nat @ A @ B2 )
& ( bot_bo228742789529271731at_nat
= ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6339_insert__disjoint_I2_J,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( bot_bo1488462491386950373nt_int
= ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ A2 ) @ B2 ) )
= ( ~ ( member2340774599025711042nt_int @ A @ B2 )
& ( bot_bo1488462491386950373nt_int
= ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6340_insert__disjoint_I2_J,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_nat @ A @ B2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6341_insert__disjoint_I2_J,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A2 ) @ B2 ) )
= ( ~ ( member6576561426505652726_nat_o @ A @ B2 )
& ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6342_insert__disjoint_I2_J,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member8440522571783428010at_nat @ A @ B2 )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6343_insert__disjoint_I2_J,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 ) )
= ( ~ ( member_o @ A @ B2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6344_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6345_insert__disjoint_I2_J,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ B2 ) )
= ( ~ ( member_int @ A @ B2 )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_6346_disjoint__insert_I1_J,axiom,
! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( ( inf_in7913087082777306421at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ A2 ) )
= bot_bo228742789529271731at_nat )
= ( ~ ( member8757157785044589968at_nat @ A @ B2 )
& ( ( inf_in7913087082777306421at_nat @ B2 @ A2 )
= bot_bo228742789529271731at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_6347_disjoint__insert_I1_J,axiom,
! [B2: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( ( inf_in8396524679539076455nt_int @ B2 @ ( insert8897473484851387113nt_int @ A @ A2 ) )
= bot_bo1488462491386950373nt_int )
= ( ~ ( member2340774599025711042nt_int @ A @ B2 )
& ( ( inf_in8396524679539076455nt_int @ B2 @ A2 )
= bot_bo1488462491386950373nt_int ) ) ) ).
% disjoint_insert(1)
thf(fact_6348_disjoint__insert_I1_J,axiom,
! [B2: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A @ A2 ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B2 )
& ( ( inf_inf_set_set_nat @ B2 @ A2 )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_6349_disjoint__insert_I1_J,axiom,
! [B2: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ B2 @ ( insert5175938949040314269_nat_o @ A @ A2 ) )
= bot_bo7824918357723582233_nat_o )
= ( ~ ( member6576561426505652726_nat_o @ A @ B2 )
& ( ( inf_in1906310914598751387_nat_o @ B2 @ A2 )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% disjoint_insert(1)
thf(fact_6350_disjoint__insert_I1_J,axiom,
! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ( inf_in2572325071724192079at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ A2 ) )
= bot_bo2099793752762293965at_nat )
= ( ~ ( member8440522571783428010at_nat @ A @ B2 )
& ( ( inf_in2572325071724192079at_nat @ B2 @ A2 )
= bot_bo2099793752762293965at_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_6351_disjoint__insert_I1_J,axiom,
! [B2: set_o,A: $o,A2: set_o] :
( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A @ A2 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B2 )
& ( ( inf_inf_set_o @ B2 @ A2 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_6352_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_6353_disjoint__insert_I1_J,axiom,
! [B2: set_int,A: int,A2: set_int] :
( ( ( inf_inf_set_int @ B2 @ ( insert_int @ A @ A2 ) )
= bot_bot_set_int )
= ( ~ ( member_int @ A @ B2 )
& ( ( inf_inf_set_int @ B2 @ A2 )
= bot_bot_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_6354_disjoint__insert_I2_J,axiom,
! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( bot_bo228742789529271731at_nat
= ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) )
= ( ~ ( member8757157785044589968at_nat @ B @ A2 )
& ( bot_bo228742789529271731at_nat
= ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6355_disjoint__insert_I2_J,axiom,
! [A2: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
( ( bot_bo1488462491386950373nt_int
= ( inf_in8396524679539076455nt_int @ A2 @ ( insert8897473484851387113nt_int @ B @ B2 ) ) )
= ( ~ ( member2340774599025711042nt_int @ B @ A2 )
& ( bot_bo1488462491386950373nt_int
= ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6356_disjoint__insert_I2_J,axiom,
! [A2: set_set_nat,B: set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) )
= ( ~ ( member_set_nat @ B @ A2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6357_disjoint__insert_I2_J,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B2: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B2 ) ) )
= ( ~ ( member6576561426505652726_nat_o @ B @ A2 )
& ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6358_disjoint__insert_I2_J,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) )
= ( ~ ( member8440522571783428010at_nat @ B @ A2 )
& ( bot_bo2099793752762293965at_nat
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6359_disjoint__insert_I2_J,axiom,
! [A2: set_o,B: $o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ ( insert_o @ B @ B2 ) ) )
= ( ~ ( member_o @ B @ A2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6360_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6361_disjoint__insert_I2_J,axiom,
! [A2: set_int,B: int,B2: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ A2 @ ( insert_int @ B @ B2 ) ) )
= ( ~ ( member_int @ B @ A2 )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_6362_insert__Diff__single,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( insert9069300056098147895at_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6363_insert__Diff__single,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6364_insert__Diff__single,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6365_insert__Diff__single,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6366_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6367_round__1,axiom,
( ( archim7778729529865785530nd_rat @ one_one_rat )
= one_one_int ) ).
% round_1
thf(fact_6368_subset__Compl__singleton,axiom,
! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
= ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6369_subset__Compl__singleton,axiom,
! [A2: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le483042692224249369nt_int @ A2 @ ( uminus6423885277529793776nt_int @ ( insert8897473484851387113nt_int @ B @ bot_bo1488462491386950373nt_int ) ) )
= ( ~ ( member2340774599025711042nt_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6370_subset__Compl__singleton,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6371_subset__Compl__singleton,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( ~ ( member6576561426505652726_nat_o @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6372_subset__Compl__singleton,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
= ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6373_subset__Compl__singleton,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6374_subset__Compl__singleton,axiom,
! [A2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6375_subset__Compl__singleton,axiom,
! [A2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_6376_push__bit__Suc__minus__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_6377_push__bit__Suc__minus__numeral,axiom,
! [N: nat,K: num] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_6378_round__neg__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_6379_push__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A )
= ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_6380_push__bit__Suc,axiom,
! [N: nat,A: code_natural] :
( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A )
= ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_6381_push__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
= ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_6382_push__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
= ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% push_bit_Suc
thf(fact_6383_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_6384_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
= ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_6385_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ one_one_int )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_6386_push__bit__of__1,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_1
thf(fact_6387_push__bit__minus__numeral,axiom,
! [L: num,K: num] :
( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_6388_push__bit__minus__numeral,axiom,
! [L: num,K: num] :
( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_6389_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
= ( uminus1351360451143612070nteger @ L ) ) ).
% minus_integer_code(2)
thf(fact_6390_insert__Collect,axiom,
! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
= ( collec6321179662152712658at_nat
@ ^ [U2: produc3843707927480180839at_nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6391_insert__Collect,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( collec3392354462482085612at_nat
@ ^ [U2: product_prod_nat_nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6392_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U2: $o] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6393_insert__Collect,axiom,
! [A: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( insert8897473484851387113nt_int @ A @ ( collec5210948495886036740nt_int @ P ) )
= ( collec5210948495886036740nt_int
@ ^ [U2: set_Pr958786334691620121nt_int] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6394_insert__Collect,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
= ( collect_set_nat
@ ^ [U2: set_nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6395_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6396_insert__Collect,axiom,
! [A: int,P: int > $o] :
( ( insert_int @ A @ ( collect_int @ P ) )
= ( collect_int
@ ^ [U2: int] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6397_insert__Collect,axiom,
! [A: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( insert5175938949040314269_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) )
= ( collec939566748876313656_nat_o
@ ^ [U2: produc3658429121746597890et_nat > $o] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_6398_insert__compr,axiom,
( insert9069300056098147895at_nat
= ( ^ [A3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( X3 = A3 )
| ( member8757157785044589968at_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6399_insert__compr,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( X3 = A3 )
| ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6400_insert__compr,axiom,
( insert_o
= ( ^ [A3: $o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( X3 = A3 )
| ( member_o @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6401_insert__compr,axiom,
( insert8897473484851387113nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int,B4: set_se6260736226359567993nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( X3 = A3 )
| ( member2340774599025711042nt_int @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6402_insert__compr,axiom,
( insert_set_nat
= ( ^ [A3: set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A3 )
| ( member_set_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6403_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
| ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6404_insert__compr,axiom,
( insert_int
= ( ^ [A3: int,B4: set_int] :
( collect_int
@ ^ [X3: int] :
( ( X3 = A3 )
| ( member_int @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6405_insert__compr,axiom,
( insert5175938949040314269_nat_o
= ( ^ [A3: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( X3 = A3 )
| ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_6406_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_6407_exhaustive__integer_H_Ocases,axiom,
! [X: produc8763457246119570046nteger] :
~ ! [F3: code_integer > option6357759511663192854e_term,D: code_integer,I2: code_integer] :
( X
!= ( produc6137756002093451184nteger @ F3 @ ( produc1086072967326762835nteger @ D @ I2 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_6408_full__exhaustive__integer_H_Ocases,axiom,
! [X: produc1908205239877642774nteger] :
~ ! [F3: produc6241069584506657477e_term > option6357759511663192854e_term,D: code_integer,I2: code_integer] :
( X
!= ( produc8603105652947943368nteger @ F3 @ ( produc1086072967326762835nteger @ D @ I2 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_6409_ID_Opred__cong,axiom,
! [X: produc3843707927480180839at_nat,Ya: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o,Pa: produc3843707927480180839at_nat > $o] :
( ( X = Ya )
=> ( ! [Z2: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ Z2 @ ( insert9069300056098147895at_nat @ Ya @ bot_bo228742789529271731at_nat ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id1044457264471098170_nat_o @ P @ X )
= ( bNF_id1044457264471098170_nat_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6410_ID_Opred__cong,axiom,
! [X: set_Pr958786334691620121nt_int,Ya: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o,Pa: set_Pr958786334691620121nt_int > $o] :
( ( X = Ya )
=> ( ! [Z2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Z2 @ ( insert8897473484851387113nt_int @ Ya @ bot_bo1488462491386950373nt_int ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id9176111459293603464_int_o @ P @ X )
= ( bNF_id9176111459293603464_int_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6411_ID_Opred__cong,axiom,
! [X: set_nat,Ya: set_nat,P: set_nat > $o,Pa: set_nat > $o] :
( ( X = Ya )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( insert_set_nat @ Ya @ bot_bot_set_set_nat ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id_bnf_set_nat_o @ P @ X )
= ( bNF_id_bnf_set_nat_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6412_ID_Opred__cong,axiom,
! [X: produc3658429121746597890et_nat > $o,Ya: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o,Pa: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X = Ya )
=> ( ! [Z2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Z2 @ ( insert5175938949040314269_nat_o @ Ya @ bot_bo7824918357723582233_nat_o ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id4470576313039291732at_o_o @ P @ X )
= ( bNF_id4470576313039291732at_o_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6413_ID_Opred__cong,axiom,
! [X: product_prod_nat_nat,Ya: product_prod_nat_nat,P: product_prod_nat_nat > $o,Pa: product_prod_nat_nat > $o] :
( ( X = Ya )
=> ( ! [Z2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( insert8211810215607154385at_nat @ Ya @ bot_bo2099793752762293965at_nat ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id5045731369644952736_nat_o @ P @ X )
= ( bNF_id5045731369644952736_nat_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6414_ID_Opred__cong,axiom,
! [X: $o,Ya: $o,P: $o > $o,Pa: $o > $o] :
( ( X = Ya )
=> ( ! [Z2: $o] :
( ( member_o @ Z2 @ ( insert_o @ Ya @ bot_bot_set_o ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id_bnf_o_o @ P @ X )
= ( bNF_id_bnf_o_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6415_ID_Opred__cong,axiom,
! [X: nat,Ya: nat,P: nat > $o,Pa: nat > $o] :
( ( X = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( insert_nat @ Ya @ bot_bot_set_nat ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id_bnf_nat_o @ P @ X )
= ( bNF_id_bnf_nat_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6416_ID_Opred__cong,axiom,
! [X: int,Ya: int,P: int > $o,Pa: int > $o] :
( ( X = Ya )
=> ( ! [Z2: int] :
( ( member_int @ Z2 @ ( insert_int @ Ya @ bot_bot_set_int ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( bNF_id_bnf_int_o @ P @ X )
= ( bNF_id_bnf_int_o @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_6417_ID_Opred__mono__strong,axiom,
! [P: produc3843707927480180839at_nat > $o,X: produc3843707927480180839at_nat,Pa: produc3843707927480180839at_nat > $o] :
( ( bNF_id1044457264471098170_nat_o @ P @ X )
=> ( ! [Z2: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ Z2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id1044457264471098170_nat_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6418_ID_Opred__mono__strong,axiom,
! [P: set_Pr958786334691620121nt_int > $o,X: set_Pr958786334691620121nt_int,Pa: set_Pr958786334691620121nt_int > $o] :
( ( bNF_id9176111459293603464_int_o @ P @ X )
=> ( ! [Z2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Z2 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id9176111459293603464_int_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6419_ID_Opred__mono__strong,axiom,
! [P: set_nat > $o,X: set_nat,Pa: set_nat > $o] :
( ( bNF_id_bnf_set_nat_o @ P @ X )
=> ( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id_bnf_set_nat_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6420_ID_Opred__mono__strong,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,Pa: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( bNF_id4470576313039291732at_o_o @ P @ X )
=> ( ! [Z2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Z2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id4470576313039291732at_o_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6421_ID_Opred__mono__strong,axiom,
! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Pa: product_prod_nat_nat > $o] :
( ( bNF_id5045731369644952736_nat_o @ P @ X )
=> ( ! [Z2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id5045731369644952736_nat_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6422_ID_Opred__mono__strong,axiom,
! [P: $o > $o,X: $o,Pa: $o > $o] :
( ( bNF_id_bnf_o_o @ P @ X )
=> ( ! [Z2: $o] :
( ( member_o @ Z2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id_bnf_o_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6423_ID_Opred__mono__strong,axiom,
! [P: nat > $o,X: nat,Pa: nat > $o] :
( ( bNF_id_bnf_nat_o @ P @ X )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id_bnf_nat_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6424_ID_Opred__mono__strong,axiom,
! [P: int > $o,X: int,Pa: int > $o] :
( ( bNF_id_bnf_int_o @ P @ X )
=> ( ! [Z2: int] :
( ( member_int @ Z2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( bNF_id_bnf_int_o @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_6425_ID_Orel__cong,axiom,
! [X: $o,Ya: $o,Y: $o,Xa: $o,R: $o > $o > $o,Ra: $o > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: $o,Yb: $o] :
( ( member_o @ Z2 @ ( insert_o @ Ya @ bot_bot_set_o ) )
=> ( ( member_o @ Yb @ ( insert_o @ Xa @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_o_o_o @ R @ X @ Y )
= ( bNF_id_bnf_o_o_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6426_ID_Orel__cong,axiom,
! [X: $o,Ya: $o,Y: nat,Xa: nat,R: $o > nat > $o,Ra: $o > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: $o,Yb: nat] :
( ( member_o @ Z2 @ ( insert_o @ Ya @ bot_bot_set_o ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Xa @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_o_nat_o @ R @ X @ Y )
= ( bNF_id_bnf_o_nat_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6427_ID_Orel__cong,axiom,
! [X: $o,Ya: $o,Y: int,Xa: int,R: $o > int > $o,Ra: $o > int > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: $o,Yb: int] :
( ( member_o @ Z2 @ ( insert_o @ Ya @ bot_bot_set_o ) )
=> ( ( member_int @ Yb @ ( insert_int @ Xa @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_o_int_o @ R @ X @ Y )
= ( bNF_id_bnf_o_int_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6428_ID_Orel__cong,axiom,
! [X: nat,Ya: nat,Y: $o,Xa: $o,R: nat > $o > $o,Ra: nat > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: nat,Yb: $o] :
( ( member_nat @ Z2 @ ( insert_nat @ Ya @ bot_bot_set_nat ) )
=> ( ( member_o @ Yb @ ( insert_o @ Xa @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_nat_o_o @ R @ X @ Y )
= ( bNF_id_bnf_nat_o_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6429_ID_Orel__cong,axiom,
! [X: nat,Ya: nat,Y: nat,Xa: nat,R: nat > nat > $o,Ra: nat > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: nat,Yb: nat] :
( ( member_nat @ Z2 @ ( insert_nat @ Ya @ bot_bot_set_nat ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Xa @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_nat_nat_o @ R @ X @ Y )
= ( bNF_id_bnf_nat_nat_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6430_ID_Orel__cong,axiom,
! [X: nat,Ya: nat,Y: int,Xa: int,R: nat > int > $o,Ra: nat > int > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: nat,Yb: int] :
( ( member_nat @ Z2 @ ( insert_nat @ Ya @ bot_bot_set_nat ) )
=> ( ( member_int @ Yb @ ( insert_int @ Xa @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_nat_int_o @ R @ X @ Y )
= ( bNF_id_bnf_nat_int_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6431_ID_Orel__cong,axiom,
! [X: int,Ya: int,Y: $o,Xa: $o,R: int > $o > $o,Ra: int > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: int,Yb: $o] :
( ( member_int @ Z2 @ ( insert_int @ Ya @ bot_bot_set_int ) )
=> ( ( member_o @ Yb @ ( insert_o @ Xa @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_int_o_o @ R @ X @ Y )
= ( bNF_id_bnf_int_o_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6432_ID_Orel__cong,axiom,
! [X: int,Ya: int,Y: nat,Xa: nat,R: int > nat > $o,Ra: int > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: int,Yb: nat] :
( ( member_int @ Z2 @ ( insert_int @ Ya @ bot_bot_set_int ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Xa @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_int_nat_o @ R @ X @ Y )
= ( bNF_id_bnf_int_nat_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6433_ID_Orel__cong,axiom,
! [X: int,Ya: int,Y: int,Xa: int,R: int > int > $o,Ra: int > int > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: int,Yb: int] :
( ( member_int @ Z2 @ ( insert_int @ Ya @ bot_bot_set_int ) )
=> ( ( member_int @ Yb @ ( insert_int @ Xa @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id_bnf_int_int_o @ R @ X @ Y )
= ( bNF_id_bnf_int_int_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6434_ID_Orel__cong,axiom,
! [X: set_nat,Ya: set_nat,Y: $o,Xa: $o,R: set_nat > $o > $o,Ra: set_nat > $o > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z2: set_nat,Yb: $o] :
( ( member_set_nat @ Z2 @ ( insert_set_nat @ Ya @ bot_bot_set_set_nat ) )
=> ( ( member_o @ Yb @ ( insert_o @ Xa @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( bNF_id1382368747416007662at_o_o @ R @ X @ Y )
= ( bNF_id1382368747416007662at_o_o @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6435_ID_Orel__mono__strong,axiom,
! [R: $o > $o > $o,X: $o,Y: $o,Ra: $o > $o > $o] :
( ( bNF_id_bnf_o_o_o @ R @ X @ Y )
=> ( ! [Z2: $o,Yb: $o] :
( ( member_o @ Z2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( member_o @ Yb @ ( insert_o @ Y @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_o_o_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6436_ID_Orel__mono__strong,axiom,
! [R: $o > nat > $o,X: $o,Y: nat,Ra: $o > nat > $o] :
( ( bNF_id_bnf_o_nat_o @ R @ X @ Y )
=> ( ! [Z2: $o,Yb: nat] :
( ( member_o @ Z2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Y @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_o_nat_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6437_ID_Orel__mono__strong,axiom,
! [R: $o > int > $o,X: $o,Y: int,Ra: $o > int > $o] :
( ( bNF_id_bnf_o_int_o @ R @ X @ Y )
=> ( ! [Z2: $o,Yb: int] :
( ( member_o @ Z2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( member_int @ Yb @ ( insert_int @ Y @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_o_int_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6438_ID_Orel__mono__strong,axiom,
! [R: nat > $o > $o,X: nat,Y: $o,Ra: nat > $o > $o] :
( ( bNF_id_bnf_nat_o_o @ R @ X @ Y )
=> ( ! [Z2: nat,Yb: $o] :
( ( member_nat @ Z2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( member_o @ Yb @ ( insert_o @ Y @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_nat_o_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6439_ID_Orel__mono__strong,axiom,
! [R: nat > nat > $o,X: nat,Y: nat,Ra: nat > nat > $o] :
( ( bNF_id_bnf_nat_nat_o @ R @ X @ Y )
=> ( ! [Z2: nat,Yb: nat] :
( ( member_nat @ Z2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Y @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_nat_nat_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6440_ID_Orel__mono__strong,axiom,
! [R: nat > int > $o,X: nat,Y: int,Ra: nat > int > $o] :
( ( bNF_id_bnf_nat_int_o @ R @ X @ Y )
=> ( ! [Z2: nat,Yb: int] :
( ( member_nat @ Z2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( member_int @ Yb @ ( insert_int @ Y @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_nat_int_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6441_ID_Orel__mono__strong,axiom,
! [R: int > $o > $o,X: int,Y: $o,Ra: int > $o > $o] :
( ( bNF_id_bnf_int_o_o @ R @ X @ Y )
=> ( ! [Z2: int,Yb: $o] :
( ( member_int @ Z2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( member_o @ Yb @ ( insert_o @ Y @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_int_o_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6442_ID_Orel__mono__strong,axiom,
! [R: int > nat > $o,X: int,Y: nat,Ra: int > nat > $o] :
( ( bNF_id_bnf_int_nat_o @ R @ X @ Y )
=> ( ! [Z2: int,Yb: nat] :
( ( member_int @ Z2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( member_nat @ Yb @ ( insert_nat @ Y @ bot_bot_set_nat ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_int_nat_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6443_ID_Orel__mono__strong,axiom,
! [R: int > int > $o,X: int,Y: int,Ra: int > int > $o] :
( ( bNF_id_bnf_int_int_o @ R @ X @ Y )
=> ( ! [Z2: int,Yb: int] :
( ( member_int @ Z2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( member_int @ Yb @ ( insert_int @ Y @ bot_bot_set_int ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id_bnf_int_int_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6444_ID_Orel__mono__strong,axiom,
! [R: set_nat > $o > $o,X: set_nat,Y: $o,Ra: set_nat > $o > $o] :
( ( bNF_id1382368747416007662at_o_o @ R @ X @ Y )
=> ( ! [Z2: set_nat,Yb: $o] :
( ( member_set_nat @ Z2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ( member_o @ Yb @ ( insert_o @ Y @ bot_bot_set_o ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( bNF_id1382368747416007662at_o_o @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6445_ID_Orel__refl__strong,axiom,
! [X: produc3843707927480180839at_nat,Ra: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
( ! [Z2: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ Z2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id5557877041440844106_nat_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6446_ID_Orel__refl__strong,axiom,
! [X: set_Pr958786334691620121nt_int,Ra: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o] :
( ! [Z2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Z2 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id745042670895148362_int_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6447_ID_Orel__refl__strong,axiom,
! [X: set_nat,Ra: set_nat > set_nat > $o] :
( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id293821497076346774_nat_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6448_ID_Orel__refl__strong,axiom,
! [X: produc3658429121746597890et_nat > $o,Ra: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
( ! [Z2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Z2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id2059982186938784458at_o_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6449_ID_Orel__refl__strong,axiom,
! [X: product_prod_nat_nat,Ra: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [Z2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id146787821168362314_nat_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6450_ID_Orel__refl__strong,axiom,
! [X: $o,Ra: $o > $o > $o] :
( ! [Z2: $o] :
( ( member_o @ Z2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id_bnf_o_o_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6451_ID_Orel__refl__strong,axiom,
! [X: nat,Ra: nat > nat > $o] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id_bnf_nat_nat_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6452_ID_Orel__refl__strong,axiom,
! [X: int,Ra: int > int > $o] :
( ! [Z2: int] :
( ( member_int @ Z2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( bNF_id_bnf_int_int_o @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6453_singleton__inject,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
= ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6454_singleton__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6455_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6456_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6457_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int @ A @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_6458_insert__not__empty,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( insert9069300056098147895at_nat @ A @ A2 )
!= bot_bo228742789529271731at_nat ) ).
% insert_not_empty
thf(fact_6459_insert__not__empty,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( insert8211810215607154385at_nat @ A @ A2 )
!= bot_bo2099793752762293965at_nat ) ).
% insert_not_empty
thf(fact_6460_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_6461_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_6462_insert__not__empty,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ A2 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_6463_doubleton__eq__iff,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
= ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D2 @ bot_bo228742789529271731at_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6464_doubleton__eq__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D2 @ bot_bo2099793752762293965at_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6465_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D2: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6466_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6467_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_6468_singleton__iff,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6469_singleton__iff,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ B @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6470_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6471_singleton__iff,axiom,
! [B: produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6472_singleton__iff,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6473_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6474_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6475_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_6476_singletonD,axiom,
! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6477_singletonD,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ B @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6478_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6479_singletonD,axiom,
! [B: produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6480_singletonD,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6481_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6482_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6483_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_6484_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N2: nat,K4: int] : ( bit_se6526347334894502574or_int @ K4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_6485_insert__UNIV,axiom,
! [X: produc3843707927480180839at_nat] :
( ( insert9069300056098147895at_nat @ X @ top_to6833984726390702231at_nat )
= top_to6833984726390702231at_nat ) ).
% insert_UNIV
thf(fact_6486_insert__UNIV,axiom,
! [X: product_prod_nat_nat] :
( ( insert8211810215607154385at_nat @ X @ top_to4669805908274784177at_nat )
= top_to4669805908274784177at_nat ) ).
% insert_UNIV
thf(fact_6487_insert__UNIV,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ top_top_set_list_nat )
= top_top_set_list_nat ) ).
% insert_UNIV
thf(fact_6488_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_6489_insert__UNIV,axiom,
! [X: rat] :
( ( insert_rat @ X @ top_top_set_rat )
= top_top_set_rat ) ).
% insert_UNIV
thf(fact_6490_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_6491_insert__UNIV,axiom,
! [X: int] :
( ( insert_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% insert_UNIV
thf(fact_6492_Int__insert__right,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6493_Int__insert__right,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6494_Int__insert__right,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( inf_in8396524679539076455nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ B2 ) )
= ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) )
& ( ~ ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( inf_in8396524679539076455nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ B2 ) )
= ( inf_in8396524679539076455nt_int @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6495_Int__insert__right,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6496_Int__insert__right,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( ( member_int @ A @ A2 )
=> ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6497_Int__insert__right,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ B2 ) )
= ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ B2 ) )
= ( inf_in1906310914598751387_nat_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6498_Int__insert__right,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6499_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_6500_Int__insert__left,axiom,
! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( member8757157785044589968at_nat @ A @ C2 )
=> ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B2 ) @ C2 )
= ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member8757157785044589968at_nat @ A @ C2 )
=> ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B2 ) @ C2 )
= ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6501_Int__insert__left,axiom,
! [A: $o,C2: set_o,B2: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C2 )
= ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6502_Int__insert__left,axiom,
! [A: set_Pr958786334691620121nt_int,C2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( ( member2340774599025711042nt_int @ A @ C2 )
=> ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B2 ) @ C2 )
= ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ B2 @ C2 ) ) ) )
& ( ~ ( member2340774599025711042nt_int @ A @ C2 )
=> ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B2 ) @ C2 )
= ( inf_in8396524679539076455nt_int @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6503_Int__insert__left,axiom,
! [A: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C2 )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_set_nat @ A @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6504_Int__insert__left,axiom,
! [A: int,C2: set_int,B2: set_int] :
( ( ( member_int @ A @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C2 )
= ( insert_int @ A @ ( inf_inf_set_int @ B2 @ C2 ) ) ) )
& ( ~ ( member_int @ A @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C2 )
= ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6505_Int__insert__left,axiom,
! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ A @ C2 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B2 ) @ C2 )
= ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B2 @ C2 ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A @ C2 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B2 ) @ C2 )
= ( inf_in1906310914598751387_nat_o @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6506_Int__insert__left,axiom,
! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B2 ) @ C2 )
= ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ C2 )
=> ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B2 ) @ C2 )
= ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6507_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_6508_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M2: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_6509_push__bit__minus,axiom,
! [N: nat,A: code_integer] :
( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ A ) )
= ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N @ A ) ) ) ).
% push_bit_minus
thf(fact_6510_push__bit__minus,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N @ A ) ) ) ).
% push_bit_minus
thf(fact_6511_Collect__conv__if2,axiom,
! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
( ( ( P @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo228742789529271731at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6512_Collect__conv__if2,axiom,
! [P: set_Pr958786334691620121nt_int > $o,A: set_Pr958786334691620121nt_int] :
( ( ( P @ A )
=> ( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
& ( ~ ( P @ A )
=> ( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo1488462491386950373nt_int ) ) ) ).
% Collect_conv_if2
thf(fact_6513_Collect__conv__if2,axiom,
! [P: set_nat > $o,A: set_nat] :
( ( ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6514_Collect__conv__if2,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,A: produc3658429121746597890et_nat > $o] :
( ( ( P @ A )
=> ( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
& ( ~ ( P @ A )
=> ( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% Collect_conv_if2
thf(fact_6515_Collect__conv__if2,axiom,
! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6516_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_6517_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_6518_Collect__conv__if2,axiom,
! [P: int > $o,A: int] :
( ( ( P @ A )
=> ( ( collect_int
@ ^ [X3: int] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_int @ A @ bot_bot_set_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_int
@ ^ [X3: int] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_int ) ) ) ).
% Collect_conv_if2
thf(fact_6519_Collect__conv__if,axiom,
! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
( ( ( P @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo228742789529271731at_nat ) ) ) ).
% Collect_conv_if
thf(fact_6520_Collect__conv__if,axiom,
! [P: set_Pr958786334691620121nt_int > $o,A: set_Pr958786334691620121nt_int] :
( ( ( P @ A )
=> ( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
& ( ~ ( P @ A )
=> ( ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo1488462491386950373nt_int ) ) ) ).
% Collect_conv_if
thf(fact_6521_Collect__conv__if,axiom,
! [P: set_nat > $o,A: set_nat] :
( ( ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_6522_Collect__conv__if,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,A: produc3658429121746597890et_nat > $o] :
( ( ( P @ A )
=> ( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
& ( ~ ( P @ A )
=> ( ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% Collect_conv_if
thf(fact_6523_Collect__conv__if,axiom,
! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_conv_if
thf(fact_6524_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_6525_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_6526_Collect__conv__if,axiom,
! [P: int > $o,A: int] :
( ( ( P @ A )
=> ( ( collect_int
@ ^ [X3: int] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_int @ A @ bot_bot_set_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_int
@ ^ [X3: int] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_int ) ) ) ).
% Collect_conv_if
thf(fact_6527_insert__def,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat] :
( sup_su6327502436637775413at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6528_insert__def,axiom,
( insert_o
= ( ^ [A3: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X3: $o] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6529_insert__def,axiom,
( insert8897473484851387113nt_int
= ( ^ [A3: set_Pr958786334691620121nt_int] :
( sup_su2047564715030645325nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6530_insert__def,axiom,
( insert_set_nat
= ( ^ [A3: set_nat] :
( sup_sup_set_set_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6531_insert__def,axiom,
( insert_int
= ( ^ [A3: int] :
( sup_sup_set_int
@ ( collect_int
@ ^ [X3: int] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6532_insert__def,axiom,
( insert5175938949040314269_nat_o
= ( ^ [A3: produc3658429121746597890et_nat > $o] :
( sup_su5209123915105501825_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6533_insert__def,axiom,
( insert6337962749363155127at_nat
= ( ^ [A3: produc4166570645942440679at_nat] :
( sup_su3035147773818789531at_nat
@ ( collec5204685387357076818at_nat
@ ^ [X3: produc4166570645942440679at_nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6534_insert__def,axiom,
( insert9069300056098147895at_nat
= ( ^ [A3: produc3843707927480180839at_nat] :
( sup_su5525570899277871387at_nat
@ ( collec6321179662152712658at_nat
@ ^ [X3: produc3843707927480180839at_nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6535_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_6536_subset__singletonD,axiom,
! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
=> ( ( A2 = bot_bo228742789529271731at_nat )
| ( A2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6537_subset__singletonD,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
=> ( ( A2 = bot_bo2099793752762293965at_nat )
| ( A2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6538_subset__singletonD,axiom,
! [A2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_6539_subset__singletonD,axiom,
! [A2: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( A2 = bot_bot_set_int )
| ( A2
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_6540_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_6541_subset__singleton__iff,axiom,
! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
= ( ( X7 = bot_bo228742789529271731at_nat )
| ( X7
= ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6542_subset__singleton__iff,axiom,
! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( ( X7 = bot_bo2099793752762293965at_nat )
| ( X7
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6543_subset__singleton__iff,axiom,
! [X7: set_o,A: $o] :
( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X7 = bot_bot_set_o )
| ( X7
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_6544_subset__singleton__iff,axiom,
! [X7: set_int,A: int] :
( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
= ( ( X7 = bot_bot_set_int )
| ( X7
= ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_6545_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_6546_singleton__Un__iff,axiom,
! [X: produc4166570645942440679at_nat,A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat] :
( ( ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat )
= ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo8422036546324065075at_nat )
& ( B2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) )
| ( ( A2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
& ( B2 = bot_bo8422036546324065075at_nat ) )
| ( ( A2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
& ( B2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6547_singleton__Un__iff,axiom,
! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
= ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo228742789529271731at_nat )
& ( B2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
| ( ( A2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
& ( B2 = bot_bo228742789529271731at_nat ) )
| ( ( A2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
& ( B2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6548_singleton__Un__iff,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
= ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo2099793752762293965at_nat )
& ( B2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
& ( B2 = bot_bo2099793752762293965at_nat ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
& ( B2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6549_singleton__Un__iff,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6550_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6551_singleton__Un__iff,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( ( insert_int @ X @ bot_bot_set_int )
= ( sup_sup_set_int @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_int )
& ( B2
= ( insert_int @ X @ bot_bot_set_int ) ) )
| ( ( A2
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B2 = bot_bot_set_int ) )
| ( ( A2
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B2
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_6552_Un__singleton__iff,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,X: produc4166570645942440679at_nat] :
( ( ( sup_su3035147773818789531at_nat @ A2 @ B2 )
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
= ( ( ( A2 = bot_bo8422036546324065075at_nat )
& ( B2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) )
| ( ( A2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
& ( B2 = bot_bo8422036546324065075at_nat ) )
| ( ( A2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
& ( B2
= ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6553_Un__singleton__iff,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
( ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
= ( ( ( A2 = bot_bo228742789529271731at_nat )
& ( B2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
| ( ( A2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
& ( B2 = bot_bo228742789529271731at_nat ) )
| ( ( A2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
& ( B2
= ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6554_Un__singleton__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ( sup_su6327502436637775413at_nat @ A2 @ B2 )
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= ( ( ( A2 = bot_bo2099793752762293965at_nat )
& ( B2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
& ( B2 = bot_bo2099793752762293965at_nat ) )
| ( ( A2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
& ( B2
= ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6555_Un__singleton__iff,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6556_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6557_Un__singleton__iff,axiom,
! [A2: set_int,B2: set_int,X: int] :
( ( ( sup_sup_set_int @ A2 @ B2 )
= ( insert_int @ X @ bot_bot_set_int ) )
= ( ( ( A2 = bot_bot_set_int )
& ( B2
= ( insert_int @ X @ bot_bot_set_int ) ) )
| ( ( A2
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B2 = bot_bot_set_int ) )
| ( ( A2
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B2
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_6558_insert__is__Un,axiom,
( insert6337962749363155127at_nat
= ( ^ [A3: produc4166570645942440679at_nat] : ( sup_su3035147773818789531at_nat @ ( insert6337962749363155127at_nat @ A3 @ bot_bo8422036546324065075at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_6559_insert__is__Un,axiom,
( insert9069300056098147895at_nat
= ( ^ [A3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_6560_insert__is__Un,axiom,
( insert8211810215607154385at_nat
= ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% insert_is_Un
thf(fact_6561_insert__is__Un,axiom,
( insert_o
= ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_6562_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_6563_insert__is__Un,axiom,
( insert_int
= ( ^ [A3: int] : ( sup_sup_set_int @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ).
% insert_is_Un
thf(fact_6564_set__minus__singleton__eq,axiom,
! [X: produc3843707927480180839at_nat,X7: set_Pr4329608150637261639at_nat] :
( ~ ( member8757157785044589968at_nat @ X @ X7 )
=> ( ( minus_3314409938677909166at_nat @ X7 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6565_set__minus__singleton__eq,axiom,
! [X: set_Pr958786334691620121nt_int,X7: set_se6260736226359567993nt_int] :
( ~ ( member2340774599025711042nt_int @ X @ X7 )
=> ( ( minus_2612819937483484256nt_int @ X7 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6566_set__minus__singleton__eq,axiom,
! [X: set_nat,X7: set_set_nat] :
( ~ ( member_set_nat @ X @ X7 )
=> ( ( minus_2163939370556025621et_nat @ X7 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6567_set__minus__singleton__eq,axiom,
! [X: produc3658429121746597890et_nat > $o,X7: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ X7 )
=> ( ( minus_1801376950450012436_nat_o @ X7 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6568_set__minus__singleton__eq,axiom,
! [X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ X7 )
=> ( ( minus_1356011639430497352at_nat @ X7 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6569_set__minus__singleton__eq,axiom,
! [X: $o,X7: set_o] :
( ~ ( member_o @ X @ X7 )
=> ( ( minus_minus_set_o @ X7 @ ( insert_o @ X @ bot_bot_set_o ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6570_set__minus__singleton__eq,axiom,
! [X: int,X7: set_int] :
( ~ ( member_int @ X @ X7 )
=> ( ( minus_minus_set_int @ X7 @ ( insert_int @ X @ bot_bot_set_int ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6571_set__minus__singleton__eq,axiom,
! [X: nat,X7: set_nat] :
( ~ ( member_nat @ X @ X7 )
=> ( ( minus_minus_set_nat @ X7 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X7 ) ) ).
% set_minus_singleton_eq
thf(fact_6572_insert__minus__eq,axiom,
! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( X != Y )
=> ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) )
= ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_6573_insert__minus__eq,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( X != Y )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_6574_insert__minus__eq,axiom,
! [X: $o,Y: $o,A2: set_o] :
( ( X != Y )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
= ( insert_o @ X @ ( minus_minus_set_o @ A2 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).
% insert_minus_eq
thf(fact_6575_insert__minus__eq,axiom,
! [X: int,Y: int,A2: set_int] :
( ( X != Y )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
= ( insert_int @ X @ ( minus_minus_set_int @ A2 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).
% insert_minus_eq
thf(fact_6576_insert__minus__eq,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( X != Y )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_6577_Diff__insert,axiom,
! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% Diff_insert
thf(fact_6578_Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% Diff_insert
thf(fact_6579_Diff__insert,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_6580_Diff__insert,axiom,
! [A2: set_int,A: int,B2: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_6581_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_6582_insert__Diff,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6583_insert__Diff,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( insert8897473484851387113nt_int @ A @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6584_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6585_insert__Diff,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( insert5175938949040314269_nat_o @ A @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6586_insert__Diff,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6587_insert__Diff,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6588_insert__Diff,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6589_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_6590_Diff__insert2,axiom,
! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_6591_Diff__insert2,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_6592_Diff__insert2,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_6593_Diff__insert2,axiom,
! [A2: set_int,A: int,B2: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_6594_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_6595_Diff__insert__absorb,axiom,
! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ~ ( member8757157785044589968at_nat @ X @ A2 )
=> ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6596_Diff__insert__absorb,axiom,
! [X: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int] :
( ~ ( member2340774599025711042nt_int @ X @ A2 )
=> ( ( minus_2612819937483484256nt_int @ ( insert8897473484851387113nt_int @ X @ A2 ) @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6597_Diff__insert__absorb,axiom,
! [X: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6598_Diff__insert__absorb,axiom,
! [X: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A2 ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6599_Diff__insert__absorb,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ A2 )
=> ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6600_Diff__insert__absorb,axiom,
! [X: $o,A2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6601_Diff__insert__absorb,axiom,
! [X: int,A2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6602_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_6603_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > $o,B: $o] :
( ( member_nat @ A @ ( vimage_nat_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6604_vimage__singleton__eq,axiom,
! [A: int,F2: int > $o,B: $o] :
( ( member_int @ A @ ( vimage_int_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6605_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6606_vimage__singleton__eq,axiom,
! [A: int,F2: int > nat,B: nat] :
( ( member_int @ A @ ( vimage_int_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6607_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > int,B: int] :
( ( member_nat @ A @ ( vimage_nat_int @ F2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6608_vimage__singleton__eq,axiom,
! [A: int,F2: int > int,B: int] :
( ( member_int @ A @ ( vimage_int_int @ F2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6609_vimage__singleton__eq,axiom,
! [A: set_nat,F2: set_nat > $o,B: $o] :
( ( member_set_nat @ A @ ( vimage_set_nat_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6610_vimage__singleton__eq,axiom,
! [A: set_nat,F2: set_nat > nat,B: nat] :
( ( member_set_nat @ A @ ( vimage_set_nat_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6611_vimage__singleton__eq,axiom,
! [A: set_nat,F2: set_nat > int,B: int] :
( ( member_set_nat @ A @ ( vimage_set_nat_int @ F2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6612_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > product_prod_nat_nat,B: product_prod_nat_nat] :
( ( member_nat @ A @ ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_6613_subset__insert__iff,axiom,
! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
= ( ( ( member8757157785044589968at_nat @ X @ A2 )
=> ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ A2 )
=> ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6614_subset__insert__iff,axiom,
! [A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
( ( ord_le483042692224249369nt_int @ A2 @ ( insert8897473484851387113nt_int @ X @ B2 ) )
= ( ( ( member2340774599025711042nt_int @ X @ A2 )
=> ( ord_le483042692224249369nt_int @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ B2 ) )
& ( ~ ( member2340774599025711042nt_int @ X @ A2 )
=> ( ord_le483042692224249369nt_int @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6615_subset__insert__iff,axiom,
! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
= ( ( ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6616_subset__insert__iff,axiom,
! [A2: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B2: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ X @ B2 ) )
= ( ( ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B2 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ord_le2965882846123202637_nat_o @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6617_subset__insert__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
= ( ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6618_subset__insert__iff,axiom,
! [A2: set_o,X: $o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6619_subset__insert__iff,axiom,
! [A2: set_int,X: int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
= ( ( ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6620_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_6621_Diff__single__insert,axiom,
! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 )
=> ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_6622_Diff__single__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 )
=> ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_6623_Diff__single__insert,axiom,
! [A2: set_o,X: $o,B2: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_6624_Diff__single__insert,axiom,
! [A2: set_int,X: int,B2: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_6625_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_6626_remove__subset,axiom,
! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ X @ S )
=> ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_6627_remove__subset,axiom,
! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ X @ S )
=> ( ord_le1924305788584680229nt_int @ ( minus_2612819937483484256nt_int @ S @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ S ) ) ).
% remove_subset
thf(fact_6628_remove__subset,axiom,
! [X: set_nat,S: set_set_nat] :
( ( member_set_nat @ X @ S )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_6629_remove__subset,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ S )
=> ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).
% remove_subset
thf(fact_6630_remove__subset,axiom,
! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ S )
=> ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_6631_remove__subset,axiom,
! [X: $o,S: set_o] :
( ( member_o @ X @ S )
=> ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).
% remove_subset
thf(fact_6632_remove__subset,axiom,
! [X: int,S: set_int] :
( ( member_int @ X @ S )
=> ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).
% remove_subset
thf(fact_6633_remove__subset,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_6634_vimage__insert,axiom,
! [F2: nat > $o,A: $o,B2: set_o] :
( ( vimage_nat_o @ F2 @ ( insert_o @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( vimage_nat_o @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6635_vimage__insert,axiom,
! [F2: nat > nat,A: nat,B2: set_nat] :
( ( vimage_nat_nat @ F2 @ ( insert_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( vimage_nat_nat @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6636_vimage__insert,axiom,
! [F2: nat > int,A: int,B2: set_int] :
( ( vimage_nat_int @ F2 @ ( insert_int @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage_nat_int @ F2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ ( vimage_nat_int @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6637_vimage__insert,axiom,
! [F2: nat > product_prod_nat_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( vimage8013328719654469172at_nat @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6638_vimage__insert,axiom,
! [F2: nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( vimage6435164912253009178at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage6435164912253009178at_nat @ F2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( vimage6435164912253009178at_nat @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6639_vimage__insert,axiom,
! [F2: produc4166570645942440679at_nat > $o,A: $o,B2: set_o] :
( ( vimage2280073292226551852_nat_o @ F2 @ ( insert_o @ A @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage2280073292226551852_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( vimage2280073292226551852_nat_o @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6640_vimage__insert,axiom,
! [F2: produc3843707927480180839at_nat > $o,A: $o,B2: set_o] :
( ( vimage3307185822755684012_nat_o @ F2 @ ( insert_o @ A @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( vimage3307185822755684012_nat_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( vimage3307185822755684012_nat_o @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6641_vimage__insert,axiom,
! [F2: produc4166570645942440679at_nat > nat,A: nat,B2: set_nat] :
( ( vimage7952471809483217980at_nat @ F2 @ ( insert_nat @ A @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage7952471809483217980at_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( vimage7952471809483217980at_nat @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6642_vimage__insert,axiom,
! [F2: produc3843707927480180839at_nat > nat,A: nat,B2: set_nat] :
( ( vimage7134676753939176892at_nat @ F2 @ ( insert_nat @ A @ B2 ) )
= ( sup_su5525570899277871387at_nat @ ( vimage7134676753939176892at_nat @ F2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( vimage7134676753939176892at_nat @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6643_vimage__insert,axiom,
! [F2: produc4166570645942440679at_nat > int,A: int,B2: set_int] :
( ( vimage7949981338974167704at_int @ F2 @ ( insert_int @ A @ B2 ) )
= ( sup_su3035147773818789531at_nat @ ( vimage7949981338974167704at_int @ F2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ ( vimage7949981338974167704at_int @ F2 @ B2 ) ) ) ).
% vimage_insert
thf(fact_6644_Compl__insert,axiom,
! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) )
= ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).
% Compl_insert
thf(fact_6645_Compl__insert,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) )
= ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% Compl_insert
thf(fact_6646_Compl__insert,axiom,
! [X: $o,A2: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_6647_Compl__insert,axiom,
! [X: int,A2: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_6648_Compl__insert,axiom,
! [X: nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_6649_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_6650_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_6651_flip__bit__eq__xor,axiom,
( bit_se1345352211410354436nteger
= ( ^ [N2: nat,A3: code_integer] : ( bit_se3222712562003087583nteger @ A3 @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_6652_flip__bit__eq__xor,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N2: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_6653_flip__bit__eq__xor,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [N2: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_6654_psubset__insert__iff,axiom,
! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B2 ) )
= ( ( ( member8757157785044589968at_nat @ X @ B2 )
=> ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ B2 )
=> ( ( ( member8757157785044589968at_nat @ X @ A2 )
=> ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
& ( ~ ( member8757157785044589968at_nat @ X @ A2 )
=> ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6655_psubset__insert__iff,axiom,
! [A2: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B2: set_se6260736226359567993nt_int] :
( ( ord_le1924305788584680229nt_int @ A2 @ ( insert8897473484851387113nt_int @ X @ B2 ) )
= ( ( ( member2340774599025711042nt_int @ X @ B2 )
=> ( ord_le1924305788584680229nt_int @ A2 @ B2 ) )
& ( ~ ( member2340774599025711042nt_int @ X @ B2 )
=> ( ( ( member2340774599025711042nt_int @ X @ A2 )
=> ( ord_le1924305788584680229nt_int @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ B2 ) )
& ( ~ ( member2340774599025711042nt_int @ X @ A2 )
=> ( ord_le483042692224249369nt_int @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6656_psubset__insert__iff,axiom,
! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
= ( ( ( member_set_nat @ X @ B2 )
=> ( ord_less_set_set_nat @ A2 @ B2 ) )
& ( ~ ( member_set_nat @ X @ B2 )
=> ( ( ( member_set_nat @ X @ A2 )
=> ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
& ( ~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6657_psubset__insert__iff,axiom,
! [A2: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B2: set_Pr4532377907799695533_nat_o] :
( ( ord_le2453136405763929_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ X @ B2 ) )
= ( ( ( member6576561426505652726_nat_o @ X @ B2 )
=> ( ord_le2453136405763929_nat_o @ A2 @ B2 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ B2 )
=> ( ( ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B2 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A2 )
=> ( ord_le2965882846123202637_nat_o @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6658_psubset__insert__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B2 ) )
= ( ( ( member8440522571783428010at_nat @ X @ B2 )
=> ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ B2 )
=> ( ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
& ( ~ ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6659_psubset__insert__iff,axiom,
! [A2: set_o,X: $o,B2: set_o] :
( ( ord_less_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ( ( member_o @ X @ B2 )
=> ( ord_less_set_o @ A2 @ B2 ) )
& ( ~ ( member_o @ X @ B2 )
=> ( ( ( member_o @ X @ A2 )
=> ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6660_psubset__insert__iff,axiom,
! [A2: set_int,X: int,B2: set_int] :
( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
= ( ( ( member_int @ X @ B2 )
=> ( ord_less_set_int @ A2 @ B2 ) )
& ( ~ ( member_int @ X @ B2 )
=> ( ( ( member_int @ X @ A2 )
=> ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6661_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_6662_push__bit__double,axiom,
! [N: nat,A: code_integer] :
( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_6663_push__bit__double,axiom,
! [N: nat,A: code_natural] :
( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
= ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_6664_push__bit__double,axiom,
! [N: nat,A: int] :
( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_6665_push__bit__double,axiom,
! [N: nat,A: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% push_bit_double
thf(fact_6666_push__bit__eq__mult,axiom,
( bit_se7788150548672797655nteger
= ( ^ [N2: nat,A3: code_integer] : ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_6667_push__bit__eq__mult,axiom,
( bit_se6611745700429515170atural
= ( ^ [N2: nat,A3: code_natural] : ( times_2397367101498566445atural @ A3 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_6668_push__bit__eq__mult,axiom,
( bit_se545348938243370406it_int
= ( ^ [N2: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_6669_push__bit__eq__mult,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N2: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_eq_mult
thf(fact_6670_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_6671_of__int__round__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_6672_of__int__round__ge,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_ge
thf(fact_6673_of__int__round__gt,axiom,
! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_gt
thf(fact_6674_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K4: int] :
( if_Code_integer @ ( ord_less_int @ K4 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K4 ) ) )
@ ( if_Code_integer @ ( K4 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_6675_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
=> ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_6676_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_6677_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_6678_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_6679_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% floor_le_neg_numeral
thf(fact_6680_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_less_floor
thf(fact_6681_floor__one,axiom,
( ( archim3151403230148437115or_rat @ one_one_rat )
= one_one_int ) ).
% floor_one
thf(fact_6682_ceiling__one,axiom,
( ( archim2889992004027027881ng_rat @ one_one_rat )
= one_one_int ) ).
% ceiling_one
thf(fact_6683_floor__uminus__of__int,axiom,
! [Z: int] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) )
= ( uminus_uminus_int @ Z ) ) ).
% floor_uminus_of_int
thf(fact_6684_zero__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% zero_less_floor
thf(fact_6685_floor__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_le_zero
thf(fact_6686_one__le__floor,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ one_one_rat @ X ) ) ).
% one_le_floor
thf(fact_6687_ceiling__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_less_one
thf(fact_6688_one__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% one_le_ceiling
thf(fact_6689_floor__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ one_one_rat ) ) ).
% floor_less_one
thf(fact_6690_ceiling__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% ceiling_le_one
thf(fact_6691_one__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ one_one_rat @ X ) ) ).
% one_less_ceiling
thf(fact_6692_floor__neg__numeral,axiom,
! [V: num] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% floor_neg_numeral
thf(fact_6693_ceiling__neg__numeral,axiom,
! [V: num] :
( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_neg_numeral
thf(fact_6694_ceiling__add__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_6695_floor__diff__one,axiom,
! [X: rat] :
( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
= ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).
% floor_diff_one
thf(fact_6696_ceiling__diff__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
= ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_6697_ceiling__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% ceiling_less_zero
thf(fact_6698_zero__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% zero_le_ceiling
thf(fact_6699_numeral__less__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_less_floor
thf(fact_6700_floor__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% floor_le_numeral
thf(fact_6701_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% ceiling_less_numeral
thf(fact_6702_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_6703_one__less__floor,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).
% one_less_floor
thf(fact_6704_floor__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% floor_le_one
thf(fact_6705_neg__numeral__le__floor,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_le_floor
thf(fact_6706_floor__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_6707_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_6708_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_6709_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y2: int] :
( ( X
= ( code_integer_of_int @ Y2 ) )
=> ~ ( member_int @ Y2 @ top_top_set_int ) ) ).
% integer_of_int_cases
thf(fact_6710_integer__of__int__induct,axiom,
! [P: code_integer > $o,X: code_integer] :
( ! [Y2: int] :
( ( member_int @ Y2 @ top_top_set_int )
=> ( P @ ( code_integer_of_int @ Y2 ) ) )
=> ( P @ X ) ) ).
% integer_of_int_induct
thf(fact_6711_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y ) )
= ( X = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_6712_uminus__integer__code_I1_J,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% uminus_integer_code(1)
thf(fact_6713_ceiling__def,axiom,
( archim2889992004027027881ng_rat
= ( ^ [X3: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X3 ) ) ) ) ) ).
% ceiling_def
thf(fact_6714_floor__minus,axiom,
! [X: rat] :
( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
= ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% floor_minus
thf(fact_6715_ceiling__minus,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
= ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).
% ceiling_minus
thf(fact_6716_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_6717_ceiling__diff__floor__le__1,axiom,
! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).
% ceiling_diff_floor_le_1
thf(fact_6718_ceiling__altdef,axiom,
( archim2889992004027027881ng_rat
= ( ^ [X3: rat] :
( if_int
@ ( X3
= ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
@ ( archim3151403230148437115or_rat @ X3 )
@ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% ceiling_altdef
thf(fact_6719_one__integer__def,axiom,
( one_one_Code_integer
= ( code_integer_of_int @ one_one_int ) ) ).
% one_integer_def
thf(fact_6720_one__add__floor,axiom,
! [X: rat] :
( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
= ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).
% one_add_floor
thf(fact_6721_floor__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
=> ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
=> ( ( archim3151403230148437115or_rat @ X )
= Z ) ) ) ).
% floor_unique
thf(fact_6722_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim3151403230148437115or_rat @ X )
= A )
= ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% floor_eq_iff
thf(fact_6723_floor__split,axiom,
! [P: int > $o,T3: rat] :
( ( P @ ( archim3151403230148437115or_rat @ T3 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T3 )
& ( ord_less_rat @ T3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
=> ( P @ I3 ) ) ) ) ).
% floor_split
thf(fact_6724_le__mult__floor,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% le_mult_floor
thf(fact_6725_less__floor__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% less_floor_iff
thf(fact_6726_floor__le__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
= ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% floor_le_iff
thf(fact_6727_floor__correct,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% floor_correct
thf(fact_6728_ceiling__correct,axiom,
! [X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).
% ceiling_correct
thf(fact_6729_ceiling__unique,axiom,
! [Z: int,X: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
=> ( ( archim2889992004027027881ng_rat @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_6730_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( ( archim2889992004027027881ng_rat @ X )
= A )
= ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
& ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_6731_ceiling__split,axiom,
! [P: int > $o,T3: rat] :
( ( P @ ( archim2889992004027027881ng_rat @ T3 ) )
= ( ! [I3: int] :
( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T3 )
& ( ord_less_eq_rat @ T3 @ ( ring_1_of_int_rat @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ).
% ceiling_split
thf(fact_6732_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_6733_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% ceiling_less_iff
thf(fact_6734_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).
% le_ceiling_iff
thf(fact_6735_floor__divide__lower,axiom,
! [Q4: rat,P4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% floor_divide_lower
thf(fact_6736_ceiling__divide__upper,axiom,
! [Q4: rat,P4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% ceiling_divide_upper
thf(fact_6737_floor__divide__upper,axiom,
! [Q4: rat,P4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).
% floor_divide_upper
thf(fact_6738_round__def,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% round_def
thf(fact_6739_ceiling__divide__lower,axiom,
! [Q4: rat,P4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q4 )
=> ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P4 ) ) ).
% ceiling_divide_lower
thf(fact_6740_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [K2: int,L3: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
=> ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_6741_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_6742_round__altdef,axiom,
( archim7778729529865785530nd_rat
= ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).
% round_altdef
thf(fact_6743_the__elem__eq,axiom,
! [X: produc3843707927480180839at_nat] :
( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
= X ) ).
% the_elem_eq
thf(fact_6744_the__elem__eq,axiom,
! [X: product_prod_nat_nat] :
( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
= X ) ).
% the_elem_eq
thf(fact_6745_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_6746_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_6747_the__elem__eq,axiom,
! [X: int] :
( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% the_elem_eq
thf(fact_6748_round__unique_H,axiom,
! [X: rat,N: int] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= N ) ) ).
% round_unique'
thf(fact_6749_of__int__round__abs__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_6750_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K4: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K4 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
@ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_6751_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K4 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_6752_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_6753_abs__idempotent,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_idempotent
thf(fact_6754_abs__idempotent,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_idempotent
thf(fact_6755_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_6756_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_6757_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_6758_abs__eq__0,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_6759_abs__eq__0,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_6760_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_6761_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_6762_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_6763_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_6764_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_add_abs
thf(fact_6765_abs__add__abs,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_add_abs
thf(fact_6766_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_6767_abs__mult__self__eq,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
= ( times_3573771949741848930nteger @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_6768_abs__mult__self__eq,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
= ( times_times_rat @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_6769_abs__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
= ( times_times_int @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_6770_abs__1,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_1
thf(fact_6771_abs__1,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_1
thf(fact_6772_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_6773_abs__minus,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus
thf(fact_6774_abs__minus,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus
thf(fact_6775_abs__minus,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus
thf(fact_6776_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_6777_abs__minus__cancel,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus_cancel
thf(fact_6778_abs__minus__cancel,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus_cancel
thf(fact_6779_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_6780_abs__le__zero__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
thf(fact_6781_abs__le__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
thf(fact_6782_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_6783_abs__le__self__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% abs_le_self_iff
thf(fact_6784_abs__le__self__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% abs_le_self_iff
thf(fact_6785_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_6786_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6787_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6788_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6789_zero__less__abs__iff,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
= ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
thf(fact_6790_zero__less__abs__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
thf(fact_6791_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_6792_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_neg_numeral
thf(fact_6793_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_neg_numeral
thf(fact_6794_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_neg_numeral
thf(fact_6795_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_6796_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_6797_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_6798_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6799_abs__power__minus,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6800_abs__power__minus,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6801_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_6802_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6803_abs__of__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6804_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6805_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_6806_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_6807_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_6808_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% abs_le_D1
thf(fact_6809_abs__le__D1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% abs_le_D1
thf(fact_6810_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_6811_abs__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_mult
thf(fact_6812_abs__mult,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_mult
thf(fact_6813_abs__mult,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_mult
thf(fact_6814_abs__one,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_one
thf(fact_6815_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_one
thf(fact_6816_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_6817_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_6818_abs__minus__commute,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_6819_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_6820_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6821_abs__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( abs_abs_Code_integer @ X )
= ( abs_abs_Code_integer @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6822_abs__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( abs_abs_rat @ X )
= ( abs_abs_rat @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_rat @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_6823_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_6824_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_6825_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_6826_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6827_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6828_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6829_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_6830_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_6831_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_6832_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6833_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6834_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6835_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6836_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6837_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6838_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6839_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6840_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6841_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6842_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6843_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6844_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D2 )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_6845_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D2 )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_6846_abs__mult__less,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D2 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_6847_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_6848_abs__ge__minus__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% abs_ge_minus_self
thf(fact_6849_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_6850_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le3102999989581377725nteger @ A @ B )
& ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6851_abs__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6852_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6853_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6854_abs__le__D2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6855_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6856_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6857_abs__leI,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6858_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6859_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6860_abs__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le6747313008572928689nteger @ A @ B )
& ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6861_abs__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_rat @ A @ B )
& ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6862_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_6863_frac__lt__1,axiom,
! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).
% frac_lt_1
thf(fact_6864_frac__1__eq,axiom,
! [X: rat] :
( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( archimedean_frac_rat @ X ) ) ).
% frac_1_eq
thf(fact_6865_dense__eq0__I,axiom,
! [X: rat] :
( ! [E3: rat] :
( ( ord_less_rat @ zero_zero_rat @ E3 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E3 ) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_6866_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
| ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
& ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
| ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6867_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( ord_less_eq_rat @ A @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B )
| ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6868_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6869_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6870_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
= ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6871_abs__mult__pos,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6872_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_6873_abs__minus__le__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% abs_minus_le_zero
thf(fact_6874_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_6875_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( abs_abs_Code_integer @ B ) )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ( B = A )
| ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6876_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A
= ( abs_abs_rat @ B ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_rat @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6877_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6878_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= B )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
& ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6879_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( ( abs_abs_rat @ A )
= B )
= ( ( ord_less_eq_rat @ zero_zero_rat @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6880_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6881_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6882_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6883_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6884_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6885_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6886_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6887_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_6888_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_6889_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_if_raw
thf(fact_6890_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_6891_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_6892_abs__of__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_neg
thf(fact_6893_abs__if,axiom,
( abs_abs_int
= ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_6894_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_6895_abs__if,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_if
thf(fact_6896_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_6897_abs__integer__code,axiom,
( abs_abs_Code_integer
= ( ^ [K4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K4 ) @ K4 ) ) ) ).
% abs_integer_code
thf(fact_6898_abs__add__one__gt__zero,axiom,
! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6899_abs__add__one__gt__zero,axiom,
! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6900_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6901_of__int__leD,axiom,
! [N: int,X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_leD
thf(fact_6902_of__int__leD,axiom,
! [N: int,X: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% of_int_leD
thf(fact_6903_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_6904_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_lessD
thf(fact_6905_of__int__lessD,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% of_int_lessD
thf(fact_6906_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_6907_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_6908_abs__square__eq__1,axiom,
! [X: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( abs_abs_Code_integer @ X )
= one_one_Code_integer ) ) ).
% abs_square_eq_1
thf(fact_6909_abs__square__eq__1,axiom,
! [X: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( abs_abs_rat @ X )
= one_one_rat ) ) ).
% abs_square_eq_1
thf(fact_6910_abs__square__eq__1,axiom,
! [X: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% abs_square_eq_1
thf(fact_6911_frac__eq,axiom,
! [X: rat] :
( ( ( archimedean_frac_rat @ X )
= X )
= ( ( ord_less_eq_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% frac_eq
thf(fact_6912_frac__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).
% frac_add
thf(fact_6913_abs__square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_le_1
thf(fact_6914_abs__square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_le_1
thf(fact_6915_abs__square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_le_1
thf(fact_6916_abs__square__less__1,axiom,
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_less_1
thf(fact_6917_abs__square__less__1,axiom,
! [X: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_less_1
thf(fact_6918_abs__square__less__1,axiom,
! [X: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_less_1
thf(fact_6919_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6920_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_6921_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_6922_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6923_floor__add,axiom,
! [X: rat,Y: rat] :
( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
& ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
=> ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_6924_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F2 @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6925_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K4 @ L2 ) @ ( modulo364778990260209775nteger @ K4 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_6926_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R4: code_integer,S2: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R4 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ S2 ) ) @ ( S2 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_6927_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K4: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K4 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_6928_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K4: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K4 ) ) )
@ ( if_int @ ( K4 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_6929_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K4: int,L2: int] :
( if_int
@ ( K4
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L2 )
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K4 )
@ ( if_int @ ( K4 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_6930_is__singleton__the__elem,axiom,
( is_sin2937591304547752795at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat] :
( A4
= ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A4 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_6931_is__singleton__the__elem,axiom,
( is_sin2850979758926227957at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat] :
( A4
= ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_6932_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
( A4
= ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_6933_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( A4
= ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_6934_is__singleton__the__elem,axiom,
( is_singleton_int
= ( ^ [A4: set_int] :
( A4
= ( insert_int @ ( the_elem_int @ A4 ) @ bot_bot_set_int ) ) ) ) ).
% is_singleton_the_elem
thf(fact_6935_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_6936_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_6937_of__int__sum,axiom,
! [F2: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F2 @ A2 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( ring_1_of_int_rat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_6938_of__int__sum,axiom,
! [F2: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( ring_1_of_int_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_sum
thf(fact_6939_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ one_one_Code_integer )
= one_one_int ) ).
% one_integer.rep_eq
thf(fact_6940_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
= ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_6941_is__singletonI,axiom,
! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).
% is_singletonI
thf(fact_6942_is__singletonI,axiom,
! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).
% is_singletonI
thf(fact_6943_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_6944_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_6945_is__singletonI,axiom,
! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% is_singletonI
thf(fact_6946_bit_Ocompl__zero,axiom,
( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.compl_zero
thf(fact_6947_bit_Ocompl__zero,axiom,
( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.compl_zero
thf(fact_6948_bit_Ocompl__one,axiom,
( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% bit.compl_one
thf(fact_6949_bit_Ocompl__one,axiom,
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% bit.compl_one
thf(fact_6950_bit_Oxor__cancel__right,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.xor_cancel_right
thf(fact_6951_bit_Oxor__cancel__right,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.xor_cancel_right
thf(fact_6952_bit_Oxor__cancel__left,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% bit.xor_cancel_left
thf(fact_6953_bit_Oxor__cancel__left,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
= ( uminus_uminus_int @ one_one_int ) ) ).
% bit.xor_cancel_left
thf(fact_6954_bit_Oxor__one__right,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_ri7632146776885996613nteger @ X ) ) ).
% bit.xor_one_right
thf(fact_6955_bit_Oxor__one__right,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ X ) ) ).
% bit.xor_one_right
thf(fact_6956_bit_Oxor__one__left,axiom,
! [X: code_integer] :
( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
= ( bit_ri7632146776885996613nteger @ X ) ) ).
% bit.xor_one_left
thf(fact_6957_bit_Oxor__one__left,axiom,
! [X: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
= ( bit_ri7919022796975470100ot_int @ X ) ) ).
% bit.xor_one_left
thf(fact_6958_minus__not__numeral__eq,axiom,
! [N: num] :
( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ).
% minus_not_numeral_eq
thf(fact_6959_minus__not__numeral__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( inc @ N ) ) ) ).
% minus_not_numeral_eq
thf(fact_6960_not__minus__numeral__eq,axiom,
! [N: num] :
( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( neg_nu5755505904847501662nteger @ N @ one ) ) ).
% not_minus_numeral_eq
thf(fact_6961_not__minus__numeral__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% not_minus_numeral_eq
thf(fact_6962_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] :
( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_6963_push__bit__minus__one__eq__not__mask,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_6964_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_6965_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_6966_not__one__eq,axiom,
( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_6967_not__one__eq,axiom,
( ( bit_ri7919022796975470100ot_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% not_one_eq
thf(fact_6968_int__of__integer,axiom,
! [X: code_integer] : ( member_int @ ( code_int_of_integer @ X ) @ top_top_set_int ) ).
% int_of_integer
thf(fact_6969_int__of__integer__cases,axiom,
! [Y: int] :
( ( member_int @ Y @ top_top_set_int )
=> ~ ! [X2: code_integer] :
( Y
!= ( code_int_of_integer @ X2 ) ) ) ).
% int_of_integer_cases
thf(fact_6970_int__of__integer__induct,axiom,
! [Y: int,P: int > $o] :
( ( member_int @ Y @ top_top_set_int )
=> ( ! [X2: code_integer] : ( P @ ( code_int_of_integer @ X2 ) )
=> ( P @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_6971_mod__sum__eq,axiom,
! [F2: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F2 @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_6972_mod__sum__eq,axiom,
! [F2: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F2 @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_6973_type__definition__integer,axiom,
type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).
% type_definition_integer
thf(fact_6974_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member_int @ Y @ top_top_set_int )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_6975_is__singletonI_H,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ( A2 != bot_bo1488462491386950373nt_int )
=> ( ! [X2: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( ( member2340774599025711042nt_int @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin6299389887212142093nt_int @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6976_is__singletonI_H,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_set_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6977_is__singletonI_H,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( A2 != bot_bo7824918357723582233_nat_o )
=> ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( ( member6576561426505652726_nat_o @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin5180296473474724033_nat_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6978_is__singletonI_H,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( A2 != bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ( member8440522571783428010at_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin2850979758926227957at_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6979_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o,Y2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ( member_o @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6980_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6981_is__singletonI_H,axiom,
! [A2: set_int] :
( ( A2 != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( member_int @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_int @ A2 ) ) ) ).
% is_singletonI'
thf(fact_6982_minus__eq__not__plus__1,axiom,
( uminus1351360451143612070nteger
= ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% minus_eq_not_plus_1
thf(fact_6983_minus__eq__not__plus__1,axiom,
( uminus_uminus_int
= ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).
% minus_eq_not_plus_1
thf(fact_6984_minus__eq__not__minus__1,axiom,
( uminus1351360451143612070nteger
= ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_6985_minus__eq__not__minus__1,axiom,
( uminus_uminus_int
= ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_6986_not__eq__complement,axiom,
( bit_ri7632146776885996613nteger
= ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).
% not_eq_complement
thf(fact_6987_not__eq__complement,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).
% not_eq_complement
thf(fact_6988_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K4: int] : ( minus_minus_int @ ( uminus_uminus_int @ K4 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_6989_and__not__numerals_I1_J,axiom,
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= zero_zero_int ) ).
% and_not_numerals(1)
thf(fact_6990_sum__power__add,axiom,
! [X: code_integer,M: nat,I4: set_nat] :
( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( power_8256067586552552935nteger @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I4 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_6991_sum__power__add,axiom,
! [X: rat,M: nat,I4: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I4 )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_6992_sum__power__add,axiom,
! [X: int,M: nat,I4: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
@ I4 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_6993_minus__numeral__inc__eq,axiom,
! [N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) )
= ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% minus_numeral_inc_eq
thf(fact_6994_minus__numeral__inc__eq,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ).
% minus_numeral_inc_eq
thf(fact_6995_unset__bit__int__def,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,K4: int] : ( bit_se725231765392027082nd_int @ K4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_6996_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% divmod_abs_code(6)
thf(fact_6997_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_6998_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_6999_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_7000_not__numeral__Bit0__eq,axiom,
! [N: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_7001_not__numeral__Bit0__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_7002_minus__numeral__eq__not__sub__one,axiom,
! [N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
= ( bit_ri7632146776885996613nteger @ ( neg_nu5755505904847501662nteger @ N @ one ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_7003_minus__numeral__eq__not__sub__one,axiom,
! [N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( bit_ri7919022796975470100ot_int @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_7004_not__numeral__BitM__eq,axiom,
! [N: num] :
( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_7005_not__numeral__BitM__eq,axiom,
! [N: num] :
( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_7006_unset__bit__eq__and__not,axiom,
( bit_se8260200283734997820nteger
= ( ^ [N2: nat,A3: code_integer] : ( bit_se3949692690581998587nteger @ A3 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_7007_unset__bit__eq__and__not,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_7008_is__singletonE,axiom,
! [A2: set_Pr4329608150637261639at_nat] :
( ( is_sin2937591304547752795at_nat @ A2 )
=> ~ ! [X2: produc3843707927480180839at_nat] :
( A2
!= ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ).
% is_singletonE
thf(fact_7009_is__singletonE,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( is_sin2850979758926227957at_nat @ A2 )
=> ~ ! [X2: product_prod_nat_nat] :
( A2
!= ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% is_singletonE
thf(fact_7010_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X2: $o] :
( A2
!= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_7011_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_7012_is__singletonE,axiom,
! [A2: set_int] :
( ( is_singleton_int @ A2 )
=> ~ ! [X2: int] :
( A2
!= ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).
% is_singletonE
thf(fact_7013_is__singleton__def,axiom,
( is_sin2937591304547752795at_nat
= ( ^ [A4: set_Pr4329608150637261639at_nat] :
? [X3: produc3843707927480180839at_nat] :
( A4
= ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% is_singleton_def
thf(fact_7014_is__singleton__def,axiom,
( is_sin2850979758926227957at_nat
= ( ^ [A4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( A4
= ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% is_singleton_def
thf(fact_7015_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
? [X3: $o] :
( A4
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_7016_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
? [X3: nat] :
( A4
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_7017_is__singleton__def,axiom,
( is_singleton_int
= ( ^ [A4: set_int] :
? [X3: int] :
( A4
= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% is_singleton_def
thf(fact_7018_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_7019_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_7020_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_7021_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_7022_minus__exp__eq__not__mask,axiom,
! [N: nat] :
( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_7023_minus__exp__eq__not__mask,axiom,
! [N: nat] :
( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_7024_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_7025_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7026_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
= ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7027_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural )
= ( groups6325495683096345652atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7028_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_7029_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_7030_sum__abs__ge__zero,axiom,
! [F2: int > int,A2: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F2 @ I3 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_7031_sum__abs,axiom,
! [F2: int > int,A2: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F2 @ I3 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_7032_sum_Oempty,axiom,
! [G2: $o > code_integer] :
( ( groups4406642042086082107nteger @ G2 @ bot_bot_set_o )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_7033_sum_Oempty,axiom,
! [G2: $o > rat] :
( ( groups7872700643590313910_o_rat @ G2 @ bot_bot_set_o )
= zero_zero_rat ) ).
% sum.empty
thf(fact_7034_sum_Oempty,axiom,
! [G2: $o > nat] :
( ( groups8507830703676809646_o_nat @ G2 @ bot_bot_set_o )
= zero_zero_nat ) ).
% sum.empty
thf(fact_7035_sum_Oempty,axiom,
! [G2: $o > int] :
( ( groups8505340233167759370_o_int @ G2 @ bot_bot_set_o )
= zero_zero_int ) ).
% sum.empty
thf(fact_7036_sum_Oempty,axiom,
! [G2: nat > code_integer] :
( ( groups7501900531339628137nteger @ G2 @ bot_bot_set_nat )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_7037_sum_Oempty,axiom,
! [G2: nat > rat] :
( ( groups2906978787729119204at_rat @ G2 @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_7038_sum_Oempty,axiom,
! [G2: nat > int] :
( ( groups3539618377306564664at_int @ G2 @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_7039_sum_Oempty,axiom,
! [G2: int > code_integer] :
( ( groups7873554091576472773nteger @ G2 @ bot_bot_set_int )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_7040_sum_Oempty,axiom,
! [G2: int > rat] :
( ( groups3906332499630173760nt_rat @ G2 @ bot_bot_set_int )
= zero_zero_rat ) ).
% sum.empty
thf(fact_7041_sum_Oempty,axiom,
! [G2: int > nat] :
( ( groups4541462559716669496nt_nat @ G2 @ bot_bot_set_int )
= zero_zero_nat ) ).
% sum.empty
thf(fact_7042_convex__sum__bound__le,axiom,
! [I4: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X @ I4 )
= one_one_Code_integer )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7043_convex__sum__bound__le,axiom,
! [I4: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X @ I4 )
= one_one_Code_integer )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7044_convex__sum__bound__le,axiom,
! [I4: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X @ I4 )
= one_one_rat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7045_convex__sum__bound__le,axiom,
! [I4: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ X @ I4 )
= one_one_rat )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups3906332499630173760nt_rat
@ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7046_convex__sum__bound__le,axiom,
! [I4: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups3539618377306564664at_int @ X @ I4 )
= one_one_int )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7047_convex__sum__bound__le,axiom,
! [I4: set_int,X: int > int,A: int > int,B: int,Delta: int] :
( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ X @ I4 )
= one_one_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7048_convex__sum__bound__le,axiom,
! [I4: set_set_nat,X: set_nat > code_integer,A: set_nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups9190459664516455967nteger @ X @ I4 )
= one_one_Code_integer )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups9190459664516455967nteger
@ ^ [I3: set_nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7049_convex__sum__bound__le,axiom,
! [I4: set_set_nat,X: set_nat > rat,A: set_nat > rat,B: rat,Delta: rat] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
=> ( ( ( groups7659867448343625626at_rat @ X @ I4 )
= one_one_rat )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups7659867448343625626at_rat
@ ^ [I3: set_nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7050_convex__sum__bound__le,axiom,
! [I4: set_set_nat,X: set_nat > int,A: set_nat > int,B: int,Delta: int] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
=> ( ( ( groups8292507037921071086at_int @ X @ I4 )
= one_one_int )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_int
@ ( abs_abs_int
@ ( minus_minus_int
@ ( groups8292507037921071086at_int
@ ^ [I3: set_nat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7051_convex__sum__bound__le,axiom,
! [I4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int > code_integer,A: set_Pr958786334691620121nt_int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
=> ( ( ( groups3738144136874984124nteger @ X @ I4 )
= one_one_Code_integer )
=> ( ! [I2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups3738144136874984124nteger
@ ^ [I3: set_Pr958786334691620121nt_int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_7052_abs__sum__abs,axiom,
! [F2: int > int,A2: set_int] :
( ( abs_abs_int
@ ( groups4538972089207619220nt_int
@ ^ [A3: int] : ( abs_abs_int @ ( F2 @ A3 ) )
@ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [A3: int] : ( abs_abs_int @ ( F2 @ A3 ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_7053_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_7054_sum_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu: int] : zero_zero_int
@ A2 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_7055_sum__diff1__nat,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat] :
( ( ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( groups3860910324918113789at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member8757157785044589968at_nat @ A @ A2 )
=> ( ( groups3860910324918113789at_nat @ F2 @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
= ( groups3860910324918113789at_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7056_sum__diff1__nat,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,F2: set_Pr958786334691620121nt_int > nat] :
( ( ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( groups185207323561075247nt_nat @ F2 @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
= ( minus_minus_nat @ ( groups185207323561075247nt_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member2340774599025711042nt_int @ A @ A2 )
=> ( ( groups185207323561075247nt_nat @ F2 @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
= ( groups185207323561075247nt_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7057_sum__diff1__nat,axiom,
! [A: set_nat,A2: set_set_nat,F2: set_nat > nat] :
( ( ( member_set_nat @ A @ A2 )
=> ( ( groups8294997508430121362at_nat @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( groups8294997508430121362at_nat @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( groups8294997508430121362at_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7058_sum__diff1__nat,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,F2: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( groups6742167275697360483_o_nat @ F2 @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
= ( minus_minus_nat @ ( groups6742167275697360483_o_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ( groups6742167275697360483_o_nat @ F2 @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
= ( groups6742167275697360483_o_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7059_sum__diff1__nat,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups977919841031483927at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups977919841031483927at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( groups977919841031483927at_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7060_sum__diff1__nat,axiom,
! [A: $o,A2: set_o,F2: $o > nat] :
( ( ( member_o @ A @ A2 )
=> ( ( groups8507830703676809646_o_nat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups8507830703676809646_o_nat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups8507830703676809646_o_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7061_sum__diff1__nat,axiom,
! [A: int,A2: set_int,F2: int > nat] :
( ( ( member_int @ A @ A2 )
=> ( ( groups4541462559716669496nt_nat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups4541462559716669496nt_nat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups4541462559716669496nt_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7062_sum__diff1__nat,axiom,
! [A: nat,A2: set_nat,F2: nat > nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3542108847815614940at_nat @ F2 @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_7063_sum_Oswap,axiom,
! [G2: nat > nat > nat,B2: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_7064_sum_Oswap,axiom,
! [G2: int > int > int,B2: set_int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [J3: int] :
( groups4538972089207619220nt_int
@ ^ [I3: int] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_7065_sum__mono,axiom,
! [K5: set_nat,F2: nat > rat,G2: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ K5 ) @ ( groups2906978787729119204at_rat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7066_sum__mono,axiom,
! [K5: set_int,F2: int > rat,G2: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ K5 ) @ ( groups3906332499630173760nt_rat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7067_sum__mono,axiom,
! [K5: set_int,F2: int > nat,G2: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ K5 ) @ ( groups4541462559716669496nt_nat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7068_sum__mono,axiom,
! [K5: set_nat,F2: nat > int,G2: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ K5 ) @ ( groups3539618377306564664at_int @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7069_sum__mono,axiom,
! [K5: set_nat,F2: nat > nat,G2: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ K5 ) @ ( groups3542108847815614940at_nat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7070_sum__mono,axiom,
! [K5: set_int,F2: int > int,G2: int > int] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F2 @ K5 ) @ ( groups4538972089207619220nt_int @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7071_sum__mono,axiom,
! [K5: set_set_nat,F2: set_nat > rat,G2: set_nat > rat] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups7659867448343625626at_rat @ F2 @ K5 ) @ ( groups7659867448343625626at_rat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7072_sum__mono,axiom,
! [K5: set_set_nat,F2: set_nat > nat,G2: set_nat > nat] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups8294997508430121362at_nat @ F2 @ K5 ) @ ( groups8294997508430121362at_nat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7073_sum__mono,axiom,
! [K5: set_set_nat,F2: set_nat > int,G2: set_nat > int] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups8292507037921071086at_int @ F2 @ K5 ) @ ( groups8292507037921071086at_int @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7074_sum__mono,axiom,
! [K5: set_se6260736226359567993nt_int,F2: set_Pr958786334691620121nt_int > rat,G2: set_Pr958786334691620121nt_int > rat] :
( ! [I2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups8773449300329355319nt_rat @ F2 @ K5 ) @ ( groups8773449300329355319nt_rat @ G2 @ K5 ) ) ) ).
% sum_mono
thf(fact_7075_sum_Odistrib,axiom,
! [G2: nat > nat,H3: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ A2 ) @ ( groups3542108847815614940at_nat @ H3 @ A2 ) ) ) ).
% sum.distrib
thf(fact_7076_sum_Odistrib,axiom,
! [G2: int > int,H3: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( plus_plus_int @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ A2 ) @ ( groups4538972089207619220nt_int @ H3 @ A2 ) ) ) ).
% sum.distrib
thf(fact_7077_sum__product,axiom,
! [F2: nat > nat,A2: set_nat,G2: nat > nat,B2: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F2 @ I3 ) @ ( G2 @ J3 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_7078_sum__product,axiom,
! [F2: int > int,A2: set_int,G2: int > int,B2: set_int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ B2 ) )
= ( groups4538972089207619220nt_int
@ ^ [I3: int] :
( groups4538972089207619220nt_int
@ ^ [J3: int] : ( times_times_int @ ( F2 @ I3 ) @ ( G2 @ J3 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_7079_sum__distrib__right,axiom,
! [F2: nat > nat,A2: set_nat,R3: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ R3 )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F2 @ N2 ) @ R3 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_7080_sum__distrib__right,axiom,
! [F2: int > int,A2: set_int,R3: int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ R3 )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ ( F2 @ N2 ) @ R3 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_7081_sum__distrib__left,axiom,
! [R3: nat,F2: nat > nat,A2: set_nat] :
( ( times_times_nat @ R3 @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R3 @ ( F2 @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_7082_sum__distrib__left,axiom,
! [R3: int,F2: int > int,A2: set_int] :
( ( times_times_int @ R3 @ ( groups4538972089207619220nt_int @ F2 @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ R3 @ ( F2 @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_7083_sum__subtractf,axiom,
! [F2: int > int,G2: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( minus_minus_int @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ A2 ) ) ) ).
% sum_subtractf
thf(fact_7084_sum__negf,axiom,
! [F2: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( uminus_uminus_int @ ( F2 @ X3 ) )
@ A2 )
= ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) ) ) ).
% sum_negf
thf(fact_7085_sum__subtractf__nat,axiom,
! [A2: set_se6260736226359567993nt_int,G2: set_Pr958786334691620121nt_int > nat,F2: set_Pr958786334691620121nt_int > nat] :
( ! [X2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups185207323561075247nt_nat
@ ^ [X3: set_Pr958786334691620121nt_int] : ( minus_minus_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups185207323561075247nt_nat @ F2 @ A2 ) @ ( groups185207323561075247nt_nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7086_sum__subtractf__nat,axiom,
! [A2: set_set_nat,G2: set_nat > nat,F2: set_nat > nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( minus_minus_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F2 @ A2 ) @ ( groups8294997508430121362at_nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7087_sum__subtractf__nat,axiom,
! [A2: set_int,G2: int > nat,F2: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( minus_minus_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7088_sum__subtractf__nat,axiom,
! [A2: set_Pr4532377907799695533_nat_o,G2: ( produc3658429121746597890et_nat > $o ) > nat,F2: ( produc3658429121746597890et_nat > $o ) > nat] :
( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups6742167275697360483_o_nat
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( minus_minus_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups6742167275697360483_o_nat @ F2 @ A2 ) @ ( groups6742167275697360483_o_nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7089_sum__subtractf__nat,axiom,
! [A2: set_nat,G2: nat > nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( G2 @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_7090_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ ( code_divmod_abs @ K4 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R4: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K4 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R4: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_7091_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_7092_arith__series__nat,axiom,
! [A: nat,D2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_7093_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ M @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_7094_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_7095_sum__power2,axiom,
! [K: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_7096_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_7097_apsnd__conv,axiom,
! [F2: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
( ( produc717174245515237944e_term @ F2 @ ( produc8603105652947943368nteger @ X @ Y ) )
= ( produc8603105652947943368nteger @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7098_apsnd__conv,axiom,
! [F2: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc8750854537940449737_nat_o @ F2 @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc5001842942810119800et_nat @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7099_apsnd__conv,axiom,
! [F2: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc7714581247149323085_nat_o @ F2 @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc2245416461498447860et_nat @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7100_apsnd__conv,axiom,
! [F2: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc1515462096303866701_nat_o @ F2 @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc5001842942810119800et_nat @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7101_apsnd__conv,axiom,
! [F2: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc969530845752564945_nat_o @ F2 @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc2245416461498447860et_nat @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7102_apsnd__conv,axiom,
! [F2: product_prod_int_int > product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
( ( produc4973430039190721449e_term @ F2 @ ( produc5700946648718959541nt_int @ X @ Y ) )
= ( produc5700946648718959541nt_int @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7103_apsnd__conv,axiom,
! [F2: product_prod_int_int > product_prod_int_int,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
( ( produc7011684061226097695e_term @ F2 @ ( produc4305682042979456191nt_int @ X @ Y ) )
= ( produc4305682042979456191nt_int @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7104_apsnd__conv,axiom,
! [F2: code_integer > code_integer,X: code_integer,Y: code_integer] :
( ( produc6499014454317279255nteger @ F2 @ ( produc1086072967326762835nteger @ X @ Y ) )
= ( produc1086072967326762835nteger @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_7105_atLeastatMost__empty__iff,axiom,
! [A: $o,B: $o] :
( ( ( set_or8904488021354931149Most_o @ A @ B )
= bot_bot_set_o )
= ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7106_atLeastatMost__empty__iff,axiom,
! [A: filter_nat,B: filter_nat] :
( ( ( set_or1955772592623580779er_nat @ A @ B )
= bot_bo498966703094740906er_nat )
= ( ~ ( ord_le2510731241096832064er_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7107_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7108_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7109_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7110_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7111_atLeastatMost__empty__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( set_or189985376899183464nteger @ A @ B )
= bot_bo3990330152332043303nteger )
= ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_7112_atLeastatMost__empty__iff2,axiom,
! [A: $o,B: $o] :
( ( bot_bot_set_o
= ( set_or8904488021354931149Most_o @ A @ B ) )
= ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7113_atLeastatMost__empty__iff2,axiom,
! [A: filter_nat,B: filter_nat] :
( ( bot_bo498966703094740906er_nat
= ( set_or1955772592623580779er_nat @ A @ B ) )
= ( ~ ( ord_le2510731241096832064er_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7114_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7115_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7116_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7117_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7118_atLeastatMost__empty__iff2,axiom,
! [A: code_integer,B: code_integer] :
( ( bot_bo3990330152332043303nteger
= ( set_or189985376899183464nteger @ A @ B ) )
= ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_7119_atLeastatMost__empty,axiom,
! [B: $o,A: $o] :
( ( ord_less_o @ B @ A )
=> ( ( set_or8904488021354931149Most_o @ A @ B )
= bot_bot_set_o ) ) ).
% atLeastatMost_empty
thf(fact_7120_atLeastatMost__empty,axiom,
! [B: literal,A: literal] :
( ( ord_less_literal @ B @ A )
=> ( ( set_or8436436507035938825iteral @ A @ B )
= bot_bot_set_literal ) ) ).
% atLeastatMost_empty
thf(fact_7121_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_7122_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_7123_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_7124_atLeastatMost__empty,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A )
=> ( ( set_or189985376899183464nteger @ A @ B )
= bot_bo3990330152332043303nteger ) ) ).
% atLeastatMost_empty
thf(fact_7125_atLeastLessThan__empty,axiom,
! [B: $o,A: $o] :
( ( ord_less_eq_o @ B @ A )
=> ( ( set_or7139685690850216873Than_o @ A @ B )
= bot_bot_set_o ) ) ).
% atLeastLessThan_empty
thf(fact_7126_atLeastLessThan__empty,axiom,
! [B: filter_nat,A: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A )
=> ( ( set_or1773934645810362255er_nat @ A @ B )
= bot_bo498966703094740906er_nat ) ) ).
% atLeastLessThan_empty
thf(fact_7127_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_7128_atLeastLessThan__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( set_or4029947393144176647an_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastLessThan_empty
thf(fact_7129_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_7130_atLeastLessThan__empty,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ( set_or8404916559141939852nteger @ A @ B )
= bot_bo3990330152332043303nteger ) ) ).
% atLeastLessThan_empty
thf(fact_7131_atLeastLessThan__empty,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( set_or4662586982721622107an_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastLessThan_empty
thf(fact_7132_atLeastLessThan__empty__iff,axiom,
! [A: $o,B: $o] :
( ( ( set_or7139685690850216873Than_o @ A @ B )
= bot_bot_set_o )
= ( ~ ( ord_less_o @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7133_atLeastLessThan__empty__iff,axiom,
! [A: literal,B: literal] :
( ( ( set_or4828121863581493221iteral @ A @ B )
= bot_bot_set_literal )
= ( ~ ( ord_less_literal @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7134_atLeastLessThan__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or4029947393144176647an_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7135_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7136_atLeastLessThan__empty__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( set_or8404916559141939852nteger @ A @ B )
= bot_bo3990330152332043303nteger )
= ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7137_atLeastLessThan__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_int @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_7138_atLeastLessThan__empty__iff2,axiom,
! [A: $o,B: $o] :
( ( bot_bot_set_o
= ( set_or7139685690850216873Than_o @ A @ B ) )
= ( ~ ( ord_less_o @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7139_atLeastLessThan__empty__iff2,axiom,
! [A: literal,B: literal] :
( ( bot_bot_set_literal
= ( set_or4828121863581493221iteral @ A @ B ) )
= ( ~ ( ord_less_literal @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7140_atLeastLessThan__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or4029947393144176647an_rat @ A @ B ) )
= ( ~ ( ord_less_rat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7141_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7142_atLeastLessThan__empty__iff2,axiom,
! [A: code_integer,B: code_integer] :
( ( bot_bo3990330152332043303nteger
= ( set_or8404916559141939852nteger @ A @ B ) )
= ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7143_atLeastLessThan__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or4662586982721622107an_int @ A @ B ) )
= ( ~ ( ord_less_int @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_7144_atLeastAtMost__singleton,axiom,
! [A: $o] :
( ( set_or8904488021354931149Most_o @ A @ A )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% atLeastAtMost_singleton
thf(fact_7145_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_7146_atLeastAtMost__singleton,axiom,
! [A: int] :
( ( set_or1266510415728281911st_int @ A @ A )
= ( insert_int @ A @ bot_bot_set_int ) ) ).
% atLeastAtMost_singleton
thf(fact_7147_atLeastAtMost__singleton,axiom,
! [A: code_integer] :
( ( set_or189985376899183464nteger @ A @ A )
= ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ).
% atLeastAtMost_singleton
thf(fact_7148_atLeastAtMost__singleton__iff,axiom,
! [A: $o,B: $o,C: $o] :
( ( ( set_or8904488021354931149Most_o @ A @ B )
= ( insert_o @ C @ bot_bot_set_o ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_7149_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_7150_atLeastAtMost__singleton__iff,axiom,
! [A: int,B: int,C: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= ( insert_int @ C @ bot_bot_set_int ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_7151_atLeastAtMost__singleton__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( ( set_or189985376899183464nteger @ A @ B )
= ( insert_Code_integer @ C @ bot_bo3990330152332043303nteger ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_7152_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_7153_ivl__disj__un__two_I7_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_7154_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_7155_ivl__disj__un__two_I7_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ M )
=> ( ( ord_le3102999989581377725nteger @ M @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
= ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_7156_ivl__disj__un__two_I7_J,axiom,
! [L: int,M: int,U: int] :
( ( ord_less_eq_int @ L @ M )
=> ( ( ord_less_eq_int @ M @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_7157_ivl__disj__int__two_I7_J,axiom,
! [L: $o,M: $o,U: $o] :
( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ M ) @ ( set_or8904488021354931149Most_o @ M @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_two(7)
thf(fact_7158_ivl__disj__int__two_I7_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(7)
thf(fact_7159_ivl__disj__int__two_I7_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( inf_in1364745209274528805nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
= bot_bo3990330152332043303nteger ) ).
% ivl_disj_int_two(7)
thf(fact_7160_ivl__disj__int__two_I7_J,axiom,
! [L: int,M: int,U: int] :
( ( inf_inf_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
= bot_bot_set_int ) ).
% ivl_disj_int_two(7)
thf(fact_7161_not__UNIV__eq__Icc,axiom,
! [L4: rat,H4: rat] :
( top_top_set_rat
!= ( set_or633870826150836451st_rat @ L4 @ H4 ) ) ).
% not_UNIV_eq_Icc
thf(fact_7162_not__UNIV__eq__Icc,axiom,
! [L4: nat,H4: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L4 @ H4 ) ) ).
% not_UNIV_eq_Icc
thf(fact_7163_not__UNIV__eq__Icc,axiom,
! [L4: int,H4: int] :
( top_top_set_int
!= ( set_or1266510415728281911st_int @ L4 @ H4 ) ) ).
% not_UNIV_eq_Icc
thf(fact_7164_ivl__disj__un__two__touch_I2_J,axiom,
! [L: literal,M: literal,U: literal] :
( ( ord_less_eq_literal @ L @ M )
=> ( ( ord_less_literal @ M @ U )
=> ( ( sup_sup_set_literal @ ( set_or8436436507035938825iteral @ L @ M ) @ ( set_or4828121863581493221iteral @ M @ U ) )
= ( set_or4828121863581493221iteral @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_7165_ivl__disj__un__two__touch_I2_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_7166_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_7167_ivl__disj__un__two__touch_I2_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ M )
=> ( ( ord_le6747313008572928689nteger @ M @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
= ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_7168_ivl__disj__un__two__touch_I2_J,axiom,
! [L: int,M: int,U: int] :
( ( ord_less_eq_int @ L @ M )
=> ( ( ord_less_int @ M @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_7169_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or7139685690850216873Than_o
= ( ^ [A3: $o,B3: $o] : ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A3 @ B3 ) @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_7170_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A3: nat,B3: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_7171_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or8404916559141939852nteger
= ( ^ [A3: code_integer,B3: code_integer] : ( minus_2355218937544613996nteger @ ( set_or189985376899183464nteger @ A3 @ B3 ) @ ( insert_Code_integer @ B3 @ bot_bo3990330152332043303nteger ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_7172_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4662586982721622107an_int
= ( ^ [A3: int,B3: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_7173_sum_Onested__swap,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% sum.nested_swap
thf(fact_7174_ivl__disj__un__singleton_I6_J,axiom,
! [L: $o,U: $o] :
( ( ord_less_eq_o @ L @ U )
=> ( ( sup_sup_set_o @ ( set_or7139685690850216873Than_o @ L @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
= ( set_or8904488021354931149Most_o @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_7175_ivl__disj__un__singleton_I6_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_7176_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_7177_ivl__disj__un__singleton_I6_J,axiom,
! [L: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ U ) @ ( insert_Code_integer @ U @ bot_bo3990330152332043303nteger ) )
= ( set_or189985376899183464nteger @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_7178_ivl__disj__un__singleton_I6_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_7179_ivl__disj__un__two__touch_I4_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_7180_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_7181_ivl__disj__un__two__touch_I4_J,axiom,
! [L: int,M: int,U: int] :
( ( ord_less_eq_int @ L @ M )
=> ( ( ord_less_eq_int @ M @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_7182_ivl__disj__un__two__touch_I4_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ M )
=> ( ( ord_le3102999989581377725nteger @ M @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M ) @ ( set_or189985376899183464nteger @ M @ U ) )
= ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_7183_atLeastAtMost__singleton_H,axiom,
! [A: $o,B: $o] :
( ( A = B )
=> ( ( set_or8904488021354931149Most_o @ A @ B )
= ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_7184_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_7185_atLeastAtMost__singleton_H,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_7186_atLeastAtMost__singleton_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
=> ( ( set_or189985376899183464nteger @ A @ B )
= ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_7187_not__UNIV__le__Icc,axiom,
! [L: rat,H3: rat] :
~ ( ord_less_eq_set_rat @ top_top_set_rat @ ( set_or633870826150836451st_rat @ L @ H3 ) ) ).
% not_UNIV_le_Icc
thf(fact_7188_not__UNIV__le__Icc,axiom,
! [L: nat,H3: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H3 ) ) ).
% not_UNIV_le_Icc
thf(fact_7189_not__UNIV__le__Icc,axiom,
! [L: int,H3: int] :
~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_or1266510415728281911st_int @ L @ H3 ) ) ).
% not_UNIV_le_Icc
thf(fact_7190_ivl__disj__un__two_I3_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_7191_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_7192_ivl__disj__un__two_I3_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ M )
=> ( ( ord_le3102999989581377725nteger @ M @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
= ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_7193_ivl__disj__un__two_I3_J,axiom,
! [L: int,M: int,U: int] :
( ( ord_less_eq_int @ L @ M )
=> ( ( ord_less_eq_int @ M @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_7194_ivl__disj__int__two_I3_J,axiom,
! [L: $o,M: $o,U: $o] :
( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ M ) @ ( set_or7139685690850216873Than_o @ M @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_two(3)
thf(fact_7195_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_7196_ivl__disj__int__two_I3_J,axiom,
! [L: code_integer,M: code_integer,U: code_integer] :
( ( inf_in1364745209274528805nteger @ ( set_or8404916559141939852nteger @ L @ M ) @ ( set_or8404916559141939852nteger @ M @ U ) )
= bot_bo3990330152332043303nteger ) ).
% ivl_disj_int_two(3)
thf(fact_7197_ivl__disj__int__two_I3_J,axiom,
! [L: int,M: int,U: int] :
( ( inf_inf_set_int @ ( set_or4662586982721622107an_int @ L @ M ) @ ( set_or4662586982721622107an_int @ M @ U ) )
= bot_bot_set_int ) ).
% ivl_disj_int_two(3)
thf(fact_7198_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7199_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_7200_sum_Oshift__bounds__Suc__ivl,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_7201_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_7202_sum_Oshift__bounds__nat__ivl,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_7203_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_7204_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( set_or2784603332384507286at_nat @ X @ Y )
= top_to7629004291339433233at_nat )
= ( ( X = bot_bo2099793752762293965at_nat )
& ( Y = top_to4669805908274784177at_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7205_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ( set_or1270514513317581473st_nat @ X @ Y )
= top_to1619013496918823676st_nat )
= ( ( X = bot_bot_set_list_nat )
& ( Y = top_top_set_list_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7206_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( set_or266616469461849773_set_o @ X @ Y )
= top_top_set_set_o )
= ( ( X = bot_bot_set_o )
& ( Y = top_top_set_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7207_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( set_or1040488700251649177et_rat @ X @ Y )
= top_top_set_set_rat )
= ( ( X = bot_bot_set_rat )
& ( Y = top_top_set_rat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7208_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4548717258645045905et_nat @ X @ Y )
= top_top_set_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7209_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( set_or370866239135849197et_int @ X @ Y )
= top_top_set_set_int )
= ( ( X = bot_bot_set_int )
& ( Y = top_top_set_int ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7210_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: $o,Y: $o] :
( ( ( set_or8904488021354931149Most_o @ X @ Y )
= top_top_set_o )
= ( ( X = bot_bot_o )
& ( Y = top_top_o ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_7211_aset_I2_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
| ( Q2 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_7212_aset_I1_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
& ( Q2 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_7213_bset_I2_J,axiom,
! [D3: int,B2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
| ( Q2 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_7214_bset_I1_J,axiom,
! [D3: int,B2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
& ( Q2 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_7215_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_7216_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_7217_sum_OatLeastAtMost__rev,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_7218_aset_I10_J,axiom,
! [D2: int,D3: int,A2: set_int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) )
=> ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% aset(10)
thf(fact_7219_aset_I9_J,axiom,
! [D2: int,D3: int,A2: set_int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) )
=> ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% aset(9)
thf(fact_7220_bset_I10_J,axiom,
! [D2: int,D3: int,B2: set_int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) )
=> ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% bset(10)
thf(fact_7221_bset_I9_J,axiom,
! [D2: int,D3: int,B2: set_int,T3: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T3 ) )
=> ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% bset(9)
thf(fact_7222_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_7223_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_7224_sum__Suc__diff_H,axiom,
! [M: nat,N: nat,F2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( minus_minus_rat @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ).
% sum_Suc_diff'
thf(fact_7225_sum__Suc__diff_H,axiom,
! [M: nat,N: nat,F2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( minus_minus_int @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ).
% sum_Suc_diff'
thf(fact_7226_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G2 @ M )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_7227_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G2 @ M )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_7228_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_7229_sum__Suc__diff,axiom,
! [M: nat,N: nat,F2: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_7230_sum__Suc__diff,axiom,
! [M: nat,N: nat,F2: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_7231_sum_OatLeastLessThan__rev,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_7232_sum__atLeastAtMost__code,axiom,
! [F2: nat > code_integer,A: nat,B: nat] :
( ( groups7501900531339628137nteger @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1084959871951514735nteger
@ ^ [A3: nat] : ( plus_p5714425477246183910nteger @ ( F2 @ A3 ) )
@ A
@ B
@ zero_z3403309356797280102nteger ) ) ).
% sum_atLeastAtMost_code
thf(fact_7233_sum__atLeastAtMost__code,axiom,
! [F2: nat > multis2468970476368604999at_nat,A: nat,B: nat] :
( ( groups6857163185585827899at_nat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo4497565046347964853at_nat
@ ^ [A3: nat] : ( plus_p7104986032573967614at_nat @ ( F2 @ A3 ) )
@ A
@ B
@ zero_z1048942125864253310at_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7234_sum__atLeastAtMost__code,axiom,
! [F2: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A3: nat] : ( plus_plus_rat @ ( F2 @ A3 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7235_sum__atLeastAtMost__code,axiom,
! [F2: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A3: nat] : ( plus_plus_int @ ( F2 @ A3 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_7236_sum__atLeastAtMost__code,axiom,
! [F2: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A3: nat] : ( plus_plus_nat @ ( F2 @ A3 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_7237_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_7238_aset_I7_J,axiom,
! [D3: int,A2: set_int,T3: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T3 @ X5 )
=> ( ord_less_int @ T3 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_7239_aset_I5_J,axiom,
! [D3: int,T3: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T3 @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T3 )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% aset(5)
thf(fact_7240_aset_I4_J,axiom,
! [D3: int,T3: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T3 @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T3 )
=> ( ( plus_plus_int @ X5 @ D3 )
!= T3 ) ) ) ) ) ).
% aset(4)
thf(fact_7241_aset_I3_J,axiom,
! [D3: int,T3: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( plus_plus_int @ T3 @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T3 )
=> ( ( plus_plus_int @ X5 @ D3 )
= T3 ) ) ) ) ) ).
% aset(3)
thf(fact_7242_bset_I7_J,axiom,
! [D3: int,T3: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T3 @ B2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T3 @ X5 )
=> ( ord_less_int @ T3 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_7243_bset_I5_J,axiom,
! [D3: int,B2: set_int,T3: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T3 )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T3 ) ) ) ) ).
% bset(5)
thf(fact_7244_bset_I4_J,axiom,
! [D3: int,T3: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T3 @ B2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T3 )
=> ( ( minus_minus_int @ X5 @ D3 )
!= T3 ) ) ) ) ) ).
% bset(4)
thf(fact_7245_bset_I3_J,axiom,
! [D3: int,T3: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( minus_minus_int @ T3 @ one_one_int ) @ B2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T3 )
=> ( ( minus_minus_int @ X5 @ D3 )
= T3 ) ) ) ) ) ).
% bset(3)
thf(fact_7246_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > rat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7247_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > int,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7248_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > nat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_7249_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_7250_aset_I8_J,axiom,
! [D3: int,A2: set_int,T3: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T3 @ X5 )
=> ( ord_less_eq_int @ T3 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% aset(8)
thf(fact_7251_aset_I6_J,axiom,
! [D3: int,T3: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( plus_plus_int @ T3 @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T3 ) ) ) ) ) ).
% aset(6)
thf(fact_7252_bset_I8_J,axiom,
! [D3: int,T3: int,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( minus_minus_int @ T3 @ one_one_int ) @ B2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T3 @ X5 )
=> ( ord_less_eq_int @ T3 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(8)
thf(fact_7253_bset_I6_J,axiom,
! [D3: int,B2: set_int,T3: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B2 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T3 ) ) ) ) ).
% bset(6)
thf(fact_7254_cpmi,axiom,
! [D3: int,P: int > $o,P7: int > $o,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z7: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z7 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( P7 @ X2 )
= ( P7 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P7 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( P @ ( plus_plus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_7255_cppi,axiom,
! [D3: int,P: int > $o,P7: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z7: int] :
! [X2: int] :
( ( ord_less_int @ Z7 @ X2 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( P7 @ X2 )
= ( P7 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ( ? [X4: int] : ( P @ X4 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P7 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ A2 )
& ( P @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_7256_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F2: nat > code_integer] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( minus_8373710615458151222nteger @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_8373710615458151222nteger @ ( F2 @ M ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( minus_8373710615458151222nteger @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_z3403309356797280102nteger ) ) ) ).
% sum_natinterval_diff
thf(fact_7257_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F2: nat > rat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( minus_minus_rat @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F2 @ M ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( minus_minus_rat @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_7258_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F2: nat > int] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( minus_minus_int @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F2 @ M ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( minus_minus_int @ ( F2 @ K4 ) @ ( F2 @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_7259_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( minus_minus_rat @ ( F2 @ K4 ) @ ( F2 @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_rat @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ).
% sum_telescope''
thf(fact_7260_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( minus_minus_int @ ( F2 @ K4 ) @ ( F2 @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_int @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ).
% sum_telescope''
thf(fact_7261_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_7262_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_7263_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_7264_sum_Oin__pairs,axiom,
! [G2: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_7265_sum_Oin__pairs,axiom,
! [G2: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_7266_sum_Oin__pairs,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_7267_sum__gp,axiom,
! [N: nat,M: nat,X: rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_7268_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_7269_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_7270_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_7271_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_7272_sum__gp__offset,axiom,
! [X: rat,M: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_7273_arith__series,axiom,
! [A: code_integer,D2: code_integer,N: nat] :
( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D2 ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7274_arith__series,axiom,
! [A: code_natural,D2: code_natural,N: nat] :
( ( groups6325495683096345652atural
@ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ D2 ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7275_arith__series,axiom,
! [A: int,D2: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7276_arith__series,axiom,
! [A: nat,D2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_7277_gauss__sum,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7278_gauss__sum,axiom,
! [N: nat] :
( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7279_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7280_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_7281_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7282_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7283_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7284_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7285_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_7286_double__arith__series,axiom,
! [A: code_integer,D2: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D2 ) ) ) ) ).
% double_arith_series
thf(fact_7287_double__arith__series,axiom,
! [A: code_natural,D2: code_natural,N: nat] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) )
@ ( groups6325495683096345652atural
@ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ D2 ) ) ) ) ).
% double_arith_series
thf(fact_7288_double__arith__series,axiom,
! [A: int,D2: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D2 ) ) ) ) ).
% double_arith_series
thf(fact_7289_double__arith__series,axiom,
! [A: rat,D2: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D2 ) ) ) ) ).
% double_arith_series
thf(fact_7290_double__arith__series,axiom,
! [A: nat,D2: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D2 ) ) ) ) ).
% double_arith_series
thf(fact_7291_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_7292_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_7293_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mult
thf(fact_7294_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_7295_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_7296_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_7297_of__nat__1,axiom,
( ( semiri4939895301339042750nteger @ one_one_nat )
= one_one_Code_integer ) ).
% of_nat_1
thf(fact_7298_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_7299_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_7300_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_7301_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer
= ( semiri4939895301339042750nteger @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7302_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7303_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7304_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_7305_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri4939895301339042750nteger @ N )
= one_one_Code_integer )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7306_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7307_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7308_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_7309_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_7310_of__nat__sum,axiom,
! [F2: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_7311_of__nat__sum,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_7312_of__nat__sum,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_7313_of__nat__sum,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_7314_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ M ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).
% of_nat_Suc
thf(fact_7315_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_7316_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_7317_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_7318_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7319_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7320_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7321_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_7322_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
= ( set_or189985376899183464nteger @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_7323_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_7324_div__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7325_div__mult2__eq_H,axiom,
! [A: code_natural,M: nat,N: nat] :
( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
= ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7326_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7327_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_7328_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_7329_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_7330_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_7331_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_7332_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_7333_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_7334_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7335_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_7336_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_7337_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_7338_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_7339_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_7340_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_7341_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_7342_int__sum,axiom,
! [F2: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_7343_int__sum,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_sum
thf(fact_7344_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7345_mod__mult2__eq_H,axiom,
! [A: code_natural,M: nat,N: nat] :
( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri3763490453095760265atural @ N ) ) )
= ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7346_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7347_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_7348_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_7349_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_7350_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_7351_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_7352_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_7353_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_7354_double__gauss__sum,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum
thf(fact_7355_double__gauss__sum,axiom,
! [N: nat] :
( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N ) @ one_one_Code_natural ) ) ) ).
% double_gauss_sum
thf(fact_7356_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_7357_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_7358_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_7359_pochhammer__double,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7360_of__nat__code,axiom,
( semiri4939895301339042750nteger
= ( ^ [N2: nat] :
( semiri4055485073559036834nteger
@ ^ [I3: code_integer] : ( plus_p5714425477246183910nteger @ I3 @ one_one_Code_integer )
@ N2
@ zero_z3403309356797280102nteger ) ) ) ).
% of_nat_code
thf(fact_7361_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_7362_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_7363_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
@ N2
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_7364_pochhammer__code,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A3: code_integer,N2: nat] :
( if_Code_integer @ ( N2 = zero_zero_nat ) @ one_one_Code_integer
@ ( set_fo1084959871951514735nteger
@ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( semiri4939895301339042750nteger @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_Code_integer ) ) ) ) ).
% pochhammer_code
thf(fact_7365_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A3: int,N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_7366_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A3: rat,N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_7367_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A3: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_7368_sum__gp0,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_7369_gchoose__row__sum__weighted,axiom,
! [R3: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( gbinomial_rat @ R3 @ K4 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R3 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_7370_case__nat__add__eq__if,axiom,
! [A: $o,F2: nat > $o,V: num,N: nat] :
( ( case_nat_o @ A @ F2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( F2 @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_7371_case__nat__add__eq__if,axiom,
! [A: nat,F2: nat > nat,V: num,N: nat] :
( ( case_nat_nat @ A @ F2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( F2 @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_7372_case__nat__add__eq__if,axiom,
! [A: option_num,F2: nat > option_num,V: num,N: nat] :
( ( case_nat_option_num @ A @ F2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( F2 @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_7373_gbinomial__0_I1_J,axiom,
! [A: code_integer] :
( ( gbinom8545251970709558553nteger @ A @ zero_zero_nat )
= one_one_Code_integer ) ).
% gbinomial_0(1)
thf(fact_7374_gbinomial__0_I1_J,axiom,
! [A: rat] :
( ( gbinomial_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_7375_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_7376_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_7377_pochhammer__0,axiom,
! [A: code_integer] :
( ( comm_s8582702949713902594nteger @ A @ zero_zero_nat )
= one_one_Code_integer ) ).
% pochhammer_0
thf(fact_7378_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_7379_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_7380_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_7381_case__nat__numeral,axiom,
! [A: $o,F2: nat > $o,V: num] :
( ( case_nat_o @ A @ F2 @ ( numeral_numeral_nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_7382_case__nat__numeral,axiom,
! [A: nat,F2: nat > nat,V: num] :
( ( case_nat_nat @ A @ F2 @ ( numeral_numeral_nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_7383_case__nat__numeral,axiom,
! [A: option_num,F2: nat > option_num,V: num] :
( ( case_nat_option_num @ A @ F2 @ ( numeral_numeral_nat @ V ) )
= ( F2 @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_7384_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_7385_nat_Ocase__distrib,axiom,
! [H3: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( H3 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7386_nat_Ocase__distrib,axiom,
! [H3: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
( ( H3 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7387_nat_Ocase__distrib,axiom,
! [H3: $o > option_num,F1: $o,F22: nat > $o,Nat: nat] :
( ( H3 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_option_num @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7388_nat_Ocase__distrib,axiom,
! [H3: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( H3 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7389_nat_Ocase__distrib,axiom,
! [H3: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
( ( H3 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7390_nat_Ocase__distrib,axiom,
! [H3: nat > option_num,F1: nat,F22: nat > nat,Nat: nat] :
( ( H3 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_option_num @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7391_nat_Ocase__distrib,axiom,
! [H3: option_num > $o,F1: option_num,F22: nat > option_num,Nat: nat] :
( ( H3 @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7392_nat_Ocase__distrib,axiom,
! [H3: option_num > nat,F1: option_num,F22: nat > option_num,Nat: nat] :
( ( H3 @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7393_nat_Ocase__distrib,axiom,
! [H3: option_num > option_num,F1: option_num,F22: nat > option_num,Nat: nat] :
( ( H3 @ ( case_nat_option_num @ F1 @ F22 @ Nat ) )
= ( case_nat_option_num @ ( H3 @ F1 )
@ ^ [X3: nat] : ( H3 @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_7394_not__empty__eq__Iic__eq__empty,axiom,
! [H3: $o] :
( bot_bot_set_o
!= ( set_ord_atMost_o @ H3 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7395_not__empty__eq__Iic__eq__empty,axiom,
! [H3: int] :
( bot_bot_set_int
!= ( set_ord_atMost_int @ H3 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7396_not__empty__eq__Iic__eq__empty,axiom,
! [H3: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H3 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_7397_not__UNIV__eq__Iic,axiom,
! [H4: rat] :
( top_top_set_rat
!= ( set_ord_atMost_rat @ H4 ) ) ).
% not_UNIV_eq_Iic
thf(fact_7398_not__UNIV__eq__Iic,axiom,
! [H4: int] :
( top_top_set_int
!= ( set_ord_atMost_int @ H4 ) ) ).
% not_UNIV_eq_Iic
thf(fact_7399_not__UNIV__eq__Iic,axiom,
! [H4: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H4 ) ) ).
% not_UNIV_eq_Iic
thf(fact_7400_atMost__def,axiom,
( set_or2459421552957432928nt_int
= ( ^ [U2: set_Pr958786334691620121nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7401_atMost__def,axiom,
( set_or1537382583742010260_nat_o
= ( ^ [U2: produc3658429121746597890et_nat > $o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_le729326519192465773_nat_o @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7402_atMost__def,axiom,
( set_or9144418160755794905er_nat
= ( ^ [U2: filter_nat] :
( collect_filter_nat
@ ^ [X3: filter_nat] : ( ord_le2510731241096832064er_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7403_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7404_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7405_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7406_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_7407_gbinomial__parallel__sum,axiom,
! [A: rat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K4 ) ) @ K4 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_7408_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_7409_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_7410_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_7411_not__UNIV__le__Iic,axiom,
! [H3: rat] :
~ ( ord_less_eq_set_rat @ top_top_set_rat @ ( set_ord_atMost_rat @ H3 ) ) ).
% not_UNIV_le_Iic
thf(fact_7412_not__UNIV__le__Iic,axiom,
! [H3: int] :
~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_ord_atMost_int @ H3 ) ) ).
% not_UNIV_le_Iic
thf(fact_7413_not__UNIV__le__Iic,axiom,
! [H3: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H3 ) ) ).
% not_UNIV_le_Iic
thf(fact_7414_atMost__eq__UNIV__iff,axiom,
! [X: set_list_nat] :
( ( ( set_or2492388921469580815st_nat @ X )
= top_to1619013496918823676st_nat )
= ( X = top_top_set_list_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7415_atMost__eq__UNIV__iff,axiom,
! [X: set_o] :
( ( ( set_ord_atMost_set_o @ X )
= top_top_set_set_o )
= ( X = top_top_set_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7416_atMost__eq__UNIV__iff,axiom,
! [X: set_rat] :
( ( ( set_or728397472755099399et_rat @ X )
= top_top_set_set_rat )
= ( X = top_top_set_rat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7417_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7418_atMost__eq__UNIV__iff,axiom,
! [X: set_int] :
( ( ( set_or58775011639299419et_int @ X )
= top_top_set_set_int )
= ( X = top_top_set_int ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7419_atMost__eq__UNIV__iff,axiom,
! [X: $o] :
( ( ( set_ord_atMost_o @ X )
= top_top_set_o )
= ( X = top_top_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_7420_gbinomial__sum__lower__neg,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K4 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_7421_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_7422_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_7423_gbinomial__mult__1,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7424_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7425_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: rat,X: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K4 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K4 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K4 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K4 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_7426_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
= one_one_Code_integer ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
= zero_z3403309356797280102nteger ) ) ) ).
% pochhammer_0_left
thf(fact_7427_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_7428_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_7429_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_7430_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: rat,X: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K4 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K4 ) @ A ) @ one_one_rat ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K4 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_7431_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K4 ) ) @ K4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K4 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_7432_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_7433_gbinomial__absorption,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_7434_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_7435_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: rat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7436_sum_OatMost__Suc__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G2 @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7437_sum_OatMost__Suc__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G2 @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7438_sum_OatMost__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_7439_sum__telescope,axiom,
! [F2: nat > rat,I: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_rat @ ( F2 @ zero_zero_nat ) @ ( F2 @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_7440_sum__telescope,axiom,
! [F2: nat > int,I: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_int @ ( F2 @ zero_zero_nat ) @ ( F2 @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_7441_pochhammer__rec,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ A @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7442_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7443_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7444_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7445_pochhammer__Suc,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7446_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7447_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7448_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7449_pochhammer__rec_H,axiom,
! [Z: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7450_pochhammer__rec_H,axiom,
! [Z: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7451_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7452_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7453_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7454_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7455_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7456_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7457_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7458_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7459_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K4 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7460_gbinomial__rec,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_7461_gbinomial__factors,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_7462_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7463_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7464_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7465_pochhammer__product_H,axiom,
! [Z: code_integer,N: nat,M: nat] :
( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7466_pochhammer__product_H,axiom,
! [Z: int,N: nat,M: nat] :
( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7467_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M: nat] :
( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7468_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M: nat] :
( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7469_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K4 ) @ A3 ) @ one_one_rat ) @ K4 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_7470_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_7471_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_7472_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K4: nat] : K4
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_7473_gbinomial__partial__row__sum,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K4 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K4 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_7474_Nitpick_Ocase__nat__unfold,axiom,
( case_nat_o
= ( ^ [X3: $o,F: nat > $o,N2: nat] :
( ( ( N2 = zero_zero_nat )
=> X3 )
& ( ( N2 != zero_zero_nat )
=> ( F @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_7475_Nitpick_Ocase__nat__unfold,axiom,
( case_nat_nat
= ( ^ [X3: nat,F: nat > nat,N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ X3 @ ( F @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_7476_Nitpick_Ocase__nat__unfold,axiom,
( case_nat_option_num
= ( ^ [X3: option_num,F: nat > option_num,N2: nat] : ( if_option_num @ ( N2 = zero_zero_nat ) @ X3 @ ( F @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_7477_gbinomial__minus,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_7478_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_7479_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A @ K )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_7480_sum_Otriangle__reindex__eq,axiom,
! [G2: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_7481_pochhammer__product,axiom,
! [M: nat,N: nat,Z: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s8582702949713902594nteger @ Z @ N )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7482_pochhammer__product,axiom,
! [M: nat,N: nat,Z: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4660882817536571857er_int @ Z @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7483_pochhammer__product,axiom,
! [M: nat,N: nat,Z: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7484_pochhammer__product,axiom,
! [M: nat,N: nat,Z: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7485_sum__gp__basic,axiom,
! [X: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7486_sum__gp__basic,axiom,
! [X: rat,N: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7487_sum__gp__basic,axiom,
! [X: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_7488_sum__power__shift,axiom,
! [M: nat,N: nat,X: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7489_sum__power__shift,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7490_sum__power__shift,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_7491_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_7492_pochhammer__absorb__comp,axiom,
! [R3: code_integer,K: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R3 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R3 ) @ K ) )
= ( times_3573771949741848930nteger @ R3 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7493_pochhammer__absorb__comp,axiom,
! [R3: int,K: nat] :
( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K ) )
= ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7494_pochhammer__absorb__comp,axiom,
! [R3: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K ) )
= ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7495_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A @ K )
= ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_7496_sum_Oin__pairs__0,axiom,
! [G2: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7497_sum_Oin__pairs__0,axiom,
! [G2: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7498_sum_Oin__pairs__0,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_7499_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7500_pochhammer__minus_H,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7501_pochhammer__minus_H,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7502_pochhammer__minus,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7503_pochhammer__minus,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7504_pochhammer__minus,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7505_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > code_integer,H3: nat > code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups7501900531339628137nteger
@ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_Code_integer @ ( J3 = K ) @ zero_z3403309356797280102nteger @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups7501900531339628137nteger
@ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7506_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > multis2468970476368604999at_nat,H3: nat > multis2468970476368604999at_nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups6857163185585827899at_nat
@ ^ [J3: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_mul8430962117462786573at_nat @ ( J3 = K ) @ zero_z1048942125864253310at_nat @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups6857163185585827899at_nat
@ ^ [J3: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7507_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > rat,H3: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7508_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > int,H3: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7509_sum_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > nat,H3: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_7510_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7511_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] :
( if_Code_integer
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) )
@ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7512_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7513_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_7514_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) @ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7515_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7516_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_7517_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] :
( if_rat @ ( K4 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K4 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ) ).
% gbinomial_code
thf(fact_7518_fact__double,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_double
thf(fact_7519_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_sum
thf(fact_7520_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_7521_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_7522_atMost__UNIV__triv,axiom,
( ( set_or2492388921469580815st_nat @ top_top_set_list_nat )
= top_to1619013496918823676st_nat ) ).
% atMost_UNIV_triv
thf(fact_7523_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_o @ top_top_set_o )
= top_top_set_set_o ) ).
% atMost_UNIV_triv
thf(fact_7524_atMost__UNIV__triv,axiom,
( ( set_or728397472755099399et_rat @ top_top_set_rat )
= top_top_set_set_rat ) ).
% atMost_UNIV_triv
thf(fact_7525_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_7526_atMost__UNIV__triv,axiom,
( ( set_or58775011639299419et_int @ top_top_set_int )
= top_top_set_set_int ) ).
% atMost_UNIV_triv
thf(fact_7527_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_7528_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [Uu: nat] : one_one_nat
@ A2 )
= one_one_nat ) ).
% prod.neutral_const
thf(fact_7529_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [Uu: nat] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7530_prod_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [Uu: int] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7531_of__nat__prod,axiom,
! [F2: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_7532_of__nat__prod,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ F2 @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( semiri681578069525770553at_rat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_7533_of__nat__prod,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_7534_of__nat__prod,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F2 @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_nat_prod
thf(fact_7535_of__int__prod,axiom,
! [F2: nat > int,A2: set_nat] :
( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F2 @ A2 ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( ring_1_of_int_rat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_7536_of__int__prod,axiom,
! [F2: nat > int,A2: set_nat] :
( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F2 @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( ring_1_of_int_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_7537_of__int__prod,axiom,
! [F2: int > int,A2: set_int] :
( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F2 @ A2 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X3: int] : ( ring_1_of_int_rat @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_7538_of__int__prod,axiom,
! [F2: int > int,A2: set_int] :
( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( ring_1_of_int_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% of_int_prod
thf(fact_7539_prod_Oempty,axiom,
! [G2: $o > code_integer] :
( ( groups7694694392188491536nteger @ G2 @ bot_bot_set_o )
= one_one_Code_integer ) ).
% prod.empty
thf(fact_7540_prod_Oempty,axiom,
! [G2: $o > assn] :
( ( groups5301882518646026715o_assn @ G2 @ bot_bot_set_o )
= one_one_assn ) ).
% prod.empty
thf(fact_7541_prod_Oempty,axiom,
! [G2: $o > rat] :
( ( groups2869687844427037835_o_rat @ G2 @ bot_bot_set_o )
= one_one_rat ) ).
% prod.empty
thf(fact_7542_prod_Oempty,axiom,
! [G2: $o > nat] :
( ( groups3504817904513533571_o_nat @ G2 @ bot_bot_set_o )
= one_one_nat ) ).
% prod.empty
thf(fact_7543_prod_Oempty,axiom,
! [G2: $o > int] :
( ( groups3502327434004483295_o_int @ G2 @ bot_bot_set_o )
= one_one_int ) ).
% prod.empty
thf(fact_7544_prod_Oempty,axiom,
! [G2: nat > code_integer] :
( ( groups3455450783089532116nteger @ G2 @ bot_bot_set_nat )
= one_one_Code_integer ) ).
% prod.empty
thf(fact_7545_prod_Oempty,axiom,
! [G2: nat > assn] :
( ( groups6906906614972039071t_assn @ G2 @ bot_bot_set_nat )
= one_one_assn ) ).
% prod.empty
thf(fact_7546_prod_Oempty,axiom,
! [G2: nat > rat] :
( ( groups73079841787564623at_rat @ G2 @ bot_bot_set_nat )
= one_one_rat ) ).
% prod.empty
thf(fact_7547_prod_Oempty,axiom,
! [G2: int > code_integer] :
( ( groups3827104343326376752nteger @ G2 @ bot_bot_set_int )
= one_one_Code_integer ) ).
% prod.empty
thf(fact_7548_prod_Oempty,axiom,
! [G2: int > assn] :
( ( groups7882442080178216443t_assn @ G2 @ bot_bot_set_int )
= one_one_assn ) ).
% prod.empty
thf(fact_7549_fact__0,axiom,
( ( semiri3624122377584611663nteger @ zero_zero_nat )
= one_one_Code_integer ) ).
% fact_0
thf(fact_7550_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_7551_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_7552_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_7553_fact__1,axiom,
( ( semiri3624122377584611663nteger @ one_one_nat )
= one_one_Code_integer ) ).
% fact_1
thf(fact_7554_fact__1,axiom,
( ( semiri773545260158071498ct_rat @ one_one_nat )
= one_one_rat ) ).
% fact_1
thf(fact_7555_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_7556_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_7557_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_7558_fact__Suc__0,axiom,
( ( semiri3624122377584611663nteger @ ( suc @ zero_zero_nat ) )
= one_one_Code_integer ) ).
% fact_Suc_0
thf(fact_7559_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_7560_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_7561_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_7562_fact__Suc,axiom,
! [N: nat] :
( ( semiri3624122377584611663nteger @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% fact_Suc
thf(fact_7563_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_7564_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_7565_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_7566_prod_OatMost__Suc,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7567_prod_OatMost__Suc,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7568_prod_OatMost__Suc,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7569_prod_OatMost__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7570_prod_OatMost__Suc,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_7571_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > code_integer] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_Code_integer ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7572_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > assn] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_assn ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7573_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7574_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7575_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7576_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > code_integer] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_Code_integer ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_7577_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > assn] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_assn ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_7578_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_7579_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_7580_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_7581_prod_Oswap,axiom,
! [G2: nat > nat > nat,B2: set_nat,A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% prod.swap
thf(fact_7582_prod_Oswap,axiom,
! [G2: nat > nat > int,B2: set_nat,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% prod.swap
thf(fact_7583_prod_Oswap,axiom,
! [G2: nat > int > int,B2: set_int,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups1705073143266064639nt_int @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups1705073143266064639nt_int
@ ^ [J3: int] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% prod.swap
thf(fact_7584_prod_Oswap,axiom,
! [G2: int > nat > int,B2: set_nat,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( groups705719431365010083at_int @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups1705073143266064639nt_int
@ ^ [I3: int] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% prod.swap
thf(fact_7585_prod_Oswap,axiom,
! [G2: int > int > int,B2: set_int,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( groups1705073143266064639nt_int @ ( G2 @ I3 ) @ B2 )
@ A2 )
= ( groups1705073143266064639nt_int
@ ^ [J3: int] :
( groups1705073143266064639nt_int
@ ^ [I3: int] : ( G2 @ I3 @ J3 )
@ A2 )
@ B2 ) ) ).
% prod.swap
thf(fact_7586_prod_Oneutral,axiom,
! [A2: set_nat,G2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ A2 )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_7587_prod_Oneutral,axiom,
! [A2: set_nat,G2: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G2 @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_7588_prod_Oneutral,axiom,
! [A2: set_int,G2: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_7589_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > code_integer,A2: set_nat] :
( ( ( groups3455450783089532116nteger @ G2 @ A2 )
!= one_one_Code_integer )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7590_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: int > code_integer,A2: set_int] :
( ( ( groups3827104343326376752nteger @ G2 @ A2 )
!= one_one_Code_integer )
=> ~ ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7591_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > assn,A2: set_nat] :
( ( ( groups6906906614972039071t_assn @ G2 @ A2 )
!= one_one_assn )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7592_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: int > assn,A2: set_int] :
( ( ( groups7882442080178216443t_assn @ G2 @ A2 )
!= one_one_assn )
=> ~ ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7593_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > rat,A2: set_nat] :
( ( ( groups73079841787564623at_rat @ G2 @ A2 )
!= one_one_rat )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7594_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: int > rat,A2: set_int] :
( ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
!= one_one_rat )
=> ~ ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7595_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: int > nat,A2: set_int] :
( ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
!= one_one_nat )
=> ~ ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7596_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > nat,A2: set_nat] :
( ( ( groups708209901874060359at_nat @ G2 @ A2 )
!= one_one_nat )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7597_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > int,A2: set_nat] :
( ( ( groups705719431365010083at_int @ G2 @ A2 )
!= one_one_int )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_int ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7598_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: int > int,A2: set_int] :
( ( ( groups1705073143266064639nt_int @ G2 @ A2 )
!= one_one_int )
=> ~ ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ( G2 @ A6 )
= one_one_int ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7599_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_7600_prod_Odistrib,axiom,
! [G2: nat > nat,H3: nat > nat,A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( times_times_nat @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ A2 ) @ ( groups708209901874060359at_nat @ H3 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7601_prod_Odistrib,axiom,
! [G2: nat > int,H3: nat > int,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [X3: nat] : ( times_times_int @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ A2 ) @ ( groups705719431365010083at_int @ H3 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7602_prod_Odistrib,axiom,
! [G2: int > int,H3: int > int,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( times_times_int @ ( G2 @ X3 ) @ ( H3 @ X3 ) )
@ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ A2 ) @ ( groups1705073143266064639nt_int @ H3 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7603_prod__power__distrib,axiom,
! [F2: nat > nat,A2: set_nat,N: nat] :
( ( power_power_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) @ N )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( power_power_nat @ ( F2 @ X3 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7604_prod__power__distrib,axiom,
! [F2: nat > int,A2: set_nat,N: nat] :
( ( power_power_int @ ( groups705719431365010083at_int @ F2 @ A2 ) @ N )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( power_power_int @ ( F2 @ X3 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7605_prod__power__distrib,axiom,
! [F2: int > int,A2: set_int,N: nat] :
( ( power_power_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) @ N )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( power_power_int @ ( F2 @ X3 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7606_mod__prod__eq,axiom,
! [F2: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( modulo_modulo_nat @ ( F2 @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_7607_mod__prod__eq,axiom,
! [F2: nat > int,A: int,A2: set_nat] :
( ( modulo_modulo_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( modulo_modulo_int @ ( F2 @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups705719431365010083at_int @ F2 @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_7608_mod__prod__eq,axiom,
! [F2: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups1705073143266064639nt_int
@ ^ [I3: int] : ( modulo_modulo_int @ ( F2 @ I3 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_7609_fact__prod,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] :
( semiri1314217659103216013at_int
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_7610_fact__prod,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] :
( semiri681578069525770553at_rat
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_7611_fact__prod,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] :
( semiri1316708129612266289at_nat
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_7612_prod__ge__1,axiom,
! [A2: set_nat,F2: nat > code_integer] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X2 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7613_prod__ge__1,axiom,
! [A2: set_int,F2: int > code_integer] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X2 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7614_prod__ge__1,axiom,
! [A2: set_nat,F2: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7615_prod__ge__1,axiom,
! [A2: set_int,F2: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7616_prod__ge__1,axiom,
! [A2: set_int,F2: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7617_prod__ge__1,axiom,
! [A2: set_nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7618_prod__ge__1,axiom,
! [A2: set_nat,F2: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7619_prod__ge__1,axiom,
! [A2: set_int,F2: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_int @ one_one_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7620_prod__ge__1,axiom,
! [A2: set_set_nat,F2: set_nat > code_integer] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X2 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups2761809935038513290nteger @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7621_prod__ge__1,axiom,
! [A2: set_set_nat,F2: set_nat > rat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ X2 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups3613417700093529605at_rat @ F2 @ A2 ) ) ) ).
% prod_ge_1
thf(fact_7622_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1408675320244567234ct_nat @ M )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_7623_fact__prod__rev,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% fact_prod_rev
thf(fact_7624_fact__prod__rev,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% fact_prod_rev
thf(fact_7625_fact__prod__rev,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ).
% fact_prod_rev
thf(fact_7626_fact__ge__1,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N ) ) ).
% fact_ge_1
thf(fact_7627_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_7628_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_7629_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_7630_pochhammer__fact,axiom,
( semiri3624122377584611663nteger
= ( comm_s8582702949713902594nteger @ one_one_Code_integer ) ) ).
% pochhammer_fact
thf(fact_7631_pochhammer__fact,axiom,
( semiri773545260158071498ct_rat
= ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% pochhammer_fact
thf(fact_7632_pochhammer__fact,axiom,
( semiri1406184849735516958ct_int
= ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% pochhammer_fact
thf(fact_7633_pochhammer__fact,axiom,
( semiri1408675320244567234ct_nat
= ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% pochhammer_fact
thf(fact_7634_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7635_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G2: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7636_prod_Oshift__bounds__Suc__ivl,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_7637_prod_Oshift__bounds__Suc__ivl,axiom,
! [G2: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_7638_power__sum,axiom,
! [C: nat,F2: nat > nat,A2: set_nat] :
( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [A3: nat] : ( power_power_nat @ C @ ( F2 @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_7639_power__sum,axiom,
! [C: int,F2: nat > nat,A2: set_nat] :
( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F2 @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [A3: nat] : ( power_power_int @ C @ ( F2 @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_7640_power__sum,axiom,
! [C: int,F2: int > nat,A2: set_int] :
( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [A3: int] : ( power_power_int @ C @ ( F2 @ A3 ) )
@ A2 ) ) ).
% power_sum
thf(fact_7641_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7642_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G2: nat > int,M: nat,K: nat,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7643_prod_Oshift__bounds__nat__ivl,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_7644_prod_Oshift__bounds__nat__ivl,axiom,
! [G2: nat > int,M: nat,K: nat,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( plus_plus_nat @ I3 @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_7645_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_7646_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_7647_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : X3
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_7648_prod__le__1,axiom,
! [A2: set_nat,F2: nat > code_integer] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) )
& ( ord_le3102999989581377725nteger @ ( F2 @ X2 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F2 @ A2 ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7649_prod__le__1,axiom,
! [A2: set_int,F2: int > code_integer] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) )
& ( ord_le3102999989581377725nteger @ ( F2 @ X2 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F2 @ A2 ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7650_prod__le__1,axiom,
! [A2: set_nat,F2: nat > rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) )
& ( ord_less_eq_rat @ ( F2 @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7651_prod__le__1,axiom,
! [A2: set_int,F2: int > rat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) )
& ( ord_less_eq_rat @ ( F2 @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7652_prod__le__1,axiom,
! [A2: set_int,F2: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X2 ) )
& ( ord_less_eq_nat @ ( F2 @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7653_prod__le__1,axiom,
! [A2: set_nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X2 ) )
& ( ord_less_eq_nat @ ( F2 @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7654_prod__le__1,axiom,
! [A2: set_nat,F2: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X2 ) )
& ( ord_less_eq_int @ ( F2 @ X2 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups705719431365010083at_int @ F2 @ A2 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_7655_prod__le__1,axiom,
! [A2: set_int,F2: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X2 ) )
& ( ord_less_eq_int @ ( F2 @ X2 ) @ one_one_int ) ) )
=> ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) @ one_one_int ) ) ).
% prod_le_1
thf(fact_7656_prod__le__1,axiom,
! [A2: set_set_nat,F2: set_nat > code_integer] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) )
& ( ord_le3102999989581377725nteger @ ( F2 @ X2 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups2761809935038513290nteger @ F2 @ A2 ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7657_prod__le__1,axiom,
! [A2: set_set_nat,F2: set_nat > rat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) )
& ( ord_less_eq_rat @ ( F2 @ X2 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups3613417700093529605at_rat @ F2 @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7658_fact__split,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri3624122377584611663nteger @ N )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_split
thf(fact_7659_fact__split,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_split
thf(fact_7660_fact__split,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_split
thf(fact_7661_fact__split,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N @ K ) @ N ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_split
thf(fact_7662_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P4: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P4 )
=> ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
= ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_7663_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P4: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P4 )
=> ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
= ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_7664_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P4: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P4 )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
= ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_7665_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P4: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P4 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
= ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_7666_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P4: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P4 )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
= ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_7667_binomial__altdef__of__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_7668_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_7669_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7670_gbinomial__mult__fact_H,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( semiri773545260158071498ct_rat @ K ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).
% gbinomial_mult_fact'
thf(fact_7671_gbinomial__mult__fact,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( gbinomial_rat @ A @ K ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).
% gbinomial_mult_fact
thf(fact_7672_gbinomial__prod__rev,axiom,
( gbinom8545251970709558553nteger
= ( ^ [A3: code_integer,K4: nat] :
( divide6298287555418463151nteger
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( minus_8373710615458151222nteger @ A3 @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) )
@ ( semiri3624122377584611663nteger @ K4 ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_7673_gbinomial__prod__rev,axiom,
( gbinom7368847122466276068atural
= ( ^ [A3: code_natural,K4: nat] :
( divide5121882707175180666atural
@ ( groups2279045934846249631atural
@ ^ [I3: nat] : ( minus_7197305767214868737atural @ A3 @ ( semiri3763490453095760265atural @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) )
@ ( semiri2447717529341329178atural @ K4 ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_7674_gbinomial__prod__rev,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] :
( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) )
@ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_7675_gbinomial__prod__rev,axiom,
( gbinomial_nat
= ( ^ [A3: nat,K4: nat] :
( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) )
@ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_7676_gbinomial__prod__rev,axiom,
( gbinomial_int
= ( ^ [A3: int,K4: nat] :
( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A3 @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) )
@ ( semiri1406184849735516958ct_int @ K4 ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_7677_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7678_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7679_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7680_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7681_prod_OatLeastAtMost__rev,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7682_prod_OatLeastAtMost__rev,axiom,
! [G2: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7683_gbinomial__Suc,axiom,
! [A: code_integer,K: nat] :
( ( gbinom8545251970709558553nteger @ A @ ( suc @ K ) )
= ( divide6298287555418463151nteger
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( minus_8373710615458151222nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri3624122377584611663nteger @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_7684_gbinomial__Suc,axiom,
! [A: code_natural,K: nat] :
( ( gbinom7368847122466276068atural @ A @ ( suc @ K ) )
= ( divide5121882707175180666atural
@ ( groups2279045934846249631atural
@ ^ [I3: nat] : ( minus_7197305767214868737atural @ A @ ( semiri3763490453095760265atural @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri2447717529341329178atural @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_7685_gbinomial__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_7686_gbinomial__Suc,axiom,
! [A: nat,K: nat] :
( ( gbinomial_nat @ A @ ( suc @ K ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_7687_gbinomial__Suc,axiom,
! [A: int,K: nat] :
( ( gbinomial_int @ A @ ( suc @ K ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_7688_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( binomial @ K4 @ M )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_7689_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7690_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7691_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7692_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7693_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7694_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_7695_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_7696_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_7697_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_7698_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_7699_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > code_integer] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( G2 @ M ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_7700_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > assn] :
( ( ord_less_nat @ M @ N )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_times_assn @ ( G2 @ M ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_7701_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_times_rat @ ( G2 @ M ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_7702_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ M ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_7703_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_times_int @ ( G2 @ M ) @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_7704_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( G2 @ ( suc @ N ) ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7705_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_assn @ ( G2 @ ( suc @ N ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7706_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( G2 @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7707_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7708_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( G2 @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7709_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( G2 @ M ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7710_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_assn @ ( G2 @ M ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7711_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( G2 @ M ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7712_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ M ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7713_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( G2 @ M ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7714_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_7715_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_7716_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_7717_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_7718_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_7719_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( G2 @ N ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_7720_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_assn @ ( G2 @ N ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_7721_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( G2 @ N ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_7722_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ N ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_7723_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( G2 @ N ) @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_7724_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_7725_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( G2 @ M )
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7726_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > assn] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times_assn @ ( G2 @ M )
@ ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7727_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( G2 @ M )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7728_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ M )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7729_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G2: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) )
= ( times_times_int @ ( G2 @ M )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7730_prod_OatMost__Suc__shift,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( G2 @ zero_zero_nat )
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7731_prod_OatMost__Suc__shift,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( G2 @ zero_zero_nat )
@ ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7732_prod_OatMost__Suc__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G2 @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7733_prod_OatMost__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7734_prod_OatMost__Suc__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G2 @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_7735_prod_OatLeastLessThan__rev,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_7736_prod_OatLeastLessThan__rev,axiom,
! [G2: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_7737_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% choose_dvd
thf(fact_7738_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% choose_dvd
thf(fact_7739_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% choose_dvd
thf(fact_7740_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% choose_dvd
thf(fact_7741_prod_Onested__swap,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% prod.nested_swap
thf(fact_7742_prod_Onested__swap,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% prod.nested_swap
thf(fact_7743_fact__numeral,axiom,
! [K: num] :
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
= ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7744_fact__numeral,axiom,
! [K: num] :
( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ K ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ K ) @ ( semiri3624122377584611663nteger @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7745_fact__numeral,axiom,
! [K: num] :
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
= ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7746_fact__numeral,axiom,
! [K: num] :
( ( semiri2447717529341329178atural @ ( numeral_numeral_nat @ K ) )
= ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ K ) @ ( semiri2447717529341329178atural @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7747_fact__numeral,axiom,
! [K: num] :
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
= ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7748_prod__atLeastAtMost__code,axiom,
! [F2: nat > code_integer,A: nat,B: nat] :
( ( groups3455450783089532116nteger @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1084959871951514735nteger
@ ^ [A3: nat] : ( times_3573771949741848930nteger @ ( F2 @ A3 ) )
@ A
@ B
@ one_one_Code_integer ) ) ).
% prod_atLeastAtMost_code
thf(fact_7749_prod__atLeastAtMost__code,axiom,
! [F2: nat > assn,A: nat,B: nat] :
( ( groups6906906614972039071t_assn @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1959793692361082170t_assn
@ ^ [A3: nat] : ( times_times_assn @ ( F2 @ A3 ) )
@ A
@ B
@ one_one_assn ) ) ).
% prod_atLeastAtMost_code
thf(fact_7750_prod__atLeastAtMost__code,axiom,
! [F2: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A3: nat] : ( times_times_rat @ ( F2 @ A3 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7751_prod__atLeastAtMost__code,axiom,
! [F2: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A3: nat] : ( times_times_nat @ ( F2 @ A3 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7752_prod__atLeastAtMost__code,axiom,
! [F2: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A3: nat] : ( times_times_int @ ( F2 @ A3 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_7753_sum__choose__lower,axiom,
! [R3: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K4 ) @ K4 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_7754_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_7755_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_7756_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K4: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K4 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K4 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K4 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ) ) ).
% binomial_code
thf(fact_7757_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > code_integer] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_Code_integer ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_7758_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > assn] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_assn ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_7759_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_7760_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_7761_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_7762_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > code_integer,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7763_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > assn,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7764_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > rat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7765_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > nat,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7766_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > int,P4: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7767_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_7768_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_7769_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G2: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7770_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [G2: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7771_pochhammer__prod,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A3: rat,N2: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% pochhammer_prod
thf(fact_7772_pochhammer__prod,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A3: nat,N2: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% pochhammer_prod
thf(fact_7773_pochhammer__prod,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A3: int,N2: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% pochhammer_prod
thf(fact_7774_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( binomial @ ( minus_minus_nat @ N @ K4 ) @ ( minus_minus_nat @ M @ K4 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_7775_vandermonde,axiom,
! [M: nat,N: nat,R3: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( times_times_nat @ ( binomial @ M @ K4 ) @ ( binomial @ N @ ( minus_minus_nat @ R3 @ K4 ) ) )
@ ( set_ord_atMost_nat @ R3 ) )
= ( binomial @ ( plus_plus_nat @ M @ N ) @ R3 ) ) ).
% vandermonde
thf(fact_7776_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_7777_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7778_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7779_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7780_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A3: rat,N2: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7781_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A3: nat,N2: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7782_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A3: int,N2: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7783_fact__num__eq__if,axiom,
( semiri3624122377584611663nteger
= ( ^ [M2: nat] : ( if_Code_integer @ ( M2 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7784_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7785_fact__num__eq__if,axiom,
( semiri773545260158071498ct_rat
= ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7786_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7787_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri3624122377584611663nteger @ N )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7788_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7789_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7790_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7791_fact__code,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7792_fact__code,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7793_fact__code,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_7794_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% pochhammer_same
thf(fact_7795_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% pochhammer_same
thf(fact_7796_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% pochhammer_same
thf(fact_7797_choose__row__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% choose_row_sum
thf(fact_7798_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K4 ) ) @ ( power_power_nat @ A @ K4 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_7799_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_7800_prod_Oin__pairs,axiom,
! [G2: nat > code_integer,M: nat,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_7801_prod_Oin__pairs,axiom,
! [G2: nat > assn,M: nat,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( times_times_assn @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_7802_prod_Oin__pairs,axiom,
! [G2: nat > rat,M: nat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_7803_prod_Oin__pairs,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_7804_prod_Oin__pairs,axiom,
! [G2: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_7805_prod_Oin__pairs__0,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7806_prod_Oin__pairs__0,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( times_times_assn @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7807_prod_Oin__pairs__0,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7808_prod_Oin__pairs__0,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7809_prod_Oin__pairs__0,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_7810_gbinomial__altdef__of__nat,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K4 @ I3 ) ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K4 ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_7811_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7812_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7813_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7814_binomial__ring,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N )
= ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N @ K4 ) ) @ ( power_8256067586552552935nteger @ A @ K4 ) ) @ ( power_8256067586552552935nteger @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7815_binomial__ring,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K4 ) ) @ ( power_power_int @ A @ K4 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7816_binomial__ring,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K4 ) ) @ ( power_power_rat @ A @ K4 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7817_binomial__ring,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K4 ) ) @ ( power_power_nat @ A @ K4 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_7818_pochhammer__binomial__sum,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N )
= ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N @ K4 ) ) @ ( comm_s8582702949713902594nteger @ A @ K4 ) ) @ ( comm_s8582702949713902594nteger @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7819_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K4 ) ) @ ( comm_s4660882817536571857er_int @ A @ K4 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7820_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K4 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K4 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_7821_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K4 ) ) @ one_one_rat ) @ K4 ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7822_choose__square__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( power_power_nat @ ( binomial @ N @ K4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_7823_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A3: rat,K4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A3 ) @ K4 ) ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_7824_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > code_integer,H3: nat > code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups3455450783089532116nteger
@ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_Code_integer @ ( J3 = K ) @ one_one_Code_integer @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups3455450783089532116nteger
@ ^ [J3: nat] : ( if_Code_integer @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7825_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > assn,H3: nat > assn] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups6906906614972039071t_assn
@ ^ [J3: nat] : ( if_assn @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_assn @ ( J3 = K ) @ one_one_assn @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups6906906614972039071t_assn
@ ^ [J3: nat] : ( if_assn @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7826_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > rat,H3: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7827_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > nat,H3: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7828_prod_Ozero__middle,axiom,
! [P4: nat,K: nat,G2: nat > int,H3: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P4 )
=> ( ( ord_less_eq_nat @ K @ P4 )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P4 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G2 @ J3 ) @ ( H3 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_7829_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_linear_sum
thf(fact_7830_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_7831_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_7832_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_7833_choose__linear__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_7834_sum_Otriangle__reindex,axiom,
! [G2: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_7835_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_7836_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > rat,D2: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D2 @ I3 ) )
@ A2 )
= ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D2 @ I3 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_7837_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_7838_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_7839_sgn__1,axiom,
( ( sgn_sgn_int @ one_one_int )
= one_one_int ) ).
% sgn_1
thf(fact_7840_sgn__1,axiom,
( ( sgn_sgn_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% sgn_1
thf(fact_7841_sgn__1,axiom,
( ( sgn_sgn_rat @ one_one_rat )
= one_one_rat ) ).
% sgn_1
thf(fact_7842_sgn__minus,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% sgn_minus
thf(fact_7843_sgn__minus,axiom,
! [A: code_integer] :
( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).
% sgn_minus
thf(fact_7844_sgn__minus,axiom,
! [A: rat] :
( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
= ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).
% sgn_minus
thf(fact_7845_prod_Oinfinite,axiom,
! [A2: set_nat,G2: nat > code_integer] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ A2 )
= one_one_Code_integer ) ) ).
% prod.infinite
thf(fact_7846_prod_Oinfinite,axiom,
! [A2: set_int,G2: int > code_integer] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ A2 )
= one_one_Code_integer ) ) ).
% prod.infinite
thf(fact_7847_prod_Oinfinite,axiom,
! [A2: set_nat,G2: nat > assn] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= one_one_assn ) ) ).
% prod.infinite
thf(fact_7848_prod_Oinfinite,axiom,
! [A2: set_int,G2: int > assn] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= one_one_assn ) ) ).
% prod.infinite
thf(fact_7849_prod_Oinfinite,axiom,
! [A2: set_nat,G2: nat > rat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ A2 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7850_prod_Oinfinite,axiom,
! [A2: set_int,G2: int > rat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7851_prod_Oinfinite,axiom,
! [A2: set_int,G2: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7852_prod_Oinfinite,axiom,
! [A2: set_nat,G2: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2 @ A2 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7853_prod_Oinfinite,axiom,
! [A2: set_nat,G2: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2 @ A2 )
= one_one_int ) ) ).
% prod.infinite
thf(fact_7854_prod_Oinfinite,axiom,
! [A2: set_int,G2: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= one_one_int ) ) ).
% prod.infinite
thf(fact_7855_divide__sgn,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
= ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).
% divide_sgn
thf(fact_7856_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_7857_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_7858_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > code_integer] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta'
thf(fact_7859_sum_Odelta_H,axiom,
! [S: set_int,A: int,B: int > code_integer] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups7873554091576472773nteger
@ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups7873554091576472773nteger
@ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta'
thf(fact_7860_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_7861_sum_Odelta_H,axiom,
! [S: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_7862_sum_Odelta_H,axiom,
! [S: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_7863_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_7864_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_7865_sum_Odelta_H,axiom,
! [S: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_7866_sum_Odelta_H,axiom,
! [S: set_set_nat,A: set_nat,B: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A @ S )
=> ( ( groups9190459664516455967nteger
@ ^ [K4: set_nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S )
=> ( ( groups9190459664516455967nteger
@ ^ [K4: set_nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta'
thf(fact_7867_sum_Odelta_H,axiom,
! [S: set_set_nat,A: set_nat,B: set_nat > rat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_7868_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > code_integer] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups7501900531339628137nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta
thf(fact_7869_sum_Odelta,axiom,
! [S: set_int,A: int,B: int > code_integer] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups7873554091576472773nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups7873554091576472773nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta
thf(fact_7870_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_7871_sum_Odelta,axiom,
! [S: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_7872_sum_Odelta,axiom,
! [S: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_7873_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_7874_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_7875_sum_Odelta,axiom,
! [S: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_7876_sum_Odelta,axiom,
! [S: set_set_nat,A: set_nat,B: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A @ S )
=> ( ( groups9190459664516455967nteger
@ ^ [K4: set_nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S )
=> ( ( groups9190459664516455967nteger
@ ^ [K4: set_nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ zero_z3403309356797280102nteger )
@ S )
= zero_z3403309356797280102nteger ) ) ) ) ).
% sum.delta
thf(fact_7877_sum_Odelta,axiom,
! [S: set_set_nat,A: set_nat,B: set_nat > rat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_rat )
@ S )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_7878_prod__eq__1__iff,axiom,
! [A2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups1707563613775114915nt_nat @ F2 @ A2 )
= one_one_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( F2 @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7879_prod__eq__1__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( ( groups4077766827762148844at_nat @ F2 @ A2 )
= one_one_nat )
= ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7880_prod__eq__1__iff,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups708209901874060359at_nat @ F2 @ A2 )
= one_one_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F2 @ X3 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_7881_prod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > code_integer] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= one_one_Code_integer ) ) ) ) ).
% prod.delta
thf(fact_7882_prod_Odelta,axiom,
! [S: set_int,A: int,B: int > code_integer] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= one_one_Code_integer ) ) ) ) ).
% prod.delta
thf(fact_7883_prod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > assn] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= one_one_assn ) ) ) ) ).
% prod.delta
thf(fact_7884_prod_Odelta,axiom,
! [S: set_int,A: int,B: int > assn] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= one_one_assn ) ) ) ) ).
% prod.delta
thf(fact_7885_prod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7886_prod_Odelta,axiom,
! [S: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7887_prod_Odelta,axiom,
! [S: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_7888_prod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_7889_prod_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups705719431365010083at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ one_one_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups705719431365010083at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ one_one_int )
@ S )
= one_one_int ) ) ) ) ).
% prod.delta
thf(fact_7890_prod_Odelta,axiom,
! [S: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1705073143266064639nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ one_one_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1705073143266064639nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ one_one_int )
@ S )
= one_one_int ) ) ) ) ).
% prod.delta
thf(fact_7891_prod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > code_integer] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= one_one_Code_integer ) ) ) ) ).
% prod.delta'
thf(fact_7892_prod_Odelta_H,axiom,
! [S: set_int,A: int,B: int > code_integer] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
@ S )
= one_one_Code_integer ) ) ) ) ).
% prod.delta'
thf(fact_7893_prod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > assn] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= one_one_assn ) ) ) ) ).
% prod.delta'
thf(fact_7894_prod_Odelta_H,axiom,
! [S: set_int,A: int,B: int > assn] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
@ S )
= one_one_assn ) ) ) ) ).
% prod.delta'
thf(fact_7895_prod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7896_prod_Odelta_H,axiom,
! [S: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
@ S )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7897_prod_Odelta_H,axiom,
! [S: set_int,A: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K4: int] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1707563613775114915nt_nat
@ ^ [K4: int] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_7898_prod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_7899_prod_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups705719431365010083at_int
@ ^ [K4: nat] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ one_one_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups705719431365010083at_int
@ ^ [K4: nat] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ one_one_int )
@ S )
= one_one_int ) ) ) ) ).
% prod.delta'
thf(fact_7900_prod_Odelta_H,axiom,
! [S: set_int,A: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1705073143266064639nt_int
@ ^ [K4: int] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ one_one_int )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1705073143266064639nt_int
@ ^ [K4: int] : ( if_int @ ( A = K4 ) @ ( B @ K4 ) @ one_one_int )
@ S )
= one_one_int ) ) ) ) ).
% prod.delta'
thf(fact_7901_sgn__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( sgn_sgn_Code_integer @ A )
= one_one_Code_integer ) ) ).
% sgn_pos
thf(fact_7902_sgn__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( sgn_sgn_rat @ A )
= one_one_rat ) ) ).
% sgn_pos
thf(fact_7903_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_7904_prod_Oinsert,axiom,
! [A2: set_o,X: $o,G2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X @ A2 )
=> ( ( groups7694694392188491536nteger @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups7694694392188491536nteger @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7905_prod_Oinsert,axiom,
! [A2: set_nat,X: nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3455450783089532116nteger @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7906_prod_Oinsert,axiom,
! [A2: set_int,X: int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ~ ( member_int @ X @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3827104343326376752nteger @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7907_prod_Oinsert,axiom,
! [A2: set_o,X: $o,G2: $o > assn] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X @ A2 )
=> ( ( groups5301882518646026715o_assn @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups5301882518646026715o_assn @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7908_prod_Oinsert,axiom,
! [A2: set_nat,X: nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups6906906614972039071t_assn @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7909_prod_Oinsert,axiom,
! [A2: set_int,X: int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ~ ( member_int @ X @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups7882442080178216443t_assn @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7910_prod_Oinsert,axiom,
! [A2: set_o,X: $o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X @ A2 )
=> ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7911_prod_Oinsert,axiom,
! [A2: set_nat,X: nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7912_prod_Oinsert,axiom,
! [A2: set_int,X: int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ~ ( member_int @ X @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7913_prod_Oinsert,axiom,
! [A2: set_o,X: $o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X @ A2 )
=> ( ( groups3504817904513533571_o_nat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_nat @ ( G2 @ X ) @ ( groups3504817904513533571_o_nat @ G2 @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_7914_abs__sgn__eq__1,axiom,
! [A: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= one_one_Code_integer ) ) ).
% abs_sgn_eq_1
thf(fact_7915_abs__sgn__eq__1,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= one_one_rat ) ) ).
% abs_sgn_eq_1
thf(fact_7916_abs__sgn__eq__1,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ).
% abs_sgn_eq_1
thf(fact_7917_single__Diff__lessThan,axiom,
! [K: $o] :
( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
= ( insert_o @ K @ bot_bot_set_o ) ) ).
% single_Diff_lessThan
thf(fact_7918_single__Diff__lessThan,axiom,
! [K: int] :
( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
= ( insert_int @ K @ bot_bot_set_int ) ) ).
% single_Diff_lessThan
thf(fact_7919_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_7920_sgn__mult__self__eq,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
= ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).
% sgn_mult_self_eq
thf(fact_7921_sgn__mult__self__eq,axiom,
! [A: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
= ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).
% sgn_mult_self_eq
thf(fact_7922_sgn__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
= ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).
% sgn_mult_self_eq
thf(fact_7923_drop__bit__of__1,axiom,
! [N: nat] :
( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_1
thf(fact_7924_drop__bit__of__1,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_1
thf(fact_7925_drop__bit__of__1,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ one_one_int )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_1
thf(fact_7926_prod_OlessThan__Suc,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7927_prod_OlessThan__Suc,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7928_prod_OlessThan__Suc,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7929_prod_OlessThan__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7930_prod_OlessThan__Suc,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7931_sgn__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_7932_sgn__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% sgn_neg
thf(fact_7933_sgn__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% sgn_neg
thf(fact_7934_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_7935_sum__mult__of__bool__eq,axiom,
! [A2: set_int,F2: int > code_integer,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups7873554091576472773nteger
@ ^ [X3: int] : ( times_3573771949741848930nteger @ ( F2 @ X3 ) @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) )
@ A2 )
= ( groups7873554091576472773nteger @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7936_sum__mult__of__bool__eq,axiom,
! [A2: set_nat,F2: nat > code_integer,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups7501900531339628137nteger
@ ^ [X3: nat] : ( times_3573771949741848930nteger @ ( F2 @ X3 ) @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) )
@ A2 )
= ( groups7501900531339628137nteger @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7937_sum__mult__of__bool__eq,axiom,
! [A2: set_int,F2: int > rat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( times_times_rat @ ( F2 @ X3 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) )
@ A2 )
= ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7938_sum__mult__of__bool__eq,axiom,
! [A2: set_nat,F2: nat > rat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( times_times_rat @ ( F2 @ X3 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) )
@ A2 )
= ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7939_sum__mult__of__bool__eq,axiom,
! [A2: set_nat,F2: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( times_times_int @ ( F2 @ X3 ) @ ( zero_n2684676970156552555ol_int @ ( P @ X3 ) ) )
@ A2 )
= ( groups3539618377306564664at_int @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7940_sum__mult__of__bool__eq,axiom,
! [A2: set_int,F2: int > nat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( times_times_nat @ ( F2 @ X3 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) ) )
@ A2 )
= ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7941_sum__mult__of__bool__eq,axiom,
! [A2: set_nat,F2: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( times_times_nat @ ( F2 @ X3 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) ) )
@ A2 )
= ( groups3542108847815614940at_nat @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7942_sum__mult__of__bool__eq,axiom,
! [A2: set_int,F2: int > int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( times_times_int @ ( F2 @ X3 ) @ ( zero_n2684676970156552555ol_int @ ( P @ X3 ) ) )
@ A2 )
= ( groups4538972089207619220nt_int @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7943_sum__mult__of__bool__eq,axiom,
! [A2: set_set_nat,F2: set_nat > code_integer,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups9190459664516455967nteger
@ ^ [X3: set_nat] : ( times_3573771949741848930nteger @ ( F2 @ X3 ) @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) )
@ A2 )
= ( groups9190459664516455967nteger @ F2 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7944_sum__mult__of__bool__eq,axiom,
! [A2: set_set_nat,F2: set_nat > rat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat
@ ^ [X3: set_nat] : ( times_times_rat @ ( F2 @ X3 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) )
@ A2 )
= ( groups7659867448343625626at_rat @ F2 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_7945_sum__of__bool__mult__eq,axiom,
! [A2: set_int,P: int > $o,F2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( groups7873554091576472773nteger
@ ^ [X3: int] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups7873554091576472773nteger @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7946_sum__of__bool__mult__eq,axiom,
! [A2: set_nat,P: nat > $o,F2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( groups7501900531339628137nteger
@ ^ [X3: nat] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups7501900531339628137nteger @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7947_sum__of__bool__mult__eq,axiom,
! [A2: set_int,P: int > $o,F2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7948_sum__of__bool__mult__eq,axiom,
! [A2: set_nat,P: nat > $o,F2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7949_sum__of__bool__mult__eq,axiom,
! [A2: set_nat,P: nat > $o,F2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups3539618377306564664at_int @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7950_sum__of__bool__mult__eq,axiom,
! [A2: set_int,P: int > $o,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7951_sum__of__bool__mult__eq,axiom,
! [A2: set_nat,P: nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups3542108847815614940at_nat @ F2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7952_sum__of__bool__mult__eq,axiom,
! [A2: set_int,P: int > $o,F2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups4538972089207619220nt_int @ F2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7953_sum__of__bool__mult__eq,axiom,
! [A2: set_set_nat,P: set_nat > $o,F2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups9190459664516455967nteger
@ ^ [X3: set_nat] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups9190459664516455967nteger @ F2 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7954_sum__of__bool__mult__eq,axiom,
! [A2: set_set_nat,P: set_nat > $o,F2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat
@ ^ [X3: set_nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) ) @ ( F2 @ X3 ) )
@ A2 )
= ( groups7659867448343625626at_rat @ F2 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_7955_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_7956_sum__zero__power,axiom,
! [A2: set_nat,C: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_7957_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_7958_int__prod,axiom,
! [F2: int > nat,A2: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_prod
thf(fact_7959_int__prod,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F2 @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F2 @ X3 ) )
@ A2 ) ) ).
% int_prod
thf(fact_7960_finite__if__eq__beyond__finite,axiom,
! [S: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( finite2998713641127702882nt_int @ S )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [S2: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ S2 @ S )
= ( minus_1052850069191792384nt_int @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_7961_finite__if__eq__beyond__finite,axiom,
! [S: set_int,S5: set_int] :
( ( finite_finite_int @ S )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [S2: set_int] :
( ( minus_minus_set_int @ S2 @ S )
= ( minus_minus_set_int @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_7962_finite__if__eq__beyond__finite,axiom,
! [S: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ S )
=> ( finite9047747110432174090at_nat
@ ( collec5514110066124741708at_nat
@ ^ [S2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat @ S2 @ S )
= ( minus_1356011639430497352at_nat @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_7963_finite__if__eq__beyond__finite,axiom,
! [S: set_nat,S5: set_nat] :
( ( finite_finite_nat @ S )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [S2: set_nat] :
( ( minus_minus_set_nat @ S2 @ S )
= ( minus_minus_set_nat @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_7964_sgn__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).
% sgn_mult
thf(fact_7965_sgn__mult,axiom,
! [A: rat,B: rat] :
( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).
% sgn_mult
thf(fact_7966_sgn__mult,axiom,
! [A: int,B: int] :
( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).
% sgn_mult
thf(fact_7967_lessThan__non__empty,axiom,
! [X: int] :
( ( set_ord_lessThan_int @ X )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_7968_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less_nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_7969_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_7970_sum_Oswap__restrict,axiom,
! [A2: set_int,B2: set_nat,G2: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] :
( groups3542108847815614940at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7971_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_int,G2: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] :
( groups4538972089207619220nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups3539618377306564664at_int
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7972_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_int,G2: nat > int > nat,R: nat > int > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups4541462559716669496nt_nat @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups4541462559716669496nt_nat
@ ^ [Y3: int] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7973_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat > nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups3542108847815614940at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7974_sum_Oswap__restrict,axiom,
! [A2: set_int,B2: set_nat,G2: int > nat > int,R: int > nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] :
( groups3539618377306564664at_int @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups3539618377306564664at_int
@ ^ [Y3: nat] :
( groups4538972089207619220nt_int
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7975_sum_Oswap__restrict,axiom,
! [A2: set_int,B2: set_int,G2: int > int > int,R: int > int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] :
( groups4538972089207619220nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups4538972089207619220nt_int
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7976_sum_Oswap__restrict,axiom,
! [A2: set_set_nat,B2: set_nat,G2: set_nat > nat > nat,R: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] :
( groups3542108847815614940at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( G2 @ X3 @ Y3 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7977_sum_Oswap__restrict,axiom,
! [A2: set_set_nat,B2: set_int,G2: set_nat > int > int,R: set_nat > int > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups8292507037921071086at_int
@ ^ [X3: set_nat] :
( groups4538972089207619220nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups4538972089207619220nt_int
@ ^ [Y3: int] :
( groups8292507037921071086at_int
@ ^ [X3: set_nat] : ( G2 @ X3 @ Y3 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7978_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_set_nat,G2: nat > set_nat > nat,R: nat > set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] :
( groups8294997508430121362at_nat @ ( G2 @ X3 )
@ ( collect_set_nat
@ ^ [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups8294997508430121362at_nat
@ ^ [Y3: set_nat] :
( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7979_sum_Oswap__restrict,axiom,
! [A2: set_int,B2: set_set_nat,G2: int > set_nat > int,R: int > set_nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] :
( groups8292507037921071086at_int @ ( G2 @ X3 )
@ ( collect_set_nat
@ ^ [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups8292507037921071086at_int
@ ^ [Y3: set_nat] :
( groups4538972089207619220nt_int
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_7980_prod_Oswap__restrict,axiom,
! [A2: set_int,B2: set_nat,G2: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [X3: int] :
( groups708209901874060359at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups1707563613775114915nt_nat
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7981_prod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_int,G2: nat > int > nat,R: nat > int > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] :
( groups1707563613775114915nt_nat @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups1707563613775114915nt_nat
@ ^ [Y3: int] :
( groups708209901874060359at_nat
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7982_prod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat > nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] :
( groups708209901874060359at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups708209901874060359at_nat
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7983_prod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat > int,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups705719431365010083at_int
@ ^ [X3: nat] :
( groups705719431365010083at_int @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups705719431365010083at_int
@ ^ [Y3: nat] :
( groups705719431365010083at_int
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7984_prod_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_int,G2: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups705719431365010083at_int
@ ^ [X3: nat] :
( groups1705073143266064639nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups1705073143266064639nt_int
@ ^ [Y3: int] :
( groups705719431365010083at_int
@ ^ [X3: nat] : ( G2 @ X3 @ Y3 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7985_prod_Oswap__restrict,axiom,
! [A2: set_int,B2: set_nat,G2: int > nat > int,R: int > nat > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups1705073143266064639nt_int
@ ^ [X3: int] :
( groups705719431365010083at_int @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups705719431365010083at_int
@ ^ [Y3: nat] :
( groups1705073143266064639nt_int
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7986_prod_Oswap__restrict,axiom,
! [A2: set_int,B2: set_int,G2: int > int > int,R: int > int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups1705073143266064639nt_int
@ ^ [X3: int] :
( groups1705073143266064639nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups1705073143266064639nt_int
@ ^ [Y3: int] :
( groups1705073143266064639nt_int
@ ^ [X3: int] : ( G2 @ X3 @ Y3 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7987_prod_Oswap__restrict,axiom,
! [A2: set_set_nat,B2: set_nat,G2: set_nat > nat > nat,R: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4248547760180025341at_nat
@ ^ [X3: set_nat] :
( groups708209901874060359at_nat @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups708209901874060359at_nat
@ ^ [Y3: nat] :
( groups4248547760180025341at_nat
@ ^ [X3: set_nat] : ( G2 @ X3 @ Y3 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7988_prod_Oswap__restrict,axiom,
! [A2: set_set_nat,B2: set_nat,G2: set_nat > nat > int,R: set_nat > nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4246057289670975065at_int
@ ^ [X3: set_nat] :
( groups705719431365010083at_int @ ( G2 @ X3 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups705719431365010083at_int
@ ^ [Y3: nat] :
( groups4246057289670975065at_int
@ ^ [X3: set_nat] : ( G2 @ X3 @ Y3 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7989_prod_Oswap__restrict,axiom,
! [A2: set_set_nat,B2: set_int,G2: set_nat > int > int,R: set_nat > int > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4246057289670975065at_int
@ ^ [X3: set_nat] :
( groups1705073143266064639nt_int @ ( G2 @ X3 )
@ ( collect_int
@ ^ [Y3: int] :
( ( member_int @ Y3 @ B2 )
& ( R @ X3 @ Y3 ) ) ) )
@ A2 )
= ( groups1705073143266064639nt_int
@ ^ [Y3: int] :
( groups4246057289670975065at_int
@ ^ [X3: set_nat] : ( G2 @ X3 @ Y3 )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( R @ X3 @ Y3 ) ) ) )
@ B2 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7990_lessThan__def,axiom,
( set_or4940836740269066044nt_int
= ( ^ [U2: set_Pr958786334691620121nt_int] :
( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_le7563427860532173253nt_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7991_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7992_lessThan__def,axiom,
( set_or7240481858991947888_nat_o
= ( ^ [U2: produc3658429121746597890et_nat > $o] :
( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_le8176553561544679545_nat_o @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7993_lessThan__def,axiom,
( set_or2436086697161274615iteral
= ( ^ [U2: literal] :
( collect_literal
@ ^ [X3: literal] : ( ord_less_literal @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7994_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7995_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7996_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_7997_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_7998_sgn__not__eq__imp,axiom,
! [B: int,A: int] :
( ( ( sgn_sgn_int @ B )
!= ( sgn_sgn_int @ A ) )
=> ( ( ( sgn_sgn_int @ A )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_7999_sgn__not__eq__imp,axiom,
! [B: code_integer,A: code_integer] :
( ( ( sgn_sgn_Code_integer @ B )
!= ( sgn_sgn_Code_integer @ A ) )
=> ( ( ( sgn_sgn_Code_integer @ A )
!= zero_z3403309356797280102nteger )
=> ( ( ( sgn_sgn_Code_integer @ B )
!= zero_z3403309356797280102nteger )
=> ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_8000_sgn__not__eq__imp,axiom,
! [B: rat,A: rat] :
( ( ( sgn_sgn_rat @ B )
!= ( sgn_sgn_rat @ A ) )
=> ( ( ( sgn_sgn_rat @ A )
!= zero_zero_rat )
=> ( ( ( sgn_sgn_rat @ B )
!= zero_zero_rat )
=> ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_8001_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_8002_sgn__minus__1,axiom,
( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% sgn_minus_1
thf(fact_8003_sgn__minus__1,axiom,
( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% sgn_minus_1
thf(fact_8004_mult__sgn__abs,axiom,
! [X: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_8005_mult__sgn__abs,axiom,
! [X: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_8006_mult__sgn__abs,axiom,
! [X: int] :
( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
= X ) ).
% mult_sgn_abs
thf(fact_8007_sgn__mult__abs,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_8008_sgn__mult__abs,axiom,
! [A: rat] :
( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_8009_sgn__mult__abs,axiom,
! [A: int] :
( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
= A ) ).
% sgn_mult_abs
thf(fact_8010_abs__mult__sgn,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_8011_abs__mult__sgn,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_8012_abs__mult__sgn,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
= A ) ).
% abs_mult_sgn
thf(fact_8013_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_Code_integer
= ( ^ [K4: code_integer] : ( times_3573771949741848930nteger @ K4 @ ( sgn_sgn_Code_integer @ K4 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_8014_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_rat
= ( ^ [K4: rat] : ( times_times_rat @ K4 @ ( sgn_sgn_rat @ K4 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_8015_linordered__idom__class_Oabs__sgn,axiom,
( abs_abs_int
= ( ^ [K4: int] : ( times_times_int @ K4 @ ( sgn_sgn_int @ K4 ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_8016_Iio__eq__empty__iff,axiom,
! [N: $o] :
( ( ( set_ord_lessThan_o @ N )
= bot_bot_set_o )
= ( N = bot_bot_o ) ) ).
% Iio_eq_empty_iff
thf(fact_8017_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_8018_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > code_integer,Y: nat > code_integer] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_z3403309356797280102nteger ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8019_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > code_integer,Y: int > code_integer] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_z3403309356797280102nteger ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8020_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8021_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > rat,Y: int > rat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8022_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8023_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8024_sum_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( plus_plus_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8025_sum_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( plus_plus_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8026_sum_Ofinite__Collect__op,axiom,
! [I4: set_set_nat,X: set_nat > code_integer,Y: set_nat > code_integer] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_z3403309356797280102nteger ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8027_sum_Ofinite__Collect__op,axiom,
! [I4: set_set_nat,X: set_nat > rat,Y: set_nat > rat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_8028_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > code_integer,Y: nat > code_integer] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_Code_integer ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_Code_integer ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_Code_integer ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8029_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > code_integer,Y: int > code_integer] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_Code_integer ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_Code_integer ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_Code_integer ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8030_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > assn,Y: nat > assn] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_assn ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_assn ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_assn ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8031_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > assn,Y: int > assn] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_assn ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_assn ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_assn ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8032_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8033_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > rat,Y: int > rat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8034_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8035_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8036_prod_Ofinite__Collect__op,axiom,
! [I4: set_nat,X: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8037_prod_Ofinite__Collect__op,axiom,
! [I4: set_int,X: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( X @ I3 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( times_times_int @ ( X @ I3 ) @ ( Y @ I3 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_8038_sum_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > code_integer,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups7501900531339628137nteger @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7501900531339628137nteger
@ ^ [X3: nat] : ( if_Code_integer @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8039_sum_Ointer__filter,axiom,
! [A2: set_int,G2: int > code_integer,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups7873554091576472773nteger @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7873554091576472773nteger
@ ^ [X3: int] : ( if_Code_integer @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8040_sum_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > rat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8041_sum_Ointer__filter,axiom,
! [A2: set_int,G2: int > rat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8042_sum_Ointer__filter,axiom,
! [A2: set_int,G2: int > nat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8043_sum_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( if_int @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8044_sum_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8045_sum_Ointer__filter,axiom,
! [A2: set_int,G2: int > int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8046_sum_Ointer__filter,axiom,
! [A2: set_set_nat,G2: set_nat > code_integer,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups9190459664516455967nteger @ G2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups9190459664516455967nteger
@ ^ [X3: set_nat] : ( if_Code_integer @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8047_sum_Ointer__filter,axiom,
! [A2: set_set_nat,G2: set_nat > rat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat @ G2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7659867448343625626at_rat
@ ^ [X3: set_nat] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_8048_prod_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > code_integer,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3455450783089532116nteger
@ ^ [X3: nat] : ( if_Code_integer @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_Code_integer )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8049_prod_Ointer__filter,axiom,
! [A2: set_int,G2: int > code_integer,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups3827104343326376752nteger
@ ^ [X3: int] : ( if_Code_integer @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_Code_integer )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8050_prod_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > assn,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups6906906614972039071t_assn
@ ^ [X3: nat] : ( if_assn @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_assn )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8051_prod_Ointer__filter,axiom,
! [A2: set_int,G2: int > assn,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups7882442080178216443t_assn
@ ^ [X3: int] : ( if_assn @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_assn )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8052_prod_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > rat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8053_prod_Ointer__filter,axiom,
! [A2: set_int,G2: int > rat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups1072433553688619179nt_rat
@ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8054_prod_Ointer__filter,axiom,
! [A2: set_int,G2: int > nat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups1707563613775114915nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_nat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8055_prod_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_nat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8056_prod_Ointer__filter,axiom,
! [A2: set_nat,G2: nat > int,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( if_int @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_int )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8057_prod_Ointer__filter,axiom,
! [A2: set_int,G2: int > int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( G2 @ X3 ) @ one_one_int )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8058_sum__strict__mono,axiom,
! [A2: set_o,F2: $o > rat,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F2 @ A2 ) @ ( groups7872700643590313910_o_rat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8059_sum__strict__mono,axiom,
! [A2: set_nat,F2: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F2 @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8060_sum__strict__mono,axiom,
! [A2: set_int,F2: int > rat,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F2 @ A2 ) @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8061_sum__strict__mono,axiom,
! [A2: set_o,F2: $o > nat,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F2 @ A2 ) @ ( groups8507830703676809646_o_nat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8062_sum__strict__mono,axiom,
! [A2: set_int,F2: int > nat,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8063_sum__strict__mono,axiom,
! [A2: set_o,F2: $o > int,G2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_int @ ( groups8505340233167759370_o_int @ F2 @ A2 ) @ ( groups8505340233167759370_o_int @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8064_sum__strict__mono,axiom,
! [A2: set_nat,F2: nat > int,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( groups3539618377306564664at_int @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8065_sum__strict__mono,axiom,
! [A2: set_nat,F2: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8066_sum__strict__mono,axiom,
! [A2: set_int,F2: int > int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ord_less_int @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8067_sum__strict__mono,axiom,
! [A2: set_set_nat,F2: set_nat > rat,G2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_rat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_less_rat @ ( groups7659867448343625626at_rat @ F2 @ A2 ) @ ( groups7659867448343625626at_rat @ G2 @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_8068_sgn__1__pos,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= one_one_Code_integer )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% sgn_1_pos
thf(fact_8069_sgn__1__pos,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= one_one_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% sgn_1_pos
thf(fact_8070_sgn__1__pos,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_1_pos
thf(fact_8071_prod_Orelated,axiom,
! [R: code_integer > code_integer > $o,S: set_nat,H3: nat > code_integer,G2: nat > code_integer] :
( ( R @ one_one_Code_integer @ one_one_Code_integer )
=> ( ! [X13: code_integer,Y12: code_integer,X24: code_integer,Y23: code_integer] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_3573771949741848930nteger @ X13 @ Y12 ) @ ( times_3573771949741848930nteger @ X24 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups3455450783089532116nteger @ H3 @ S ) @ ( groups3455450783089532116nteger @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8072_prod_Orelated,axiom,
! [R: code_integer > code_integer > $o,S: set_int,H3: int > code_integer,G2: int > code_integer] :
( ( R @ one_one_Code_integer @ one_one_Code_integer )
=> ( ! [X13: code_integer,Y12: code_integer,X24: code_integer,Y23: code_integer] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_3573771949741848930nteger @ X13 @ Y12 ) @ ( times_3573771949741848930nteger @ X24 @ Y23 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups3827104343326376752nteger @ H3 @ S ) @ ( groups3827104343326376752nteger @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8073_prod_Orelated,axiom,
! [R: assn > assn > $o,S: set_nat,H3: nat > assn,G2: nat > assn] :
( ( R @ one_one_assn @ one_one_assn )
=> ( ! [X13: assn,Y12: assn,X24: assn,Y23: assn] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_assn @ X13 @ Y12 ) @ ( times_times_assn @ X24 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups6906906614972039071t_assn @ H3 @ S ) @ ( groups6906906614972039071t_assn @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8074_prod_Orelated,axiom,
! [R: assn > assn > $o,S: set_int,H3: int > assn,G2: int > assn] :
( ( R @ one_one_assn @ one_one_assn )
=> ( ! [X13: assn,Y12: assn,X24: assn,Y23: assn] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_assn @ X13 @ Y12 ) @ ( times_times_assn @ X24 @ Y23 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups7882442080178216443t_assn @ H3 @ S ) @ ( groups7882442080178216443t_assn @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8075_prod_Orelated,axiom,
! [R: rat > rat > $o,S: set_nat,H3: nat > rat,G2: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X13: rat,Y12: rat,X24: rat,Y23: rat] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_rat @ X13 @ Y12 ) @ ( times_times_rat @ X24 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H3 @ S ) @ ( groups73079841787564623at_rat @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8076_prod_Orelated,axiom,
! [R: rat > rat > $o,S: set_int,H3: int > rat,G2: int > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X13: rat,Y12: rat,X24: rat,Y23: rat] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_rat @ X13 @ Y12 ) @ ( times_times_rat @ X24 @ Y23 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups1072433553688619179nt_rat @ H3 @ S ) @ ( groups1072433553688619179nt_rat @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8077_prod_Orelated,axiom,
! [R: nat > nat > $o,S: set_int,H3: int > nat,G2: int > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X13: nat,Y12: nat,X24: nat,Y23: nat] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_nat @ X13 @ Y12 ) @ ( times_times_nat @ X24 @ Y23 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups1707563613775114915nt_nat @ H3 @ S ) @ ( groups1707563613775114915nt_nat @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8078_prod_Orelated,axiom,
! [R: nat > nat > $o,S: set_nat,H3: nat > nat,G2: nat > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X13: nat,Y12: nat,X24: nat,Y23: nat] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_nat @ X13 @ Y12 ) @ ( times_times_nat @ X24 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups708209901874060359at_nat @ H3 @ S ) @ ( groups708209901874060359at_nat @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8079_prod_Orelated,axiom,
! [R: int > int > $o,S: set_nat,H3: nat > int,G2: nat > int] :
( ( R @ one_one_int @ one_one_int )
=> ( ! [X13: int,Y12: int,X24: int,Y23: int] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_int @ X13 @ Y12 ) @ ( times_times_int @ X24 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups705719431365010083at_int @ H3 @ S ) @ ( groups705719431365010083at_int @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8080_prod_Orelated,axiom,
! [R: int > int > $o,S: set_int,H3: int > int,G2: int > int] :
( ( R @ one_one_int @ one_one_int )
=> ( ! [X13: int,Y12: int,X24: int,Y23: int] :
( ( ( R @ X13 @ X24 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times_int @ X13 @ Y12 ) @ ( times_times_int @ X24 @ Y23 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H3 @ X2 ) @ ( G2 @ X2 ) ) )
=> ( R @ ( groups1705073143266064639nt_int @ H3 @ S ) @ ( groups1705073143266064639nt_int @ G2 @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_8081_prod_Oinsert__if,axiom,
! [A2: set_o,X: $o,G2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X @ A2 )
=> ( ( groups7694694392188491536nteger @ G2 @ ( insert_o @ X @ A2 ) )
= ( groups7694694392188491536nteger @ G2 @ A2 ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( groups7694694392188491536nteger @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups7694694392188491536nteger @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8082_prod_Oinsert__if,axiom,
! [A2: set_nat,X: nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( insert_nat @ X @ A2 ) )
= ( groups3455450783089532116nteger @ G2 @ A2 ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3455450783089532116nteger @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8083_prod_Oinsert__if,axiom,
! [A2: set_int,X: int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( ( member_int @ X @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( insert_int @ X @ A2 ) )
= ( groups3827104343326376752nteger @ G2 @ A2 ) ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3827104343326376752nteger @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8084_prod_Oinsert__if,axiom,
! [A2: set_o,X: $o,G2: $o > assn] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X @ A2 )
=> ( ( groups5301882518646026715o_assn @ G2 @ ( insert_o @ X @ A2 ) )
= ( groups5301882518646026715o_assn @ G2 @ A2 ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( groups5301882518646026715o_assn @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups5301882518646026715o_assn @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8085_prod_Oinsert__if,axiom,
! [A2: set_nat,X: nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( insert_nat @ X @ A2 ) )
= ( groups6906906614972039071t_assn @ G2 @ A2 ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups6906906614972039071t_assn @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8086_prod_Oinsert__if,axiom,
! [A2: set_int,X: int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( ( member_int @ X @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( insert_int @ X @ A2 ) )
= ( groups7882442080178216443t_assn @ G2 @ A2 ) ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups7882442080178216443t_assn @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8087_prod_Oinsert__if,axiom,
! [A2: set_o,X: $o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X @ A2 )
=> ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
= ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8088_prod_Oinsert__if,axiom,
! [A2: set_nat,X: nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( groups73079841787564623at_rat @ G2 @ A2 ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8089_prod_Oinsert__if,axiom,
! [A2: set_int,X: int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( ( member_int @ X @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
= ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) )
& ( ~ ( member_int @ X @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8090_prod_Oinsert__if,axiom,
! [A2: set_o,X: $o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X @ A2 )
=> ( ( groups3504817904513533571_o_nat @ G2 @ ( insert_o @ X @ A2 ) )
= ( groups3504817904513533571_o_nat @ G2 @ A2 ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( groups3504817904513533571_o_nat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_nat @ ( G2 @ X ) @ ( groups3504817904513533571_o_nat @ G2 @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_8091_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S: set_int,I: int > int,J: int > int,T: set_int,G2: int > code_integer,H3: int > code_integer] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3827104343326376752nteger @ G2 @ S )
= ( groups3827104343326376752nteger @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8092_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S: set_int,I: int > int,J: int > int,T: set_int,G2: int > assn,H3: int > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ S )
= ( groups7882442080178216443t_assn @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8093_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S: set_int,I: int > int,J: int > int,T: set_int,G2: int > rat,H3: int > rat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ S )
= ( groups1072433553688619179nt_rat @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8094_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S: set_int,I: int > int,J: int > int,T: set_int,G2: int > nat,H3: int > nat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_nat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_nat ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ S )
= ( groups1707563613775114915nt_nat @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8095_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_nat,S: set_int,I: nat > int,J: int > nat,T: set_nat,G2: int > code_integer,H3: nat > code_integer] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_nat @ ( J @ A6 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3827104343326376752nteger @ G2 @ S )
= ( groups3455450783089532116nteger @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8096_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_nat,S: set_int,I: nat > int,J: int > nat,T: set_nat,G2: int > assn,H3: nat > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_nat @ ( J @ A6 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ S )
= ( groups6906906614972039071t_assn @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8097_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_nat,S: set_int,I: nat > int,J: int > nat,T: set_nat,G2: int > rat,H3: nat > rat] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_nat @ ( J @ A6 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_int @ ( I @ B6 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ S )
= ( groups73079841787564623at_rat @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8098_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_int,S: set_nat,I: int > nat,J: nat > int,T: set_int,G2: nat > code_integer,H3: int > code_integer] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_nat @ ( I @ B6 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3455450783089532116nteger @ G2 @ S )
= ( groups3827104343326376752nteger @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8099_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_int,S: set_nat,I: int > nat,J: nat > int,T: set_int,G2: nat > assn,H3: int > assn] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_nat @ ( I @ B6 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ S )
= ( groups7882442080178216443t_assn @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8100_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_int,S: set_nat,I: int > nat,J: nat > int,T: set_int,G2: nat > rat,H3: int > rat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_int @ ( J @ A6 ) @ ( minus_minus_set_int @ T @ T4 ) ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ T @ T4 ) )
=> ( member_nat @ ( I @ B6 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S4 )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ T4 )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S )
=> ( ( H3 @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups73079841787564623at_rat @ G2 @ S )
= ( groups1072433553688619179nt_rat @ H3 @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8101_abs__sgn__eq,axiom,
! [A: code_integer] :
( ( ( A = zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= zero_z3403309356797280102nteger ) )
& ( ( A != zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
= one_one_Code_integer ) ) ) ).
% abs_sgn_eq
thf(fact_8102_abs__sgn__eq,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
= one_one_rat ) ) ) ).
% abs_sgn_eq
thf(fact_8103_abs__sgn__eq,axiom,
! [A: int] :
( ( ( A = zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= zero_zero_int ) )
& ( ( A != zero_zero_int )
=> ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
= one_one_int ) ) ) ).
% abs_sgn_eq
thf(fact_8104_sum_Onat__diff__reindex,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_8105_prod_Onat__diff__reindex,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8106_prod_Onat__diff__reindex,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8107_sum__eq__1__iff,axiom,
! [A2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ( F2 @ X3 )
= one_one_nat )
& ! [Y3: int] :
( ( member_int @ Y3 @ A2 )
=> ( ( X3 != Y3 )
=> ( ( F2 @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_8108_sum__eq__1__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( ( groups977919841031483927at_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
& ( ( F2 @ X3 )
= one_one_nat )
& ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ A2 )
=> ( ( X3 != Y3 )
=> ( ( F2 @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_8109_sum__eq__1__iff,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ A2 )
= one_one_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( F2 @ X3 )
= one_one_nat )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( X3 != Y3 )
=> ( ( F2 @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_8110_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_8111_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: code_integer,N: nat] :
( ( divide6298287555418463151nteger @ A @ ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer ) )
= ( bit_se3928097537394005634nteger @ N @ A ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_8112_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: code_natural,N: nat] :
( ( divide5121882707175180666atural @ A @ ( bit_se6611745700429515170atural @ N @ one_one_Code_natural ) )
= ( bit_se2751692689150723149atural @ N @ A ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_8113_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: int,N: nat] :
( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
= ( bit_se8568078237143864401it_int @ N @ A ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_8114_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: nat,N: nat] :
( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
= ( bit_se8570568707652914677it_nat @ N @ A ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_8115_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F2: nat > code_integer,B2: code_integer,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ F2 @ S3 )
= B2 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8116_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F2: int > code_integer,B2: code_integer,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ F2 @ S3 )
= B2 )
=> ( ( member_int @ I @ S3 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8117_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F2: nat > rat,B2: rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F2 @ S3 )
= B2 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8118_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F2: int > rat,B2: rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F2 @ S3 )
= B2 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8119_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F2: int > nat,B2: nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ S3 )
= B2 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8120_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F2: nat > int,B2: int,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F2 @ S3 )
= B2 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8121_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F2: nat > nat,B2: nat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ S3 )
= B2 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8122_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F2: int > int,B2: int,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F2 @ S3 )
= B2 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_int @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8123_sum__nonneg__leq__bound,axiom,
! [S3: set_set_nat,F2: set_nat > code_integer,B2: code_integer,I: set_nat] :
( ( finite1152437895449049373et_nat @ S3 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups9190459664516455967nteger @ F2 @ S3 )
= B2 )
=> ( ( member_set_nat @ I @ S3 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8124_sum__nonneg__leq__bound,axiom,
! [S3: set_set_nat,F2: set_nat > rat,B2: rat,I: set_nat] :
( ( finite1152437895449049373et_nat @ S3 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups7659867448343625626at_rat @ F2 @ S3 )
= B2 )
=> ( ( member_set_nat @ I @ S3 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_8125_sum__nonneg__0,axiom,
! [S3: set_nat,F2: nat > code_integer,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ F2 @ S3 )
= zero_z3403309356797280102nteger )
=> ( ( member_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_z3403309356797280102nteger ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8126_sum__nonneg__0,axiom,
! [S3: set_int,F2: int > code_integer,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ F2 @ S3 )
= zero_z3403309356797280102nteger )
=> ( ( member_int @ I @ S3 )
=> ( ( F2 @ I )
= zero_z3403309356797280102nteger ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8127_sum__nonneg__0,axiom,
! [S3: set_nat,F2: nat > rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F2 @ S3 )
= zero_zero_rat )
=> ( ( member_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8128_sum__nonneg__0,axiom,
! [S3: set_int,F2: int > rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F2 @ S3 )
= zero_zero_rat )
=> ( ( member_int @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8129_sum__nonneg__0,axiom,
! [S3: set_int,F2: int > nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F2 @ S3 )
= zero_zero_nat )
=> ( ( member_int @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8130_sum__nonneg__0,axiom,
! [S3: set_nat,F2: nat > int,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ( ( groups3539618377306564664at_int @ F2 @ S3 )
= zero_zero_int )
=> ( ( member_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8131_sum__nonneg__0,axiom,
! [S3: set_nat,F2: nat > nat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F2 @ S3 )
= zero_zero_nat )
=> ( ( member_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8132_sum__nonneg__0,axiom,
! [S3: set_int,F2: int > int,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F2 @ S3 )
= zero_zero_int )
=> ( ( member_int @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8133_sum__nonneg__0,axiom,
! [S3: set_set_nat,F2: set_nat > code_integer,I: set_nat] :
( ( finite1152437895449049373et_nat @ S3 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S3 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ( ( groups9190459664516455967nteger @ F2 @ S3 )
= zero_z3403309356797280102nteger )
=> ( ( member_set_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_z3403309356797280102nteger ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8134_sum__nonneg__0,axiom,
! [S3: set_set_nat,F2: set_nat > rat,I: set_nat] :
( ( finite1152437895449049373et_nat @ S3 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ( ( groups7659867448343625626at_rat @ F2 @ S3 )
= zero_zero_rat )
=> ( ( member_set_nat @ I @ S3 )
=> ( ( F2 @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_8135_ivl__disj__un__one_I2_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or4029947393144176647an_rat @ L @ U ) )
= ( set_ord_lessThan_rat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_8136_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_8137_ivl__disj__un__one_I2_J,axiom,
! [L: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or8404916559141939852nteger @ L @ U ) )
= ( set_or5754767410780653050nteger @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_8138_ivl__disj__un__one_I2_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( set_ord_lessThan_int @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_8139_ivl__disj__int__one_I4_J,axiom,
! [L: $o,U: $o] :
( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_one(4)
thf(fact_8140_ivl__disj__int__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(4)
thf(fact_8141_ivl__disj__int__one_I4_J,axiom,
! [L: int,U: int] :
( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
= bot_bot_set_int ) ).
% ivl_disj_int_one(4)
thf(fact_8142_ivl__disj__int__one_I4_J,axiom,
! [L: code_integer,U: code_integer] :
( ( inf_in1364745209274528805nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or189985376899183464nteger @ L @ U ) )
= bot_bo3990330152332043303nteger ) ).
% ivl_disj_int_one(4)
thf(fact_8143_prod_Otriangle__reindex,axiom,
! [G2: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K4: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8144_prod_Otriangle__reindex,axiom,
! [G2: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K4: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8145_ivl__disj__int__one_I2_J,axiom,
! [L: $o,U: $o] :
( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or7139685690850216873Than_o @ L @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_one(2)
thf(fact_8146_ivl__disj__int__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(2)
thf(fact_8147_ivl__disj__int__one_I2_J,axiom,
! [L: code_integer,U: code_integer] :
( ( inf_in1364745209274528805nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or8404916559141939852nteger @ L @ U ) )
= bot_bo3990330152332043303nteger ) ).
% ivl_disj_int_one(2)
thf(fact_8148_ivl__disj__int__one_I2_J,axiom,
! [L: int,U: int] :
( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or4662586982721622107an_int @ L @ U ) )
= bot_bot_set_int ) ).
% ivl_disj_int_one(2)
thf(fact_8149_sum_Ointer__restrict,axiom,
! [A2: set_int,G2: int > code_integer,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups7873554091576472773nteger @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups7873554091576472773nteger
@ ^ [X3: int] : ( if_Code_integer @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8150_sum_Ointer__restrict,axiom,
! [A2: set_int,G2: int > rat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( if_rat @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8151_sum_Ointer__restrict,axiom,
! [A2: set_int,G2: int > nat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8152_sum_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > code_integer,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups7501900531339628137nteger @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups7501900531339628137nteger
@ ^ [X3: nat] : ( if_Code_integer @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8153_sum_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > rat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( if_rat @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8154_sum_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > int,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8155_sum_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8156_sum_Ointer__restrict,axiom,
! [A2: set_int,G2: int > int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_int )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8157_sum_Ointer__restrict,axiom,
! [A2: set_set_nat,G2: set_nat > code_integer,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups9190459664516455967nteger @ G2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( groups9190459664516455967nteger
@ ^ [X3: set_nat] : ( if_Code_integer @ ( member_set_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_z3403309356797280102nteger )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8158_sum_Ointer__restrict,axiom,
! [A2: set_set_nat,G2: set_nat > rat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat @ G2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( groups7659867448343625626at_rat
@ ^ [X3: set_nat] : ( if_rat @ ( member_set_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ zero_zero_rat )
@ A2 ) ) ) ).
% sum.inter_restrict
thf(fact_8159_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( groups7873554091576472773nteger @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= zero_z3403309356797280102nteger ) ) ) )
= ( groups7873554091576472773nteger @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8160_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= zero_zero_rat ) ) ) )
= ( groups3906332499630173760nt_rat @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8161_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8162_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( groups7501900531339628137nteger @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= zero_z3403309356797280102nteger ) ) ) )
= ( groups7501900531339628137nteger @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8163_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= zero_zero_rat ) ) ) )
= ( groups2906978787729119204at_rat @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8164_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= zero_zero_int ) ) ) )
= ( groups3539618377306564664at_int @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8165_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8166_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= zero_zero_int ) ) ) )
= ( groups4538972089207619220nt_int @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8167_sum_Osetdiff__irrelevant,axiom,
! [A2: set_set_nat,G2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups9190459664516455967nteger @ G2
@ ( minus_2163939370556025621et_nat @ A2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( G2 @ X3 )
= zero_z3403309356797280102nteger ) ) ) )
= ( groups9190459664516455967nteger @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8168_sum_Osetdiff__irrelevant,axiom,
! [A2: set_set_nat,G2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat @ G2
@ ( minus_2163939370556025621et_nat @ A2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( G2 @ X3 )
= zero_zero_rat ) ) ) )
= ( groups7659867448343625626at_rat @ G2 @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_8169_prod_Ointer__restrict,axiom,
! [A2: set_int,G2: int > code_integer,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups3827104343326376752nteger
@ ^ [X3: int] : ( if_Code_integer @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_Code_integer )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8170_prod_Ointer__restrict,axiom,
! [A2: set_int,G2: int > assn,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups7882442080178216443t_assn
@ ^ [X3: int] : ( if_assn @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_assn )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8171_prod_Ointer__restrict,axiom,
! [A2: set_int,G2: int > rat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X3: int] : ( if_rat @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8172_prod_Ointer__restrict,axiom,
! [A2: set_int,G2: int > nat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups1707563613775114915nt_nat
@ ^ [X3: int] : ( if_nat @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_nat )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8173_prod_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > code_integer,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups3455450783089532116nteger
@ ^ [X3: nat] : ( if_Code_integer @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_Code_integer )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8174_prod_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > assn,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups6906906614972039071t_assn
@ ^ [X3: nat] : ( if_assn @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_assn )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8175_prod_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > rat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( if_rat @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8176_prod_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_nat )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8177_prod_Ointer__restrict,axiom,
! [A2: set_nat,G2: nat > int,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( groups705719431365010083at_int
@ ^ [X3: nat] : ( if_int @ ( member_nat @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_int )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8178_prod_Ointer__restrict,axiom,
! [A2: set_int,G2: int > int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( if_int @ ( member_int @ X3 @ B2 ) @ ( G2 @ X3 ) @ one_one_int )
@ A2 ) ) ) ).
% prod.inter_restrict
thf(fact_8179_prod_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= one_one_Code_integer ) ) ) )
= ( groups3827104343326376752nteger @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8180_prod_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= one_one_assn ) ) ) )
= ( groups7882442080178216443t_assn @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8181_prod_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= one_one_rat ) ) ) )
= ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8182_prod_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= one_one_nat ) ) ) )
= ( groups1707563613775114915nt_nat @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8183_prod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= one_one_Code_integer ) ) ) )
= ( groups3455450783089532116nteger @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8184_prod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= one_one_assn ) ) ) )
= ( groups6906906614972039071t_assn @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8185_prod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= one_one_rat ) ) ) )
= ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8186_prod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= one_one_nat ) ) ) )
= ( groups708209901874060359at_nat @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8187_prod_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X3: nat] :
( ( G2 @ X3 )
= one_one_int ) ) ) )
= ( groups705719431365010083at_int @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8188_prod_Osetdiff__irrelevant,axiom,
! [A2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2
@ ( minus_minus_set_int @ A2
@ ( collect_int
@ ^ [X3: int] :
( ( G2 @ X3 )
= one_one_int ) ) ) )
= ( groups1705073143266064639nt_int @ G2 @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8189_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_8190_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_8191_sum__diff__distrib,axiom,
! [Q2: nat > nat,P: nat > nat,N: nat] :
( ! [X2: nat] : ( ord_less_eq_nat @ ( Q2 @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q2 @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q2 @ X3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_8192_sum__pos,axiom,
! [I4: set_o,F2: $o > code_integer] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups4406642042086082107nteger @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8193_sum__pos,axiom,
! [I4: set_nat,F2: nat > code_integer] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8194_sum__pos,axiom,
! [I4: set_int,F2: int > code_integer] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8195_sum__pos,axiom,
! [I4: set_o,F2: $o > rat] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8196_sum__pos,axiom,
! [I4: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8197_sum__pos,axiom,
! [I4: set_int,F2: int > rat] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8198_sum__pos,axiom,
! [I4: set_o,F2: $o > nat] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8199_sum__pos,axiom,
! [I4: set_int,F2: int > nat] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F2 @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8200_sum__pos,axiom,
! [I4: set_o,F2: $o > int] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_less_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8201_sum__pos,axiom,
! [I4: set_nat,F2: nat > int] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_int @ zero_zero_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F2 @ I4 ) ) ) ) ) ).
% sum_pos
thf(fact_8202_less__1__prod2,axiom,
! [I4: set_nat,I: nat,F2: nat > code_integer] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I @ I4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8203_less__1__prod2,axiom,
! [I4: set_int,I: int,F2: int > code_integer] :
( ( finite_finite_int @ I4 )
=> ( ( member_int @ I @ I4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8204_less__1__prod2,axiom,
! [I4: set_nat,I: nat,F2: nat > rat] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I @ I4 )
=> ( ( ord_less_rat @ one_one_rat @ ( F2 @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8205_less__1__prod2,axiom,
! [I4: set_int,I: int,F2: int > rat] :
( ( finite_finite_int @ I4 )
=> ( ( member_int @ I @ I4 )
=> ( ( ord_less_rat @ one_one_rat @ ( F2 @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8206_less__1__prod2,axiom,
! [I4: set_nat,I: nat,F2: nat > int] :
( ( finite_finite_nat @ I4 )
=> ( ( member_nat @ I @ I4 )
=> ( ( ord_less_int @ one_one_int @ ( F2 @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8207_less__1__prod2,axiom,
! [I4: set_int,I: int,F2: int > int] :
( ( finite_finite_int @ I4 )
=> ( ( member_int @ I @ I4 )
=> ( ( ord_less_int @ one_one_int @ ( F2 @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8208_less__1__prod2,axiom,
! [I4: set_set_nat,I: set_nat,F2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I @ I4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups2761809935038513290nteger @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8209_less__1__prod2,axiom,
! [I4: set_set_nat,I: set_nat,F2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I @ I4 )
=> ( ( ord_less_rat @ one_one_rat @ ( F2 @ I ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups3613417700093529605at_rat @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8210_less__1__prod2,axiom,
! [I4: set_set_nat,I: set_nat,F2: set_nat > int] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( member_set_nat @ I @ I4 )
=> ( ( ord_less_int @ one_one_int @ ( F2 @ I ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4246057289670975065at_int @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8211_less__1__prod2,axiom,
! [I4: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat,F2: product_prod_nat_nat > code_integer] :
( ( finite6177210948735845034at_nat @ I4 )
=> ( ( member8440522571783428010at_nat @ I @ I4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I ) )
=> ( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I4 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups1230400874837758585nteger @ F2 @ I4 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8212_less__1__prod,axiom,
! [I4: set_o,F2: $o > code_integer] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8213_less__1__prod,axiom,
! [I4: set_nat,F2: nat > code_integer] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8214_less__1__prod,axiom,
! [I4: set_int,F2: int > code_integer] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8215_less__1__prod,axiom,
! [I4: set_o,F2: $o > rat] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_less_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8216_less__1__prod,axiom,
! [I4: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8217_less__1__prod,axiom,
! [I4: set_int,F2: int > rat] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_rat @ one_one_rat @ ( F2 @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8218_less__1__prod,axiom,
! [I4: set_o,F2: $o > int] :
( ( finite_finite_o @ I4 )
=> ( ( I4 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I4 )
=> ( ord_less_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8219_less__1__prod,axiom,
! [I4: set_nat,F2: nat > int] :
( ( finite_finite_nat @ I4 )
=> ( ( I4 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I4 )
=> ( ord_less_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8220_less__1__prod,axiom,
! [I4: set_int,F2: int > int] :
( ( finite_finite_int @ I4 )
=> ( ( I4 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I4 )
=> ( ord_less_int @ one_one_int @ ( F2 @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8221_less__1__prod,axiom,
! [I4: set_set_nat,F2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ I4 )
=> ( ( I4 != bot_bot_set_set_nat )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups2761809935038513290nteger @ F2 @ I4 ) ) ) ) ) ).
% less_1_prod
thf(fact_8222_sum_OlessThan__Suc__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G2 @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8223_sum_OlessThan__Suc__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G2 @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8224_sum_OlessThan__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_8225_sum__lessThan__telescope_H,axiom,
! [F2: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F2 @ zero_zero_nat ) @ ( F2 @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8226_sum__lessThan__telescope_H,axiom,
! [F2: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F2 @ zero_zero_nat ) @ ( F2 @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_8227_sum__lessThan__telescope,axiom,
! [F2: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F2 @ M ) @ ( F2 @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8228_sum__lessThan__telescope,axiom,
! [F2: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F2 @ M ) @ ( F2 @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_8229_sgn__1__neg,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_8230_sgn__1__neg,axiom,
! [A: code_integer] :
( ( ( sgn_sgn_Code_integer @ A )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% sgn_1_neg
thf(fact_8231_sgn__1__neg,axiom,
! [A: rat] :
( ( ( sgn_sgn_rat @ A )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% sgn_1_neg
thf(fact_8232_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_8233_sgn__if,axiom,
( sgn_sgn_Code_integer
= ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% sgn_if
thf(fact_8234_sgn__if,axiom,
( sgn_sgn_rat
= ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_if
thf(fact_8235_prod_OlessThan__Suc__shift,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( G2 @ zero_zero_nat )
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8236_prod_OlessThan__Suc__shift,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_assn @ ( G2 @ zero_zero_nat )
@ ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8237_prod_OlessThan__Suc__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G2 @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8238_prod_OlessThan__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8239_prod_OlessThan__Suc__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G2 @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8240_sum_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > code_integer,G2: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7873554091576472773nteger @ G2 @ S )
= ( groups7873554091576472773nteger @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8241_sum_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > rat,G2: int > rat] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_rat ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_rat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3906332499630173760nt_rat @ G2 @ S )
= ( groups3906332499630173760nt_rat @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8242_sum_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > nat,G2: int > nat] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_nat ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_nat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat @ G2 @ S )
= ( groups4541462559716669496nt_nat @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8243_sum_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > code_integer,G2: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7501900531339628137nteger @ G2 @ S )
= ( groups7501900531339628137nteger @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8244_sum_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_rat ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_rat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups2906978787729119204at_rat @ G2 @ S )
= ( groups2906978787729119204at_rat @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8245_sum_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > int,G2: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_int ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_int ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3539618377306564664at_int @ G2 @ S )
= ( groups3539618377306564664at_int @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8246_sum_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S )
= ( groups3542108847815614940at_nat @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8247_sum_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > int,G2: int > int] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_int ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_int ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups4538972089207619220nt_int @ G2 @ S )
= ( groups4538972089207619220nt_int @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8248_sum_Omono__neutral__cong,axiom,
! [T: set_set_nat,S: set_set_nat,H3: set_nat > code_integer,G2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( finite1152437895449049373et_nat @ S )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ ( minus_2163939370556025621et_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_z3403309356797280102nteger ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups9190459664516455967nteger @ G2 @ S )
= ( groups9190459664516455967nteger @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8249_sum_Omono__neutral__cong,axiom,
! [T: set_set_nat,S: set_set_nat,H3: set_nat > rat,G2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ T )
=> ( ( finite1152437895449049373et_nat @ S )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ ( minus_2163939370556025621et_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= zero_zero_rat ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ ( minus_2163939370556025621et_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= zero_zero_rat ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7659867448343625626at_rat @ G2 @ S )
= ( groups7659867448343625626at_rat @ H3 @ T ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_8250_prod_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G2: int > code_integer] :
( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3827104343326376752nteger @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8251_prod_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G2: int > assn] :
( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups7882442080178216443t_assn @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8252_prod_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G2: int > rat] :
( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8253_prod_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G2: int > nat] :
( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8254_prod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G2: nat > code_integer] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3455450783089532116nteger @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8255_prod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G2: nat > assn] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups6906906614972039071t_assn @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8256_prod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G2: nat > rat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ A2 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups73079841787564623at_rat @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8257_prod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G2: nat > nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2 @ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups708209901874060359at_nat @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8258_prod_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G2: nat > int] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2 @ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups705719431365010083at_int @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8259_prod_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G2: int > int] :
( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1705073143266064639nt_int @ G2 @ B2 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8260_prod_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G2: int > code_integer,H3: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3827104343326376752nteger @ G2 @ T )
= ( groups3827104343326376752nteger @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8261_prod_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G2: int > assn,H3: int > assn] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ T )
= ( groups7882442080178216443t_assn @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8262_prod_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G2: int > rat,H3: int > rat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ T )
= ( groups1072433553688619179nt_rat @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8263_prod_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G2: int > nat,H3: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ T )
= ( groups1707563613775114915nt_nat @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8264_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > code_integer,H3: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3455450783089532116nteger @ G2 @ T )
= ( groups3455450783089532116nteger @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8265_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > assn,H3: nat > assn] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ T )
= ( groups6906906614972039071t_assn @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8266_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > rat,H3: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups73079841787564623at_rat @ G2 @ T )
= ( groups73079841787564623at_rat @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8267_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ T )
= ( groups708209901874060359at_nat @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8268_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > int,H3: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups705719431365010083at_int @ G2 @ T )
= ( groups705719431365010083at_int @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8269_prod_Omono__neutral__cong__right,axiom,
! [T: set_int,S: set_int,G2: int > int,H3: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ T )
= ( groups1705073143266064639nt_int @ H3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8270_prod_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H3: int > code_integer,G2: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_Code_integer ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3827104343326376752nteger @ G2 @ S )
= ( groups3827104343326376752nteger @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8271_prod_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H3: int > assn,G2: int > assn] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_assn ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ S )
= ( groups7882442080178216443t_assn @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8272_prod_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H3: int > rat,G2: int > rat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_rat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ S )
= ( groups1072433553688619179nt_rat @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8273_prod_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H3: int > nat,G2: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_nat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ S )
= ( groups1707563613775114915nt_nat @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8274_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H3: nat > code_integer,G2: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_Code_integer ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3455450783089532116nteger @ G2 @ S )
= ( groups3455450783089532116nteger @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8275_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H3: nat > assn,G2: nat > assn] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_assn ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ S )
= ( groups6906906614972039071t_assn @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8276_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H3: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_rat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups73079841787564623at_rat @ G2 @ S )
= ( groups73079841787564623at_rat @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8277_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H3: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S )
= ( groups708209901874060359at_nat @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8278_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H3: nat > int,G2: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_int ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups705719431365010083at_int @ G2 @ S )
= ( groups705719431365010083at_int @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8279_prod_Omono__neutral__cong__left,axiom,
! [T: set_int,S: set_int,H3: int > int,G2: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ X2 )
= one_one_int ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ S )
= ( groups1705073143266064639nt_int @ H3 @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8280_prod_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G2: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3827104343326376752nteger @ G2 @ T )
= ( groups3827104343326376752nteger @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8281_prod_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G2: int > assn] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ T )
= ( groups7882442080178216443t_assn @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8282_prod_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G2: int > rat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ T )
= ( groups1072433553688619179nt_rat @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8283_prod_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G2: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ T )
= ( groups1707563613775114915nt_nat @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8284_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G2 @ T )
= ( groups3455450783089532116nteger @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8285_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ T )
= ( groups6906906614972039071t_assn @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8286_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G2 @ T )
= ( groups73079841787564623at_rat @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8287_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ T )
= ( groups708209901874060359at_nat @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8288_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G2: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G2 @ T )
= ( groups705719431365010083at_int @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8289_prod_Omono__neutral__right,axiom,
! [T: set_int,S: set_int,G2: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ T )
= ( groups1705073143266064639nt_int @ G2 @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8290_prod_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G2: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3827104343326376752nteger @ G2 @ S )
= ( groups3827104343326376752nteger @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8291_prod_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G2: int > assn] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ S )
= ( groups7882442080178216443t_assn @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8292_prod_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G2: int > rat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ S )
= ( groups1072433553688619179nt_rat @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8293_prod_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G2: int > nat] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ S )
= ( groups1707563613775114915nt_nat @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8294_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G2 @ S )
= ( groups3455450783089532116nteger @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8295_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ S )
= ( groups6906906614972039071t_assn @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8296_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G2 @ S )
= ( groups73079841787564623at_rat @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8297_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S )
= ( groups708209901874060359at_nat @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8298_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G2: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G2 @ S )
= ( groups705719431365010083at_int @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8299_prod_Omono__neutral__left,axiom,
! [T: set_int,S: set_int,G2: int > int] :
( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ S @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ S )
= ( groups1705073143266064639nt_int @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8300_prod_Osame__carrierI,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > code_integer,H3: int > code_integer] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ( ( groups3827104343326376752nteger @ G2 @ C2 )
= ( groups3827104343326376752nteger @ H3 @ C2 ) )
=> ( ( groups3827104343326376752nteger @ G2 @ A2 )
= ( groups3827104343326376752nteger @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8301_prod_Osame__carrierI,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > assn,H3: int > assn] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ( ( groups7882442080178216443t_assn @ G2 @ C2 )
= ( groups7882442080178216443t_assn @ H3 @ C2 ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= ( groups7882442080178216443t_assn @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8302_prod_Osame__carrierI,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > rat,H3: int > rat] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ( ( groups1072433553688619179nt_rat @ G2 @ C2 )
= ( groups1072433553688619179nt_rat @ H3 @ C2 ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= ( groups1072433553688619179nt_rat @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8303_prod_Osame__carrierI,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > nat,H3: int > nat] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_nat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G2 @ C2 )
= ( groups1707563613775114915nt_nat @ H3 @ C2 ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
= ( groups1707563613775114915nt_nat @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8304_prod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > code_integer,H3: nat > code_integer] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ( ( groups3455450783089532116nteger @ G2 @ C2 )
= ( groups3455450783089532116nteger @ H3 @ C2 ) )
=> ( ( groups3455450783089532116nteger @ G2 @ A2 )
= ( groups3455450783089532116nteger @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8305_prod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > assn,H3: nat > assn] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ( ( groups6906906614972039071t_assn @ G2 @ C2 )
= ( groups6906906614972039071t_assn @ H3 @ C2 ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= ( groups6906906614972039071t_assn @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8306_prod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > rat,H3: nat > rat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ( ( groups73079841787564623at_rat @ G2 @ C2 )
= ( groups73079841787564623at_rat @ H3 @ C2 ) )
=> ( ( groups73079841787564623at_rat @ G2 @ A2 )
= ( groups73079841787564623at_rat @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8307_prod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_nat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G2 @ C2 )
= ( groups708209901874060359at_nat @ H3 @ C2 ) )
=> ( ( groups708209901874060359at_nat @ G2 @ A2 )
= ( groups708209901874060359at_nat @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8308_prod_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > int,H3: nat > int] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_int ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_int ) )
=> ( ( ( groups705719431365010083at_int @ G2 @ C2 )
= ( groups705719431365010083at_int @ H3 @ C2 ) )
=> ( ( groups705719431365010083at_int @ G2 @ A2 )
= ( groups705719431365010083at_int @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8309_prod_Osame__carrierI,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > int,H3: int > int] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_int ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_int ) )
=> ( ( ( groups1705073143266064639nt_int @ G2 @ C2 )
= ( groups1705073143266064639nt_int @ H3 @ C2 ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= ( groups1705073143266064639nt_int @ H3 @ B2 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8310_prod_Osame__carrier,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > code_integer,H3: int > code_integer] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ( ( groups3827104343326376752nteger @ G2 @ A2 )
= ( groups3827104343326376752nteger @ H3 @ B2 ) )
= ( ( groups3827104343326376752nteger @ G2 @ C2 )
= ( groups3827104343326376752nteger @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8311_prod_Osame__carrier,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > assn,H3: int > assn] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= ( groups7882442080178216443t_assn @ H3 @ B2 ) )
= ( ( groups7882442080178216443t_assn @ G2 @ C2 )
= ( groups7882442080178216443t_assn @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8312_prod_Osame__carrier,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > rat,H3: int > rat] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= ( groups1072433553688619179nt_rat @ H3 @ B2 ) )
= ( ( groups1072433553688619179nt_rat @ G2 @ C2 )
= ( groups1072433553688619179nt_rat @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8313_prod_Osame__carrier,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > nat,H3: int > nat] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_nat ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
= ( groups1707563613775114915nt_nat @ H3 @ B2 ) )
= ( ( groups1707563613775114915nt_nat @ G2 @ C2 )
= ( groups1707563613775114915nt_nat @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8314_prod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > code_integer,H3: nat > code_integer] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_Code_integer ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_Code_integer ) )
=> ( ( ( groups3455450783089532116nteger @ G2 @ A2 )
= ( groups3455450783089532116nteger @ H3 @ B2 ) )
= ( ( groups3455450783089532116nteger @ G2 @ C2 )
= ( groups3455450783089532116nteger @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8315_prod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > assn,H3: nat > assn] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_assn ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_assn ) )
=> ( ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= ( groups6906906614972039071t_assn @ H3 @ B2 ) )
= ( ( groups6906906614972039071t_assn @ G2 @ C2 )
= ( groups6906906614972039071t_assn @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8316_prod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > rat,H3: nat > rat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_rat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_rat ) )
=> ( ( ( groups73079841787564623at_rat @ G2 @ A2 )
= ( groups73079841787564623at_rat @ H3 @ B2 ) )
= ( ( groups73079841787564623at_rat @ G2 @ C2 )
= ( groups73079841787564623at_rat @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8317_prod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_nat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G2 @ A2 )
= ( groups708209901874060359at_nat @ H3 @ B2 ) )
= ( ( groups708209901874060359at_nat @ G2 @ C2 )
= ( groups708209901874060359at_nat @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8318_prod_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G2: nat > int,H3: nat > int] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_int ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_int ) )
=> ( ( ( groups705719431365010083at_int @ G2 @ A2 )
= ( groups705719431365010083at_int @ H3 @ B2 ) )
= ( ( groups705719431365010083at_int @ G2 @ C2 )
= ( groups705719431365010083at_int @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8319_prod_Osame__carrier,axiom,
! [C2: set_int,A2: set_int,B2: set_int,G2: int > int,H3: int > int] :
( ( finite_finite_int @ C2 )
=> ( ( ord_less_eq_set_int @ A2 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ! [A6: int] :
( ( member_int @ A6 @ ( minus_minus_set_int @ C2 @ A2 ) )
=> ( ( G2 @ A6 )
= one_one_int ) )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ C2 @ B2 ) )
=> ( ( H3 @ B6 )
= one_one_int ) )
=> ( ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= ( groups1705073143266064639nt_int @ H3 @ B2 ) )
= ( ( groups1705073143266064639nt_int @ G2 @ C2 )
= ( groups1705073143266064639nt_int @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8320_sum_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) @ ( groups3906332499630173760nt_rat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8321_sum_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8322_sum_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8323_sum_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ A2 ) @ ( groups3539618377306564664at_int @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8324_sum_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8325_sum_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8326_sum_Ounion__inter,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) ) @ ( groups342789780944988191at_rat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) )
= ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ A2 ) @ ( groups342789780944988191at_rat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8327_sum_Ounion__inter,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( plus_plus_nat @ ( groups977919841031483927at_nat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) ) @ ( groups977919841031483927at_nat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) )
= ( plus_plus_nat @ ( groups977919841031483927at_nat @ G2 @ A2 ) @ ( groups977919841031483927at_nat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8328_sum_Ounion__inter,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( plus_plus_int @ ( groups975429370522433651at_int @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) ) @ ( groups975429370522433651at_int @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) )
= ( plus_plus_int @ ( groups975429370522433651at_int @ G2 @ A2 ) @ ( groups975429370522433651at_int @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8329_sum_Ounion__inter,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,G2: produc4166570645942440679at_nat > rat] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ( plus_plus_rat @ ( groups6424767709179651461at_rat @ G2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) ) @ ( groups6424767709179651461at_rat @ G2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) )
= ( plus_plus_rat @ ( groups6424767709179651461at_rat @ G2 @ A2 ) @ ( groups6424767709179651461at_rat @ G2 @ B2 ) ) ) ) ) ).
% sum.union_inter
thf(fact_8330_sum_OInt__Diff,axiom,
! [A2: set_int,G2: int > rat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8331_sum_OInt__Diff,axiom,
! [A2: set_int,G2: int > nat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8332_sum_OInt__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat,B2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( groups342789780944988191at_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) @ ( groups342789780944988191at_rat @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8333_sum_OInt__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,B2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( groups977919841031483927at_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( groups977919841031483927at_nat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) @ ( groups977919841031483927at_nat @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8334_sum_OInt__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > int,B2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( groups975429370522433651at_int @ G2 @ A2 )
= ( plus_plus_int @ ( groups975429370522433651at_int @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) @ ( groups975429370522433651at_int @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8335_sum_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > rat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8336_sum_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > int,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G2 @ A2 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8337_sum_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8338_sum_OInt__Diff,axiom,
! [A2: set_int,G2: int > int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2 @ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_8339_sum_OatLeast1__atMost__eq,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( G2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_8340_prod_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups3827104343326376752nteger @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ A2 ) @ ( groups3827104343326376752nteger @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8341_prod_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups3455450783089532116nteger @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ A2 ) @ ( groups3455450783089532116nteger @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8342_prod_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups7882442080178216443t_assn @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ A2 ) @ ( groups7882442080178216443t_assn @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8343_prod_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ A2 ) @ ( groups6906906614972039071t_assn @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8344_prod_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) @ ( groups1072433553688619179nt_rat @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8345_prod_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ A2 ) @ ( groups73079841787564623at_rat @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8346_prod_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ A2 ) @ ( groups1707563613775114915nt_nat @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8347_prod_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups708209901874060359at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ A2 ) @ ( groups708209901874060359at_nat @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8348_prod_Ounion__inter,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ A2 ) @ ( groups705719431365010083at_int @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8349_prod_Ounion__inter,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups1705073143266064639nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ A2 ) @ ( groups1705073143266064639nt_int @ G2 @ B2 ) ) ) ) ) ).
% prod.union_inter
thf(fact_8350_prod_OInt__Diff,axiom,
! [A2: set_int,G2: int > code_integer,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8351_prod_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > code_integer,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8352_prod_OInt__Diff,axiom,
! [A2: set_int,G2: int > assn,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8353_prod_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > assn,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8354_prod_OInt__Diff,axiom,
! [A2: set_int,G2: int > rat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8355_prod_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > rat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ A2 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8356_prod_OInt__Diff,axiom,
! [A2: set_int,G2: int > nat,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat @ G2 @ A2 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8357_prod_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G2 @ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups708209901874060359at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8358_prod_OInt__Diff,axiom,
! [A2: set_nat,G2: nat > int,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G2 @ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8359_prod_OInt__Diff,axiom,
! [A2: set_int,G2: int > int,B2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G2 @ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups1705073143266064639nt_int @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_8360_prod_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > code_integer,G2: int > code_integer] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_Code_integer ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_Code_integer ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3827104343326376752nteger @ G2 @ S )
= ( groups3827104343326376752nteger @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8361_prod_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > assn,G2: int > assn] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_assn ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_assn ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ S )
= ( groups7882442080178216443t_assn @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8362_prod_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > rat,G2: int > rat] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_rat ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_rat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ S )
= ( groups1072433553688619179nt_rat @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8363_prod_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > nat,G2: int > nat] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_nat ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_nat ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ S )
= ( groups1707563613775114915nt_nat @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8364_prod_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > code_integer,G2: nat > code_integer] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_Code_integer ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_Code_integer ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3455450783089532116nteger @ G2 @ S )
= ( groups3455450783089532116nteger @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8365_prod_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > assn,G2: nat > assn] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_assn ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_assn ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ S )
= ( groups6906906614972039071t_assn @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8366_prod_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_rat ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_rat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups73079841787564623at_rat @ G2 @ S )
= ( groups73079841787564623at_rat @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8367_prod_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_nat ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S )
= ( groups708209901874060359at_nat @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8368_prod_Omono__neutral__cong,axiom,
! [T: set_nat,S: set_nat,H3: nat > int,G2: nat > int] :
( ( finite_finite_nat @ T )
=> ( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_int ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_int ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups705719431365010083at_int @ G2 @ S )
= ( groups705719431365010083at_int @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8369_prod_Omono__neutral__cong,axiom,
! [T: set_int,S: set_int,H3: int > int,G2: int > int] :
( ( finite_finite_int @ T )
=> ( ( finite_finite_int @ S )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ T @ S ) )
=> ( ( H3 @ I2 )
= one_one_int ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T ) )
=> ( ( G2 @ I2 )
= one_one_int ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ S @ T ) )
=> ( ( G2 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ S )
= ( groups1705073143266064639nt_int @ H3 @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_8370_prod_OatLeast1__atMost__eq,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( G2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8371_prod_OatLeast1__atMost__eq,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups705719431365010083at_int
@ ^ [K4: nat] : ( G2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8372_sum__bounds__lt__plus1,axiom,
! [F2: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( F2 @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_8373_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X3: int] : X3
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8374_sum_Onat__group,axiom,
! [G2: nat > nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [M2: nat] : ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K ) ) ) ) ).
% sum.nat_group
thf(fact_8375_prod_Onat__group,axiom,
! [G2: nat > nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [M2: nat] : ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K ) ) ) ) ).
% prod.nat_group
thf(fact_8376_prod_Onat__group,axiom,
! [G2: nat > int,K: nat,N: nat] :
( ( groups705719431365010083at_int
@ ^ [M2: nat] : ( groups705719431365010083at_int @ G2 @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K ) ) ) ) ).
% prod.nat_group
thf(fact_8377_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8378_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8379_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8380_Iio__Int__singleton,axiom,
! [X: $o,K: $o] :
( ( ( ord_less_o @ X @ K )
=> ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X @ bot_bot_set_o ) )
= ( insert_o @ X @ bot_bot_set_o ) ) )
& ( ~ ( ord_less_o @ X @ K )
=> ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o ) ) ) ).
% Iio_Int_singleton
thf(fact_8381_Iio__Int__singleton,axiom,
! [X: literal,K: literal] :
( ( ( ord_less_literal @ X @ K )
=> ( ( inf_inf_set_literal @ ( set_or2436086697161274615iteral @ K ) @ ( insert_literal @ X @ bot_bot_set_literal ) )
= ( insert_literal @ X @ bot_bot_set_literal ) ) )
& ( ~ ( ord_less_literal @ X @ K )
=> ( ( inf_inf_set_literal @ ( set_or2436086697161274615iteral @ K ) @ ( insert_literal @ X @ bot_bot_set_literal ) )
= bot_bot_set_literal ) ) ) ).
% Iio_Int_singleton
thf(fact_8382_Iio__Int__singleton,axiom,
! [X: rat,K: rat] :
( ( ( ord_less_rat @ X @ K )
=> ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
= ( insert_rat @ X @ bot_bot_set_rat ) ) )
& ( ~ ( ord_less_rat @ X @ K )
=> ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
= bot_bot_set_rat ) ) ) ).
% Iio_Int_singleton
thf(fact_8383_Iio__Int__singleton,axiom,
! [X: int,K: int] :
( ( ( ord_less_int @ X @ K )
=> ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X @ bot_bot_set_int ) )
= ( insert_int @ X @ bot_bot_set_int ) ) )
& ( ~ ( ord_less_int @ X @ K )
=> ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int ) ) ) ).
% Iio_Int_singleton
thf(fact_8384_Iio__Int__singleton,axiom,
! [X: nat,K: nat] :
( ( ( ord_less_nat @ X @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ) ).
% Iio_Int_singleton
thf(fact_8385_ivl__disj__un__singleton_I2_J,axiom,
! [U: $o] :
( ( sup_sup_set_o @ ( set_ord_lessThan_o @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
= ( set_ord_atMost_o @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_8386_ivl__disj__un__singleton_I2_J,axiom,
! [U: int] :
( ( sup_sup_set_int @ ( set_ord_lessThan_int @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
= ( set_ord_atMost_int @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_8387_ivl__disj__un__singleton_I2_J,axiom,
! [U: nat] :
( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_ord_atMost_nat @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_8388_ivl__disj__un__one_I4_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( set_ord_atMost_rat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_8389_ivl__disj__un__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_8390_ivl__disj__un__one_I4_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( set_ord_atMost_int @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_8391_ivl__disj__un__one_I4_J,axiom,
! [L: code_integer,U: code_integer] :
( ( ord_le3102999989581377725nteger @ L @ U )
=> ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or189985376899183464nteger @ L @ U ) )
= ( set_or9101266186257409494nteger @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_8392_sum_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > rat,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8393_sum_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > nat,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8394_sum_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8395_sum_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > int,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( if_int @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups3539618377306564664at_int @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8396_sum_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8397_sum_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups4538972089207619220nt_int @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8398_sum_OIf__cases,axiom,
! [A2: set_set_nat,P: set_nat > $o,H3: set_nat > rat,G2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups7659867448343625626at_rat
@ ^ [X3: set_nat] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_rat @ ( groups7659867448343625626at_rat @ H3 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) @ ( groups7659867448343625626at_rat @ G2 @ ( inf_inf_set_set_nat @ A2 @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8399_sum_OIf__cases,axiom,
! [A2: set_set_nat,P: set_nat > $o,H3: set_nat > nat,G2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : ( if_nat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_nat @ ( groups8294997508430121362at_nat @ H3 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) @ ( groups8294997508430121362at_nat @ G2 @ ( inf_inf_set_set_nat @ A2 @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8400_sum_OIf__cases,axiom,
! [A2: set_set_nat,P: set_nat > $o,H3: set_nat > int,G2: set_nat > int] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( groups8292507037921071086at_int
@ ^ [X3: set_nat] : ( if_int @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_int @ ( groups8292507037921071086at_int @ H3 @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) ) @ ( groups8292507037921071086at_int @ G2 @ ( inf_inf_set_set_nat @ A2 @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8401_sum_OIf__cases,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,H3: product_prod_nat_nat > rat,G2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( groups342789780944988191at_rat
@ ^ [X3: product_prod_nat_nat] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( plus_plus_rat @ ( groups342789780944988191at_rat @ H3 @ ( inf_in2572325071724192079at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) ) ) @ ( groups342789780944988191at_rat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_8402_prod_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > code_integer,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger
@ ^ [X3: int] : ( if_Code_integer @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups3827104343326376752nteger @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8403_prod_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > code_integer,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger
@ ^ [X3: nat] : ( if_Code_integer @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups3455450783089532116nteger @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8404_prod_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > assn,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn
@ ^ [X3: int] : ( if_assn @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups7882442080178216443t_assn @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8405_prod_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > assn,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn
@ ^ [X3: nat] : ( if_assn @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8406_prod_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > rat,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat
@ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8407_prod_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat
@ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8408_prod_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > nat,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8409_prod_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat
@ ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups708209901874060359at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8410_prod_OIf__cases,axiom,
! [A2: set_nat,P: nat > $o,H3: nat > int,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int
@ ^ [X3: nat] : ( if_int @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ H3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8411_prod_OIf__cases,axiom,
! [A2: set_int,P: int > $o,H3: int > int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G2 @ X3 ) )
@ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ H3 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups1705073143266064639nt_int @ G2 @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_8412_one__diff__power__eq,axiom,
! [X: code_integer,N: nat] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
= ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_8413_one__diff__power__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_8414_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_8415_power__diff__1__eq,axiom,
! [X: code_integer,N: nat] :
( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N ) @ one_one_Code_integer )
= ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_8416_power__diff__1__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_8417_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_8418_geometric__sum,axiom,
! [X: rat,N: nat] :
( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_8419_prod__mono__strict,axiom,
! [A2: set_o,F2: $o > code_integer,G2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) )
& ( ord_le6747313008572928689nteger @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F2 @ A2 ) @ ( groups7694694392188491536nteger @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8420_prod__mono__strict,axiom,
! [A2: set_nat,F2: nat > code_integer,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) )
& ( ord_le6747313008572928689nteger @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F2 @ A2 ) @ ( groups3455450783089532116nteger @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8421_prod__mono__strict,axiom,
! [A2: set_int,F2: int > code_integer,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I2 ) )
& ( ord_le6747313008572928689nteger @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F2 @ A2 ) @ ( groups3827104343326376752nteger @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8422_prod__mono__strict,axiom,
! [A2: set_o,F2: $o > rat,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) )
& ( ord_less_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F2 @ A2 ) @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8423_prod__mono__strict,axiom,
! [A2: set_nat,F2: nat > rat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) )
& ( ord_less_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) @ ( groups73079841787564623at_rat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8424_prod__mono__strict,axiom,
! [A2: set_int,F2: int > rat,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I2 ) )
& ( ord_less_rat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8425_prod__mono__strict,axiom,
! [A2: set_o,F2: $o > nat,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) )
& ( ord_less_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_nat @ ( groups3504817904513533571_o_nat @ F2 @ A2 ) @ ( groups3504817904513533571_o_nat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8426_prod__mono__strict,axiom,
! [A2: set_int,F2: int > nat,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) )
& ( ord_less_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) @ ( groups1707563613775114915nt_nat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8427_prod__mono__strict,axiom,
! [A2: set_o,F2: $o > int,G2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( F2 @ I2 ) )
& ( ord_less_int @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_int @ ( groups3502327434004483295_o_int @ F2 @ A2 ) @ ( groups3502327434004483295_o_int @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8428_prod__mono__strict,axiom,
! [A2: set_nat,F2: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I2 ) )
& ( ord_less_nat @ ( F2 @ I2 ) @ ( G2 @ I2 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_nat @ ( groups708209901874060359at_nat @ F2 @ A2 ) @ ( groups708209901874060359at_nat @ G2 @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8429_sum_OatMost__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat @ ( G2 @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8430_sum_OatMost__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G2 @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8431_sum_OatMost__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G2 @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8432_sum_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_z3403309356797280102nteger ) )
=> ( ( groups7873554091576472773nteger @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_p5714425477246183910nteger @ ( groups7873554091576472773nteger @ G2 @ A2 ) @ ( groups7873554091576472773nteger @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8433_sum_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) @ ( groups3906332499630173760nt_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8434_sum_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8435_sum_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_z3403309356797280102nteger ) )
=> ( ( groups7501900531339628137nteger @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G2 @ A2 ) @ ( groups7501900531339628137nteger @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8436_sum_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8437_sum_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ A2 ) @ ( groups3539618377306564664at_int @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8438_sum_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8439_sum_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8440_sum_Ounion__inter__neutral,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > code_integer] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_z3403309356797280102nteger ) )
=> ( ( groups196832835297161892nteger @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_p5714425477246183910nteger @ ( groups196832835297161892nteger @ G2 @ A2 ) @ ( groups196832835297161892nteger @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8441_sum_Ounion__inter__neutral,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= zero_zero_rat ) )
=> ( ( groups342789780944988191at_rat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ A2 ) @ ( groups342789780944988191at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_8442_sum_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups7872700643590313910_o_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups7872700643590313910_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8443_sum_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups8507830703676809646_o_nat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8444_sum_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups8505340233167759370_o_int @ G2 @ A2 )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups8505340233167759370_o_int @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8445_sum_Oremove,axiom,
! [A2: set_int,X: int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups3906332499630173760nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8446_sum_Oremove,axiom,
! [A2: set_int,X: int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups4541462559716669496nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8447_sum_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups2906978787729119204at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8448_sum_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups3539618377306564664at_int @ G2 @ A2 )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups3539618377306564664at_int @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8449_sum_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2 @ A2 )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8450_sum_Oremove,axiom,
! [A2: set_int,X: int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2 @ A2 )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups4538972089207619220nt_int @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8451_sum_Oremove,axiom,
! [A2: set_set_nat,X: set_nat,G2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( groups7659867448343625626at_rat @ G2 @ A2 )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups7659867448343625626at_rat @ G2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_8452_sum_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > rat,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups7872700643590313910_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups7872700643590313910_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8453_sum_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > nat,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G2 @ ( insert_o @ X @ A2 ) )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups8507830703676809646_o_nat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8454_sum_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > int,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups8505340233167759370_o_int @ G2 @ ( insert_o @ X @ A2 ) )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups8505340233167759370_o_int @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8455_sum_Oinsert__remove,axiom,
! [A2: set_int,G2: int > rat,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups3906332499630173760nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8456_sum_Oinsert__remove,axiom,
! [A2: set_int,G2: int > nat,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat @ G2 @ ( insert_int @ X @ A2 ) )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups4541462559716669496nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8457_sum_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > rat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups2906978787729119204at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8458_sum_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > int,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G2 @ ( insert_nat @ X @ A2 ) )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups3539618377306564664at_int @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8459_sum_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( plus_plus_nat @ ( G2 @ X ) @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8460_sum_Oinsert__remove,axiom,
! [A2: set_int,G2: int > int,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G2 @ ( insert_int @ X @ A2 ) )
= ( plus_plus_int @ ( G2 @ X ) @ ( groups4538972089207619220nt_int @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8461_sum_Oinsert__remove,axiom,
! [A2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat,X: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( groups342789780944988191at_rat @ G2 @ ( insert8211810215607154385at_nat @ X @ A2 ) )
= ( plus_plus_rat @ ( G2 @ X ) @ ( groups342789780944988191at_rat @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_8462_prod_OatMost__shift,axiom,
! [G2: nat > code_integer,N: nat] :
( ( groups3455450783089532116nteger @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( times_3573771949741848930nteger @ ( G2 @ zero_zero_nat )
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8463_prod_OatMost__shift,axiom,
! [G2: nat > assn,N: nat] :
( ( groups6906906614972039071t_assn @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( times_times_assn @ ( G2 @ zero_zero_nat )
@ ( groups6906906614972039071t_assn
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8464_prod_OatMost__shift,axiom,
! [G2: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( G2 @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8465_prod_OatMost__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ ( G2 @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8466_prod_OatMost__shift,axiom,
! [G2: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( G2 @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8467_sum__diff1,axiom,
! [A2: set_o,A: $o,F2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups7872700643590313910_o_rat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups7872700643590313910_o_rat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups7872700643590313910_o_rat @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8468_sum__diff1,axiom,
! [A2: set_int,A: int,F2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( ( member_int @ A @ A2 )
=> ( ( groups3906332499630173760nt_rat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups3906332499630173760nt_rat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups3906332499630173760nt_rat @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8469_sum__diff1,axiom,
! [A2: set_o,A: $o,F2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups8505340233167759370_o_int @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( minus_minus_int @ ( groups8505340233167759370_o_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups8505340233167759370_o_int @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups8505340233167759370_o_int @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8470_sum__diff1,axiom,
! [A2: set_nat,A: nat,F2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ A2 )
=> ( ( groups2906978787729119204at_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups2906978787729119204at_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups2906978787729119204at_rat @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8471_sum__diff1,axiom,
! [A2: set_nat,A: nat,F2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ A2 )
=> ( ( groups3539618377306564664at_int @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups3539618377306564664at_int @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3539618377306564664at_int @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8472_sum__diff1,axiom,
! [A2: set_int,A: int,F2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( ( member_int @ A @ A2 )
=> ( ( groups4538972089207619220nt_int @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups4538972089207619220nt_int @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups4538972089207619220nt_int @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8473_sum__diff1,axiom,
! [A2: set_set_nat,A: set_nat,F2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( member_set_nat @ A @ A2 )
=> ( ( groups7659867448343625626at_rat @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( minus_minus_rat @ ( groups7659867448343625626at_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( groups7659867448343625626at_rat @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( groups7659867448343625626at_rat @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8474_sum__diff1,axiom,
! [A2: set_set_nat,A: set_nat,F2: set_nat > int] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( member_set_nat @ A @ A2 )
=> ( ( groups8292507037921071086at_int @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( minus_minus_int @ ( groups8292507037921071086at_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( groups8292507037921071086at_int @ F2 @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( groups8292507037921071086at_int @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8475_sum__diff1,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,F2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups342789780944988191at_rat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( minus_minus_rat @ ( groups342789780944988191at_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups342789780944988191at_rat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( groups342789780944988191at_rat @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8476_sum__diff1,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,F2: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups975429370522433651at_int @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( minus_minus_int @ ( groups975429370522433651at_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( groups975429370522433651at_int @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
= ( groups975429370522433651at_int @ F2 @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_8477_sum__Un,axiom,
! [A2: set_int,B2: set_int,F2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups3906332499630173760nt_rat @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F2 @ A2 ) @ ( groups3906332499630173760nt_rat @ F2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8478_sum__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups342789780944988191at_rat @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F2 @ A2 ) @ ( groups342789780944988191at_rat @ F2 @ B2 ) ) @ ( groups342789780944988191at_rat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8479_sum__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F2: produc4166570645942440679at_nat > rat] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ( groups6424767709179651461at_rat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( groups6424767709179651461at_rat @ F2 @ A2 ) @ ( groups6424767709179651461at_rat @ F2 @ B2 ) ) @ ( groups6424767709179651461at_rat @ F2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8480_sum__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > rat] :
( ( finite4343798906461161616at_nat @ A2 )
=> ( ( finite4343798906461161616at_nat @ B2 )
=> ( ( groups3225780264831618053at_rat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( groups3225780264831618053at_rat @ F2 @ A2 ) @ ( groups3225780264831618053at_rat @ F2 @ B2 ) ) @ ( groups3225780264831618053at_rat @ F2 @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8481_sum__Un,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups2906978787729119204at_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F2 @ A2 ) @ ( groups2906978787729119204at_rat @ F2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8482_sum__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups975429370522433651at_int @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( groups975429370522433651at_int @ F2 @ A2 ) @ ( groups975429370522433651at_int @ F2 @ B2 ) ) @ ( groups975429370522433651at_int @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8483_sum__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F2: produc4166570645942440679at_nat > int] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ( groups7057407298757096921at_int @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( groups7057407298757096921at_int @ F2 @ A2 ) @ ( groups7057407298757096921at_int @ F2 @ B2 ) ) @ ( groups7057407298757096921at_int @ F2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8484_sum__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > int] :
( ( finite4343798906461161616at_nat @ A2 )
=> ( ( finite4343798906461161616at_nat @ B2 )
=> ( ( groups3858419854409063513at_int @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( groups3858419854409063513at_int @ F2 @ A2 ) @ ( groups3858419854409063513at_int @ F2 @ B2 ) ) @ ( groups3858419854409063513at_int @ F2 @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8485_sum__Un,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3539618377306564664at_int @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ ( groups3539618377306564664at_int @ F2 @ B2 ) ) @ ( groups3539618377306564664at_int @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8486_sum__Un,axiom,
! [A2: set_int,B2: set_int,F2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4538972089207619220nt_int @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( minus_minus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ ( groups4538972089207619220nt_int @ F2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un
thf(fact_8487_sum_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups7872700643590313910_o_rat @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G2 @ A2 ) @ ( groups7872700643590313910_o_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8488_sum_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups8507830703676809646_o_nat @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( plus_plus_nat @ ( groups8507830703676809646_o_nat @ G2 @ A2 ) @ ( groups8507830703676809646_o_nat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8489_sum_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups8505340233167759370_o_int @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( plus_plus_int @ ( groups8505340233167759370_o_int @ G2 @ A2 ) @ ( groups8505340233167759370_o_int @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8490_sum_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ A2 ) @ ( groups2906978787729119204at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8491_sum_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups3539618377306564664at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ A2 ) @ ( groups3539618377306564664at_int @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8492_sum_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups3906332499630173760nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ A2 ) @ ( groups3906332499630173760nt_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8493_sum_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups4541462559716669496nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ A2 ) @ ( groups4541462559716669496nt_nat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8494_sum_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ A2 ) @ ( groups3542108847815614940at_nat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8495_sum_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups4538972089207619220nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ A2 ) @ ( groups4538972089207619220nt_int @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8496_sum_Ounion__disjoint,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
= bot_bo2099793752762293965at_nat )
=> ( ( groups342789780944988191at_rat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ A2 ) @ ( groups342789780944988191at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_8497_prod_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups7694694392188491536nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups7694694392188491536nteger @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8498_prod_Oremove,axiom,
! [A2: set_int,X: int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8499_prod_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ A2 )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8500_prod_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > assn] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups5301882518646026715o_assn @ G2 @ A2 )
= ( times_times_assn @ ( G2 @ X ) @ ( groups5301882518646026715o_assn @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8501_prod_Oremove,axiom,
! [A2: set_int,X: int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ A2 )
= ( times_times_assn @ ( G2 @ X ) @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8502_prod_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ A2 )
= ( times_times_assn @ ( G2 @ X ) @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8503_prod_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups2869687844427037835_o_rat @ G2 @ A2 )
= ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8504_prod_Oremove,axiom,
! [A2: set_int,X: int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ A2 )
= ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8505_prod_Oremove,axiom,
! [A2: set_nat,X: nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ A2 )
= ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8506_prod_Oremove,axiom,
! [A2: set_o,X: $o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X @ A2 )
=> ( ( groups3504817904513533571_o_nat @ G2 @ A2 )
= ( times_times_nat @ ( G2 @ X ) @ ( groups3504817904513533571_o_nat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_8507_prod_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > code_integer,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups7694694392188491536nteger @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups7694694392188491536nteger @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8508_prod_Oinsert__remove,axiom,
! [A2: set_int,G2: int > code_integer,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8509_prod_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > code_integer,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_3573771949741848930nteger @ ( G2 @ X ) @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8510_prod_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > assn,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups5301882518646026715o_assn @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups5301882518646026715o_assn @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8511_prod_Oinsert__remove,axiom,
! [A2: set_int,G2: int > assn,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8512_prod_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > assn,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_assn @ ( G2 @ X ) @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8513_prod_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > rat,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups2869687844427037835_o_rat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups2869687844427037835_o_rat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8514_prod_Oinsert__remove,axiom,
! [A2: set_int,G2: int > rat,X: int] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8515_prod_Oinsert__remove,axiom,
! [A2: set_nat,G2: nat > rat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A2 ) )
= ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8516_prod_Oinsert__remove,axiom,
! [A2: set_o,G2: $o > nat,X: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups3504817904513533571_o_nat @ G2 @ ( insert_o @ X @ A2 ) )
= ( times_times_nat @ ( G2 @ X ) @ ( groups3504817904513533571_o_nat @ G2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_8517_prod_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3827104343326376752nteger @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ A2 ) @ ( groups3827104343326376752nteger @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8518_prod_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ A2 ) @ ( groups3455450783089532116nteger @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8519_prod_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ A2 ) @ ( groups7882442080178216443t_assn @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8520_prod_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ A2 ) @ ( groups6906906614972039071t_assn @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8521_prod_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) @ ( groups1072433553688619179nt_rat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8522_prod_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ A2 ) @ ( groups73079841787564623at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8523_prod_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ A2 ) @ ( groups1707563613775114915nt_nat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8524_prod_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ A2 ) @ ( groups708209901874060359at_nat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8525_prod_Ounion__inter__neutral,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G2 @ A2 ) @ ( groups705719431365010083at_int @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8526_prod_Ounion__inter__neutral,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( G2 @ X2 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ A2 ) @ ( groups1705073143266064639nt_int @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_8527_prod_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups7694694392188491536nteger @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( groups7694694392188491536nteger @ G2 @ A2 ) @ ( groups7694694392188491536nteger @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8528_prod_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups3455450783089532116nteger @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ A2 ) @ ( groups3455450783089532116nteger @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8529_prod_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups3827104343326376752nteger @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ A2 ) @ ( groups3827104343326376752nteger @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8530_prod_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > assn] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups5301882518646026715o_assn @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( times_times_assn @ ( groups5301882518646026715o_assn @ G2 @ A2 ) @ ( groups5301882518646026715o_assn @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8531_prod_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ A2 ) @ ( groups6906906614972039071t_assn @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8532_prod_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ A2 ) @ ( groups7882442080178216443t_assn @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8533_prod_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > rat] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups2869687844427037835_o_rat @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( times_times_rat @ ( groups2869687844427037835_o_rat @ G2 @ A2 ) @ ( groups2869687844427037835_o_rat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8534_prod_Ounion__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ A2 ) @ ( groups73079841787564623at_rat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8535_prod_Ounion__disjoint,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( ( inf_inf_set_int @ A2 @ B2 )
= bot_bot_set_int )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A2 ) @ ( groups1072433553688619179nt_rat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8536_prod_Ounion__disjoint,axiom,
! [A2: set_o,B2: set_o,G2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ B2 )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( groups3504817904513533571_o_nat @ G2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( times_times_nat @ ( groups3504817904513533571_o_nat @ G2 @ A2 ) @ ( groups3504817904513533571_o_nat @ G2 @ B2 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_8537_sum_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups3906332499630173760nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8538_sum_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4541462559716669496nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4541462559716669496nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8539_sum_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups2906978787729119204at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8540_sum_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3539618377306564664at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3539618377306564664at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8541_sum_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8542_sum_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4538972089207619220nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4538972089207619220nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8543_sum_Ounion__diff2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups342789780944988191at_rat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups342789780944988191at_rat @ G2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups342789780944988191at_rat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8544_sum_Ounion__diff2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups977919841031483927at_nat @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups977919841031483927at_nat @ G2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups977919841031483927at_nat @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8545_sum_Ounion__diff2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups975429370522433651at_int @ G2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups975429370522433651at_int @ G2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups975429370522433651at_int @ G2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups975429370522433651at_int @ G2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8546_sum_Ounion__diff2,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,G2: produc4166570645942440679at_nat > rat] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ( groups6424767709179651461at_rat @ G2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups6424767709179651461at_rat @ G2 @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) ) @ ( groups6424767709179651461at_rat @ G2 @ ( minus_5060654252129873198at_nat @ B2 @ A2 ) ) ) @ ( groups6424767709179651461at_rat @ G2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_8547_sum__Un2,axiom,
! [A2: set_int,B2: set_int,F2: int > rat] :
( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
=> ( ( groups3906332499630173760nt_rat @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8548_sum__Un2,axiom,
! [A2: set_int,B2: set_int,F2: int > nat] :
( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
=> ( ( groups4541462559716669496nt_nat @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8549_sum__Un2,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ( groups2906978787729119204at_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ F2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8550_sum__Un2,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ( groups3539618377306564664at_int @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ F2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3539618377306564664at_int @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8551_sum__Un2,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ( groups3542108847815614940at_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ F2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3542108847815614940at_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8552_sum__Un2,axiom,
! [A2: set_int,B2: set_int,F2: int > int] :
( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
=> ( ( groups4538972089207619220nt_int @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ F2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4538972089207619220nt_int @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8553_sum__Un2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
=> ( ( groups342789780944988191at_rat @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups342789780944988191at_rat @ F2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups342789780944988191at_rat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8554_sum__Un2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
=> ( ( groups977919841031483927at_nat @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups977919841031483927at_nat @ F2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups977919841031483927at_nat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8555_sum__Un2,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > int] :
( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
=> ( ( groups975429370522433651at_int @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( plus_plus_int @ ( groups975429370522433651at_int @ F2 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ ( groups975429370522433651at_int @ F2 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) @ ( groups975429370522433651at_int @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8556_sum__Un2,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F2: produc4166570645942440679at_nat > rat] :
( ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
=> ( ( groups6424767709179651461at_rat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( groups6424767709179651461at_rat @ F2 @ ( minus_5060654252129873198at_nat @ A2 @ B2 ) ) @ ( groups6424767709179651461at_rat @ F2 @ ( minus_5060654252129873198at_nat @ B2 @ A2 ) ) ) @ ( groups6424767709179651461at_rat @ F2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ).
% sum_Un2
thf(fact_8557_prod_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups3827104343326376752nteger @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3827104343326376752nteger @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups3827104343326376752nteger @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8558_prod_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3455450783089532116nteger @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3455450783089532116nteger @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3455450783089532116nteger @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8559_prod_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > assn] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups7882442080178216443t_assn @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_assn @ ( times_times_assn @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups7882442080178216443t_assn @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups7882442080178216443t_assn @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8560_prod_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > assn] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups6906906614972039071t_assn @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups6906906614972039071t_assn @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8561_prod_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8562_prod_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8563_prod_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_nat @ ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8564_prod_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups708209901874060359at_nat @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_nat @ ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups708209901874060359at_nat @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups708209901874060359at_nat @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8565_prod_Ounion__diff2,axiom,
! [A2: set_nat,B2: set_nat,G2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups705719431365010083at_int @ G2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( times_times_int @ ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8566_prod_Ounion__diff2,axiom,
! [A2: set_int,B2: set_int,G2: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups1705073143266064639nt_int @ G2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( times_times_int @ ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1705073143266064639nt_int @ G2 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups1705073143266064639nt_int @ G2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_8567_sum__Un__nat,axiom,
! [A2: set_int,B2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ( groups4541462559716669496nt_nat @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ ( groups4541462559716669496nt_nat @ F2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_8568_sum__Un__nat,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ( groups977919841031483927at_nat @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F2 @ A2 ) @ ( groups977919841031483927at_nat @ F2 @ B2 ) ) @ ( groups977919841031483927at_nat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_8569_sum__Un__nat,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F2: produc4166570645942440679at_nat > nat] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ( groups7059897769266147197at_nat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups7059897769266147197at_nat @ F2 @ A2 ) @ ( groups7059897769266147197at_nat @ F2 @ B2 ) ) @ ( groups7059897769266147197at_nat @ F2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_8570_sum__Un__nat,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat] :
( ( finite4343798906461161616at_nat @ A2 )
=> ( ( finite4343798906461161616at_nat @ B2 )
=> ( ( groups3860910324918113789at_nat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F2 @ A2 ) @ ( groups3860910324918113789at_nat @ F2 @ B2 ) ) @ ( groups3860910324918113789at_nat @ F2 @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_8571_sum__Un__nat,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups3542108847815614940at_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ ( groups3542108847815614940at_nat @ F2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_8572_sum_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > rat,C: $o > rat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups7872700643590313910_o_rat
@ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_rat @ ( B @ A ) @ ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups7872700643590313910_o_rat
@ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8573_sum_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > nat,C: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_nat @ ( B @ A ) @ ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8574_sum_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > int,C: $o > int] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K4: $o] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_int @ ( B @ A ) @ ( groups8505340233167759370_o_int @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K4: $o] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups8505340233167759370_o_int @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8575_sum_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > rat,C: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8576_sum_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > nat,C: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_nat @ ( B @ A ) @ ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8577_sum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > rat,C: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_rat @ ( B @ A ) @ ( groups2906978787729119204at_rat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups2906978787729119204at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups2906978787729119204at_rat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8578_sum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > int,C: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_int @ ( B @ A ) @ ( groups3539618377306564664at_int @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3539618377306564664at_int
@ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3539618377306564664at_int @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8579_sum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > nat,C: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_nat @ ( B @ A ) @ ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8580_sum_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > int,C: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_int @ ( B @ A ) @ ( groups4538972089207619220nt_int @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K4: int] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups4538972089207619220nt_int @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8581_sum_Odelta__remove,axiom,
! [S: set_set_nat,A: set_nat,B: set_nat > rat,C: set_nat > rat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( plus_plus_rat @ ( B @ A ) @ ( groups7659867448343625626at_rat @ C @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ) )
& ( ~ ( member_set_nat @ A @ S )
=> ( ( groups7659867448343625626at_rat
@ ^ [K4: set_nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups7659867448343625626at_rat @ C @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_8582_sum__div__partition,axiom,
! [A2: set_int,F2: int > nat,B: nat] :
( ( finite_finite_int @ A2 )
=> ( ( divide_divide_nat @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) @ B )
= ( plus_plus_nat
@ ( groups4541462559716669496nt_nat
@ ^ [A3: int] : ( divide_divide_nat @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] : ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_nat
@ ( groups4541462559716669496nt_nat @ F2
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] :
~ ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8583_sum__div__partition,axiom,
! [A2: set_nat,F2: nat > int,B: int] :
( ( finite_finite_nat @ A2 )
=> ( ( divide_divide_int @ ( groups3539618377306564664at_int @ F2 @ A2 ) @ B )
= ( plus_plus_int
@ ( groups3539618377306564664at_int
@ ^ [A3: nat] : ( divide_divide_int @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] : ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_int
@ ( groups3539618377306564664at_int @ F2
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] :
~ ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8584_sum__div__partition,axiom,
! [A2: set_int,F2: int > code_integer,B: code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( divide6298287555418463151nteger @ ( groups7873554091576472773nteger @ F2 @ A2 ) @ B )
= ( plus_p5714425477246183910nteger
@ ( groups7873554091576472773nteger
@ ^ [A3: int] : ( divide6298287555418463151nteger @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] : ( dvd_dvd_Code_integer @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide6298287555418463151nteger
@ ( groups7873554091576472773nteger @ F2
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] :
~ ( dvd_dvd_Code_integer @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8585_sum__div__partition,axiom,
! [A2: set_nat,F2: nat > code_integer,B: code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( divide6298287555418463151nteger @ ( groups7501900531339628137nteger @ F2 @ A2 ) @ B )
= ( plus_p5714425477246183910nteger
@ ( groups7501900531339628137nteger
@ ^ [A3: nat] : ( divide6298287555418463151nteger @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] : ( dvd_dvd_Code_integer @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide6298287555418463151nteger
@ ( groups7501900531339628137nteger @ F2
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] :
~ ( dvd_dvd_Code_integer @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8586_sum__div__partition,axiom,
! [A2: set_int,F2: int > code_natural,B: code_natural] :
( ( finite_finite_int @ A2 )
=> ( ( divide5121882707175180666atural @ ( groups6697149243333190288atural @ F2 @ A2 ) @ B )
= ( plus_p4538020629002901425atural
@ ( groups6697149243333190288atural
@ ^ [A3: int] : ( divide5121882707175180666atural @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] : ( dvd_dvd_Code_natural @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide5121882707175180666atural
@ ( groups6697149243333190288atural @ F2
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] :
~ ( dvd_dvd_Code_natural @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8587_sum__div__partition,axiom,
! [A2: set_nat,F2: nat > code_natural,B: code_natural] :
( ( finite_finite_nat @ A2 )
=> ( ( divide5121882707175180666atural @ ( groups6325495683096345652atural @ F2 @ A2 ) @ B )
= ( plus_p4538020629002901425atural
@ ( groups6325495683096345652atural
@ ^ [A3: nat] : ( divide5121882707175180666atural @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] : ( dvd_dvd_Code_natural @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide5121882707175180666atural
@ ( groups6325495683096345652atural @ F2
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] :
~ ( dvd_dvd_Code_natural @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8588_sum__div__partition,axiom,
! [A2: set_nat,F2: nat > nat,B: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( divide_divide_nat @ ( groups3542108847815614940at_nat @ F2 @ A2 ) @ B )
= ( plus_plus_nat
@ ( groups3542108847815614940at_nat
@ ^ [A3: nat] : ( divide_divide_nat @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] : ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_nat
@ ( groups3542108847815614940at_nat @ F2
@ ( inf_inf_set_nat @ A2
@ ( collect_nat
@ ^ [A3: nat] :
~ ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8589_sum__div__partition,axiom,
! [A2: set_int,F2: int > int,B: int] :
( ( finite_finite_int @ A2 )
=> ( ( divide_divide_int @ ( groups4538972089207619220nt_int @ F2 @ A2 ) @ B )
= ( plus_plus_int
@ ( groups4538972089207619220nt_int
@ ^ [A3: int] : ( divide_divide_int @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] : ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_int
@ ( groups4538972089207619220nt_int @ F2
@ ( inf_inf_set_int @ A2
@ ( collect_int
@ ^ [A3: int] :
~ ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8590_sum__div__partition,axiom,
! [A2: set_set_nat,F2: set_nat > nat,B: nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( divide_divide_nat @ ( groups8294997508430121362at_nat @ F2 @ A2 ) @ B )
= ( plus_plus_nat
@ ( groups8294997508430121362at_nat
@ ^ [A3: set_nat] : ( divide_divide_nat @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_set_nat @ A2
@ ( collect_set_nat
@ ^ [A3: set_nat] : ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_nat
@ ( groups8294997508430121362at_nat @ F2
@ ( inf_inf_set_set_nat @ A2
@ ( collect_set_nat
@ ^ [A3: set_nat] :
~ ( dvd_dvd_nat @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8591_sum__div__partition,axiom,
! [A2: set_set_nat,F2: set_nat > int,B: int] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( divide_divide_int @ ( groups8292507037921071086at_int @ F2 @ A2 ) @ B )
= ( plus_plus_int
@ ( groups8292507037921071086at_int
@ ^ [A3: set_nat] : ( divide_divide_int @ ( F2 @ A3 ) @ B )
@ ( inf_inf_set_set_nat @ A2
@ ( collect_set_nat
@ ^ [A3: set_nat] : ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ ( divide_divide_int
@ ( groups8292507037921071086at_int @ F2
@ ( inf_inf_set_set_nat @ A2
@ ( collect_set_nat
@ ^ [A3: set_nat] :
~ ( dvd_dvd_int @ B @ ( F2 @ A3 ) ) ) ) )
@ B ) ) ) ) ).
% sum_div_partition
thf(fact_8592_prod_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > code_integer,C: $o > code_integer] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups7694694392188491536nteger
@ ^ [K4: $o] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_3573771949741848930nteger @ ( B @ A ) @ ( groups7694694392188491536nteger @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups7694694392188491536nteger
@ ^ [K4: $o] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups7694694392188491536nteger @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8593_prod_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > code_integer,C: int > code_integer] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_3573771949741848930nteger @ ( B @ A ) @ ( groups3827104343326376752nteger @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups3827104343326376752nteger
@ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3827104343326376752nteger @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8594_prod_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > code_integer,C: nat > code_integer] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_3573771949741848930nteger @ ( B @ A ) @ ( groups3455450783089532116nteger @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3455450783089532116nteger
@ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3455450783089532116nteger @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8595_prod_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > assn,C: $o > assn] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups5301882518646026715o_assn
@ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_assn @ ( B @ A ) @ ( groups5301882518646026715o_assn @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups5301882518646026715o_assn
@ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups5301882518646026715o_assn @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8596_prod_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > assn,C: int > assn] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_assn @ ( B @ A ) @ ( groups7882442080178216443t_assn @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups7882442080178216443t_assn
@ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups7882442080178216443t_assn @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8597_prod_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > assn,C: nat > assn] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_assn @ ( B @ A ) @ ( groups6906906614972039071t_assn @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6906906614972039071t_assn
@ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups6906906614972039071t_assn @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8598_prod_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > rat,C: $o > rat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups2869687844427037835_o_rat
@ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_rat @ ( B @ A ) @ ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups2869687844427037835_o_rat
@ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8599_prod_Odelta__remove,axiom,
! [S: set_int,A: int,B: int > rat,C: int > rat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_rat @ ( B @ A ) @ ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ A @ S )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8600_prod_Odelta__remove,axiom,
! [S: set_nat,A: nat,B: nat > rat,C: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_rat @ ( B @ A ) @ ( groups73079841787564623at_rat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups73079841787564623at_rat
@ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups73079841787564623at_rat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8601_prod_Odelta__remove,axiom,
! [S: set_o,A: $o,B: $o > nat,C: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups3504817904513533571_o_nat
@ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( times_times_nat @ ( B @ A ) @ ( groups3504817904513533571_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups3504817904513533571_o_nat
@ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
@ S )
= ( groups3504817904513533571_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_8602_sum__gp__strict,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_8603_power__diff__sumr2,axiom,
! [X: code_integer,N: nat,Y: code_integer] :
( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
= ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8604_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8605_power__diff__sumr2,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_8606_diff__power__eq__sum,axiom,
! [X: code_integer,N: nat,Y: code_integer] :
( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) @ ( power_8256067586552552935nteger @ Y @ ( suc @ N ) ) )
= ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
@ ( groups7501900531339628137nteger
@ ^ [P6: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ P6 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8607_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X @ P6 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8608_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_8609_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn_int @ R3 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_8610_member__le__sum,axiom,
! [I: $o,A2: set_o,F2: $o > code_integer] :
( ( member_o @ I @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I @ bot_bot_set_o ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_o @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ ( groups4406642042086082107nteger @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8611_member__le__sum,axiom,
! [I: int,A2: set_int,F2: int > code_integer] :
( ( member_int @ I @ A2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_int @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ ( groups7873554091576472773nteger @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8612_member__le__sum,axiom,
! [I: nat,A2: set_nat,F2: nat > code_integer] :
( ( member_nat @ I @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_le3102999989581377725nteger @ ( F2 @ I ) @ ( groups7501900531339628137nteger @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8613_member__le__sum,axiom,
! [I: $o,A2: set_o,F2: $o > rat] :
( ( member_o @ I @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I @ bot_bot_set_o ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_o @ A2 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ ( groups7872700643590313910_o_rat @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8614_member__le__sum,axiom,
! [I: int,A2: set_int,F2: int > rat] :
( ( member_int @ I @ A2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_int @ A2 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ ( groups3906332499630173760nt_rat @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8615_member__le__sum,axiom,
! [I: nat,A2: set_nat,F2: nat > rat] :
( ( member_nat @ I @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_rat @ ( F2 @ I ) @ ( groups2906978787729119204at_rat @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8616_member__le__sum,axiom,
! [I: $o,A2: set_o,F2: $o > nat] :
( ( member_o @ I @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I @ bot_bot_set_o ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_o @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( groups8507830703676809646_o_nat @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8617_member__le__sum,axiom,
! [I: int,A2: set_int,F2: int > nat] :
( ( member_int @ I @ A2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_int @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8618_member__le__sum,axiom,
! [I: $o,A2: set_o,F2: $o > int] :
( ( member_o @ I @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I @ bot_bot_set_o ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_o @ A2 )
=> ( ord_less_eq_int @ ( F2 @ I ) @ ( groups8505340233167759370_o_int @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8619_member__le__sum,axiom,
! [I: nat,A2: set_nat,F2: nat > int] :
( ( member_nat @ I @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X2 ) ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_int @ ( F2 @ I ) @ ( groups3539618377306564664at_int @ F2 @ A2 ) ) ) ) ) ).
% member_le_sum
thf(fact_8620_prod__mono2,axiom,
! [B2: set_int,A2: set_int,F2: int > code_integer] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B6 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A6 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F2 @ A2 ) @ ( groups3827104343326376752nteger @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8621_prod__mono2,axiom,
! [B2: set_int,A2: set_int,F2: int > rat] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B6 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) @ ( groups1072433553688619179nt_rat @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8622_prod__mono2,axiom,
! [B2: set_nat,A2: set_nat,F2: nat > code_integer] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B6 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A6 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F2 @ A2 ) @ ( groups3455450783089532116nteger @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8623_prod__mono2,axiom,
! [B2: set_nat,A2: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B6 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) @ ( groups73079841787564623at_rat @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8624_prod__mono2,axiom,
! [B2: set_nat,A2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ B6 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_int @ ( groups705719431365010083at_int @ F2 @ A2 ) @ ( groups705719431365010083at_int @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8625_prod__mono2,axiom,
! [B2: set_int,A2: set_int,F2: int > int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [B6: int] :
( ( member_int @ B6 @ ( minus_minus_set_int @ B2 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ B6 ) ) )
=> ( ! [A6: int] :
( ( member_int @ A6 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F2 @ A2 ) @ ( groups1705073143266064639nt_int @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8626_prod__mono2,axiom,
! [B2: set_set_nat,A2: set_set_nat,F2: set_nat > code_integer] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B6 ) ) )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A6 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups2761809935038513290nteger @ F2 @ A2 ) @ ( groups2761809935038513290nteger @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8627_prod__mono2,axiom,
! [B2: set_set_nat,A2: set_set_nat,F2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B6 ) ) )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_rat @ ( groups3613417700093529605at_rat @ F2 @ A2 ) @ ( groups3613417700093529605at_rat @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8628_prod__mono2,axiom,
! [B2: set_set_nat,A2: set_set_nat,F2: set_nat > int] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F2 @ B6 ) ) )
=> ( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F2 @ A6 ) ) )
=> ( ord_less_eq_int @ ( groups4246057289670975065at_int @ F2 @ A2 ) @ ( groups4246057289670975065at_int @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8629_prod__mono2,axiom,
! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > code_integer] :
( ( finite6177210948735845034at_nat @ B2 )
=> ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
=> ( ! [B6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B6 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B6 ) ) )
=> ( ! [A6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A6 @ A2 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A6 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups1230400874837758585nteger @ F2 @ A2 ) @ ( groups1230400874837758585nteger @ F2 @ B2 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8630_prod__diff1,axiom,
! [A2: set_o,F2: $o > nat,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_nat )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups3504817904513533571_o_nat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( divide_divide_nat @ ( groups3504817904513533571_o_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups3504817904513533571_o_nat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups3504817904513533571_o_nat @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8631_prod__diff1,axiom,
! [A2: set_int,F2: int > nat,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_nat )
=> ( ( ( member_int @ A @ A2 )
=> ( ( groups1707563613775114915nt_nat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( divide_divide_nat @ ( groups1707563613775114915nt_nat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups1707563613775114915nt_nat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups1707563613775114915nt_nat @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8632_prod__diff1,axiom,
! [A2: set_o,F2: $o > int,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_int )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups3502327434004483295_o_int @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( divide_divide_int @ ( groups3502327434004483295_o_int @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups3502327434004483295_o_int @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups3502327434004483295_o_int @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8633_prod__diff1,axiom,
! [A2: set_o,F2: $o > code_integer,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( ( F2 @ A )
!= zero_z3403309356797280102nteger )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups7694694392188491536nteger @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( divide6298287555418463151nteger @ ( groups7694694392188491536nteger @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups7694694392188491536nteger @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups7694694392188491536nteger @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8634_prod__diff1,axiom,
! [A2: set_int,F2: int > code_integer,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( ( F2 @ A )
!= zero_z3403309356797280102nteger )
=> ( ( ( member_int @ A @ A2 )
=> ( ( groups3827104343326376752nteger @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( divide6298287555418463151nteger @ ( groups3827104343326376752nteger @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups3827104343326376752nteger @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups3827104343326376752nteger @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8635_prod__diff1,axiom,
! [A2: set_nat,F2: nat > code_integer,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( F2 @ A )
!= zero_z3403309356797280102nteger )
=> ( ( ( member_nat @ A @ A2 )
=> ( ( groups3455450783089532116nteger @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( divide6298287555418463151nteger @ ( groups3455450783089532116nteger @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups3455450783089532116nteger @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3455450783089532116nteger @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8636_prod__diff1,axiom,
! [A2: set_o,F2: $o > rat,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_rat )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups2869687844427037835_o_rat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( divide_divide_rat @ ( groups2869687844427037835_o_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups2869687844427037835_o_rat @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups2869687844427037835_o_rat @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8637_prod__diff1,axiom,
! [A2: set_int,F2: int > rat,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_rat )
=> ( ( ( member_int @ A @ A2 )
=> ( ( groups1072433553688619179nt_rat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( divide_divide_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_int @ A @ A2 )
=> ( ( groups1072433553688619179nt_rat @ F2 @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( groups1072433553688619179nt_rat @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8638_prod__diff1,axiom,
! [A2: set_nat,F2: nat > rat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( F2 @ A )
!= zero_zero_rat )
=> ( ( ( member_nat @ A @ A2 )
=> ( ( groups73079841787564623at_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( divide_divide_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups73079841787564623at_rat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups73079841787564623at_rat @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8639_prod__diff1,axiom,
! [A2: set_o,F2: $o > code_natural,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( ( F2 @ A )
!= zero_z2226904508553997617atural )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups6518289543945209051atural @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( divide5121882707175180666atural @ ( groups6518289543945209051atural @ F2 @ A2 ) @ ( F2 @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups6518289543945209051atural @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups6518289543945209051atural @ F2 @ A2 ) ) ) ) ) ) ).
% prod_diff1
thf(fact_8640_prod__Un,axiom,
! [A2: set_int,B2: set_int,F2: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( inf_inf_set_int @ A2 @ B2 ) )
=> ( ( F2 @ X2 )
!= zero_zero_rat ) )
=> ( ( groups1072433553688619179nt_rat @ F2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( divide_divide_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ F2 @ A2 ) @ ( groups1072433553688619179nt_rat @ F2 @ B2 ) ) @ ( groups1072433553688619179nt_rat @ F2 @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_8641_prod__Un,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > rat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
=> ( ( F2 @ X2 )
!= zero_zero_rat ) )
=> ( ( groups3442636767675653108at_rat @ F2 @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( times_times_rat @ ( groups3442636767675653108at_rat @ F2 @ A2 ) @ ( groups3442636767675653108at_rat @ F2 @ B2 ) ) @ ( groups3442636767675653108at_rat @ F2 @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_8642_prod__Un,axiom,
! [A2: set_Pr8551490117392284871at_nat,B2: set_Pr8551490117392284871at_nat,F2: produc4166570645942440679at_nat > rat] :
( ( finite1918287321285529104at_nat @ A2 )
=> ( ( finite1918287321285529104at_nat @ B2 )
=> ( ! [X2: produc4166570645942440679at_nat] :
( ( member6689249552917799696at_nat @ X2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) )
=> ( ( F2 @ X2 )
!= zero_zero_rat ) )
=> ( ( groups5938585286922990810at_rat @ F2 @ ( sup_su3035147773818789531at_nat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( times_times_rat @ ( groups5938585286922990810at_rat @ F2 @ A2 ) @ ( groups5938585286922990810at_rat @ F2 @ B2 ) ) @ ( groups5938585286922990810at_rat @ F2 @ ( inf_in1697001100524423349at_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_8643_prod__Un,axiom,
! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > rat] :
( ( finite4343798906461161616at_nat @ A2 )
=> ( ( finite4343798906461161616at_nat @ B2 )
=> ( ! [X2: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) )
=> ( ( F2 @ X2 )
!= zero_zero_rat ) )
=> ( ( groups8874911130973611098at_rat @ F2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( times_times_rat @ ( groups8874911130973611098at_rat @ F2 @ A2 ) @ ( groups8874911130973611098at_rat @ F2 @ B2 ) ) @ ( groups8874911130973611098at_rat @ F2 @ ( inf_in7913087082777306421at_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_8644_prod__Un,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( F2 @ X2 )
!= zero_zero_rat ) )
=> ( ( groups73079841787564623at_rat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ F2 @ A2 ) @ ( groups73079841787564623at_rat @ F2 @ B2 ) ) @ ( groups73079841787564623at_rat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_8645_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_8646_one__diff__power__eq_H,axiom,
! [X: code_integer,N: nat] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
= ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8647_one__diff__power__eq_H,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8648_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_8649_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22 = zero_zero_int )
=> ( A32
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q7: int] :
( ( A32
= ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q7 @ A22 ) ) ) )
=> ~ ! [R6: int,Q7: int] :
( ( A32
= ( product_Pair_int_int @ Q7 @ R6 ) )
=> ( ( ( sgn_sgn_int @ R6 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R6 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q7 @ A22 ) @ R6 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_8650_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod_int_int] :
( ? [K4: int] :
( ( A12 = K4 )
& ( A23 = zero_zero_int )
& ( A33
= ( product_Pair_int_int @ zero_zero_int @ K4 ) ) )
| ? [L2: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
& ( L2 != zero_zero_int )
& ( K4
= ( times_times_int @ Q5 @ L2 ) ) )
| ? [R4: int,L2: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L2 )
& ( A33
= ( product_Pair_int_int @ Q5 @ R4 ) )
& ( ( sgn_sgn_int @ R4 )
= ( sgn_sgn_int @ L2 ) )
& ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L2 ) )
& ( K4
= ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R4 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_8651_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ K @ L )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_8652_prod_Otriangle__reindex__eq,axiom,
! [G2: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K4: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8653_prod_Otriangle__reindex__eq,axiom,
! [G2: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G2 )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K4: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G2 @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
@ ( set_ord_atMost_nat @ K4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8654_drop__bit__rec,axiom,
( bit_se8570568707652914677it_nat
= ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ A3 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_8655_drop__bit__rec,axiom,
( bit_se3928097537394005634nteger
= ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ A3 @ ( bit_se3928097537394005634nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_8656_drop__bit__rec,axiom,
( bit_se2751692689150723149atural
= ( ^ [N2: nat,A3: code_natural] : ( if_Code_natural @ ( N2 = zero_zero_nat ) @ A3 @ ( bit_se2751692689150723149atural @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_8657_drop__bit__rec,axiom,
( bit_se8568078237143864401it_int
= ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ A3 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_8658_finite__compl,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ ( uminus6524753893492686040at_nat @ A2 ) )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_compl
thf(fact_8659_finite__compl,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ ( uminus3195874150345416415st_nat @ A2 ) )
= ( finite8100373058378681591st_nat @ top_top_set_list_nat ) ) ) ).
% finite_compl
thf(fact_8660_finite__compl,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ ( uminus_uminus_set_o @ A2 ) )
= ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_compl
thf(fact_8661_finite__compl,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( finite_finite_rat @ ( uminus2201863774496077783et_rat @ A2 ) )
= ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_compl
thf(fact_8662_finite__compl,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_8663_finite__compl,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ ( uminus1532241313380277803et_int @ A2 ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_compl
thf(fact_8664_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_8665_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_8666_finite__Collect__not,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P ) )
=> ( ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
~ ( P @ X3 ) ) )
= ( finite8744585540193469122nt_int @ top_to6034159466715884489nt_int ) ) ) ).
% finite_Collect_not
thf(fact_8667_finite__Collect__not,axiom,
! [P: set_nat > $o] :
( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
~ ( P @ X3 ) ) )
= ( finite1152437895449049373et_nat @ top_top_set_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_8668_finite__Collect__not,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ P ) )
=> ( ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
~ ( P @ X3 ) ) )
= ( finite3252695134891459830_nat_o @ top_to4869887016220417533_nat_o ) ) ) ).
% finite_Collect_not
thf(fact_8669_finite__Collect__not,axiom,
! [P: product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
=> ( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
~ ( P @ X3 ) ) )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Collect_not
thf(fact_8670_finite__Collect__not,axiom,
! [P: list_nat > $o] :
( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
~ ( P @ X3 ) ) )
= ( finite8100373058378681591st_nat @ top_top_set_list_nat ) ) ) ).
% finite_Collect_not
thf(fact_8671_finite__Collect__not,axiom,
! [P: $o > $o] :
( ( finite_finite_o @ ( collect_o @ P ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
~ ( P @ X3 ) ) )
= ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Collect_not
thf(fact_8672_finite__Collect__not,axiom,
! [P: rat > $o] :
( ( finite_finite_rat @ ( collect_rat @ P ) )
=> ( ( finite_finite_rat
@ ( collect_rat
@ ^ [X3: rat] :
~ ( P @ X3 ) ) )
= ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Collect_not
thf(fact_8673_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_8674_finite__Collect__not,axiom,
! [P: int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
~ ( P @ X3 ) ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Collect_not
thf(fact_8675_finite__Collect__subsets,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( finite2998713641127702882nt_int @ A2 )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [B4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_8676_finite__Collect__subsets,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_8677_finite__Collect__subsets,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( finite9047747110432174090at_nat
@ ( collec5514110066124741708at_nat
@ ^ [B4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_8678_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_8679_finite__finite__vimage__IntI,axiom,
! [F4: set_o,H3: int > $o,A2: set_int] :
( ( finite_finite_o @ F4 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ F4 )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_o @ H3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A2 ) ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_o @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8680_finite__finite__vimage__IntI,axiom,
! [F4: set_o,H3: nat > $o,A2: set_nat] :
( ( finite_finite_o @ F4 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ F4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A2 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8681_finite__finite__vimage__IntI,axiom,
! [F4: set_nat,H3: int > nat,A2: set_int] :
( ( finite_finite_nat @ F4 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ F4 )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_nat @ H3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A2 ) ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8682_finite__finite__vimage__IntI,axiom,
! [F4: set_nat,H3: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ F4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A2 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8683_finite__finite__vimage__IntI,axiom,
! [F4: set_int,H3: int > int,A2: set_int] :
( ( finite_finite_int @ F4 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ F4 )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_int @ H3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A2 ) ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_int @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8684_finite__finite__vimage__IntI,axiom,
! [F4: set_int,H3: nat > int,A2: set_nat] :
( ( finite_finite_int @ F4 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ F4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A2 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8685_finite__finite__vimage__IntI,axiom,
! [F4: set_set_nat,H3: int > set_nat,A2: set_int] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ F4 )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_set_nat @ H3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) @ A2 ) ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_set_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8686_finite__finite__vimage__IntI,axiom,
! [F4: set_set_nat,H3: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ F4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_set_nat @ H3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) @ A2 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_set_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8687_finite__finite__vimage__IntI,axiom,
! [F4: set_Pr1261947904930325089at_nat,H3: int > product_prod_nat_nat,A2: set_int] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ! [Y2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y2 @ F4 )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage1827798767263827160at_nat @ H3 @ ( insert8211810215607154385at_nat @ Y2 @ bot_bo2099793752762293965at_nat ) ) @ A2 ) ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage1827798767263827160at_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8688_finite__finite__vimage__IntI,axiom,
! [F4: set_Pr1261947904930325089at_nat,H3: nat > product_prod_nat_nat,A2: set_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ! [Y2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y2 @ F4 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ H3 @ ( insert8211810215607154385at_nat @ Y2 @ bot_bo2099793752762293965at_nat ) ) @ A2 ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ H3 @ F4 ) @ A2 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_8689_finite__Collect__conjI,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P ) )
| ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ Q2 ) ) )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8690_finite__Collect__conjI,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8691_finite__Collect__conjI,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ P ) )
| ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ Q2 ) ) )
=> ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8692_finite__Collect__conjI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8693_finite__Collect__conjI,axiom,
! [P: int > $o,Q2: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q2 ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8694_finite__Collect__conjI,axiom,
! [P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
| ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q2 ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( P @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_8695_finite__Collect__disjI,axiom,
! [P: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
( ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P ) )
& ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8696_finite__Collect__disjI,axiom,
! [P: set_nat > $o,Q2: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8697_finite__Collect__disjI,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ P ) )
& ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8698_finite__Collect__disjI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8699_finite__Collect__disjI,axiom,
! [P: int > $o,Q2: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8700_finite__Collect__disjI,axiom,
! [P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] :
( ( P @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
& ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_8701_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_8702_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_8703_finite__Plus__UNIV__iff,axiom,
( ( finite6699802884135759036um_o_o @ top_to1686961084667892491um_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8704_finite__Plus__UNIV__iff,axiom,
( ( finite2301497163391418442_o_rat @ top_to8915897456388626257_o_rat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8705_finite__Plus__UNIV__iff,axiom,
( ( finite5809725721784815170_o_nat @ top_to6072511757011528009_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8706_finite__Plus__UNIV__iff,axiom,
( ( finite1631874702275618462_o_int @ top_to3582428612079801253_o_int )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8707_finite__Plus__UNIV__iff,axiom,
( ( finite5580927318523740128_rat_o @ top_to6370680384054995943_rat_o )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8708_finite__Plus__UNIV__iff,axiom,
( ( finite2362923481445498406at_rat @ top_to5104522503381003637at_rat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8709_finite__Plus__UNIV__iff,axiom,
( ( finite5871152039838895134at_nat @ top_to2261136804003905389at_nat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8710_finite__Plus__UNIV__iff,axiom,
( ( finite1693301020329698426at_int @ top_to8994425695926954441at_int )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8711_finite__Plus__UNIV__iff,axiom,
( ( finite94888208985532392_nat_o @ top_to7120114879189831663_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8712_finite__Plus__UNIV__iff,axiom,
( ( finite2679478125380364318at_rat @ top_to281834657035230061at_rat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_8713_finite__Int,axiom,
! [F4: set_int,G3: set_int] :
( ( ( finite_finite_int @ F4 )
| ( finite_finite_int @ G3 ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_8714_finite__Int,axiom,
! [F4: set_Pr1261947904930325089at_nat,G3: set_Pr1261947904930325089at_nat] :
( ( ( finite6177210948735845034at_nat @ F4 )
| ( finite6177210948735845034at_nat @ G3 ) )
=> ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_8715_finite__Int,axiom,
! [F4: set_nat,G3: set_nat] :
( ( ( finite_finite_nat @ F4 )
| ( finite_finite_nat @ G3 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_8716_finite__Un,axiom,
! [F4: set_int,G3: set_int] :
( ( finite_finite_int @ ( sup_sup_set_int @ F4 @ G3 ) )
= ( ( finite_finite_int @ F4 )
& ( finite_finite_int @ G3 ) ) ) ).
% finite_Un
thf(fact_8717_finite__Un,axiom,
! [F4: set_Pr1261947904930325089at_nat,G3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F4 @ G3 ) )
= ( ( finite6177210948735845034at_nat @ F4 )
& ( finite6177210948735845034at_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_8718_finite__Un,axiom,
! [F4: set_Pr8551490117392284871at_nat,G3: set_Pr8551490117392284871at_nat] :
( ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F4 @ G3 ) )
= ( ( finite1918287321285529104at_nat @ F4 )
& ( finite1918287321285529104at_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_8719_finite__Un,axiom,
! [F4: set_Pr4329608150637261639at_nat,G3: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F4 @ G3 ) )
= ( ( finite4343798906461161616at_nat @ F4 )
& ( finite4343798906461161616at_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_8720_finite__Un,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) )
= ( ( finite_finite_nat @ F4 )
& ( finite_finite_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_8721_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_8722_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_8723_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_8724_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_8725_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_8726_not__finite__existsD,axiom,
! [P: set_Pr958786334691620121nt_int > $o] :
( ~ ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P ) )
=> ? [X_1: set_Pr958786334691620121nt_int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8727_not__finite__existsD,axiom,
! [P: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
=> ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8728_not__finite__existsD,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ~ ( finite3252695134891459830_nat_o @ ( collec939566748876313656_nat_o @ P ) )
=> ? [X_1: produc3658429121746597890et_nat > $o] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8729_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8730_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8731_not__finite__existsD,axiom,
! [P: product_prod_nat_nat > $o] :
( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
=> ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_8732_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8733_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8734_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8735_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8736_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_nat,R: set_nat > nat > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8737_pigeonhole__infinite__rel,axiom,
! [A2: set_set_nat,B2: set_int,R: set_nat > int > $o] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ~ ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [A3: set_nat] :
( ( member_set_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8738_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_Pr1261947904930325089at_nat,R: nat > product_prod_nat_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8739_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_Pr1261947904930325089at_nat,R: int > product_prod_nat_nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite6177210948735845034at_nat @ B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ? [Xa2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8740_pigeonhole__infinite__rel,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_nat,R: product_prod_nat_nat > nat > $o] :
( ~ ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [A3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8741_pigeonhole__infinite__rel,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_int,R: product_prod_nat_nat > int > $o] :
( ~ ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ B2 )
& ( R @ X2 @ Xa2 ) ) )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ~ ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [A3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_8742_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K4: code_integer] : ( if_Code_integer @ ( K4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_8743_finite_OemptyI,axiom,
finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% finite.emptyI
thf(fact_8744_finite_OemptyI,axiom,
finite_finite_o @ bot_bot_set_o ).
% finite.emptyI
thf(fact_8745_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_8746_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_8747_infinite__imp__nonempty,axiom,
! [S: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S )
=> ( S != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_8748_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ ( finite_finite_o @ S )
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
thf(fact_8749_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_8750_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_8751_finite__UNIV,axiom,
finite_finite_o @ top_top_set_o ).
% finite_UNIV
thf(fact_8752_ex__new__if__finite,axiom,
! [A2: set_se6260736226359567993nt_int] :
( ~ ( finite8744585540193469122nt_int @ top_to6034159466715884489nt_int )
=> ( ( finite8744585540193469122nt_int @ A2 )
=> ? [A6: set_Pr958786334691620121nt_int] :
~ ( member2340774599025711042nt_int @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8753_ex__new__if__finite,axiom,
! [A2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ top_top_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ A2 )
=> ? [A6: set_nat] :
~ ( member_set_nat @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8754_ex__new__if__finite,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ~ ( finite3252695134891459830_nat_o @ top_to4869887016220417533_nat_o )
=> ( ( finite3252695134891459830_nat_o @ A2 )
=> ? [A6: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8755_ex__new__if__finite,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( finite6177210948735845034at_nat @ A2 )
=> ? [A6: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8756_ex__new__if__finite,axiom,
! [A2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ top_top_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ A2 )
=> ? [A6: list_nat] :
~ ( member_list_nat @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8757_ex__new__if__finite,axiom,
! [A2: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A2 )
=> ? [A6: $o] :
~ ( member_o @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8758_ex__new__if__finite,axiom,
! [A2: set_rat] :
( ~ ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_rat @ A2 )
=> ? [A6: rat] :
~ ( member_rat @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8759_ex__new__if__finite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A2 )
=> ? [A6: nat] :
~ ( member_nat @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8760_ex__new__if__finite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_int @ A2 )
=> ? [A6: int] :
~ ( member_int @ A6 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_8761_infinite__UNIV__char__0,axiom,
~ ( finite_finite_rat @ top_top_set_rat ) ).
% infinite_UNIV_char_0
thf(fact_8762_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_8763_infinite__UNIV__char__0,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_char_0
thf(fact_8764_finite__UnI,axiom,
! [F4: set_int,G3: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( finite_finite_int @ G3 )
=> ( finite_finite_int @ ( sup_sup_set_int @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_8765_finite__UnI,axiom,
! [F4: set_Pr1261947904930325089at_nat,G3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( finite6177210948735845034at_nat @ G3 )
=> ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_8766_finite__UnI,axiom,
! [F4: set_Pr8551490117392284871at_nat,G3: set_Pr8551490117392284871at_nat] :
( ( finite1918287321285529104at_nat @ F4 )
=> ( ( finite1918287321285529104at_nat @ G3 )
=> ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_8767_finite__UnI,axiom,
! [F4: set_Pr4329608150637261639at_nat,G3: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( finite4343798906461161616at_nat @ G3 )
=> ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_8768_finite__UnI,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( finite_finite_nat @ G3 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_8769_Un__infinite,axiom,
! [S: set_int,T: set_int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) ) ).
% Un_infinite
thf(fact_8770_Un__infinite,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S )
=> ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_8771_Un__infinite,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ~ ( finite1918287321285529104at_nat @ S )
=> ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_8772_Un__infinite,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ~ ( finite4343798906461161616at_nat @ S )
=> ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_8773_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_8774_infinite__Un,axiom,
! [S: set_int,T: set_int] :
( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T ) ) )
= ( ~ ( finite_finite_int @ S )
| ~ ( finite_finite_int @ T ) ) ) ).
% infinite_Un
thf(fact_8775_infinite__Un,axiom,
! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) ) )
= ( ~ ( finite6177210948735845034at_nat @ S )
| ~ ( finite6177210948735845034at_nat @ T ) ) ) ).
% infinite_Un
thf(fact_8776_infinite__Un,axiom,
! [S: set_Pr8551490117392284871at_nat,T: set_Pr8551490117392284871at_nat] :
( ( ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T ) ) )
= ( ~ ( finite1918287321285529104at_nat @ S )
| ~ ( finite1918287321285529104at_nat @ T ) ) ) ).
% infinite_Un
thf(fact_8777_infinite__Un,axiom,
! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T ) ) )
= ( ~ ( finite4343798906461161616at_nat @ S )
| ~ ( finite4343798906461161616at_nat @ T ) ) ) ).
% infinite_Un
thf(fact_8778_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_8779_finite__has__minimal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A2 )
=> ( ( ord_less_eq_o @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8780_finite__has__minimal,axiom,
! [A2: set_filter_nat] :
( ( finite2119507909894593271er_nat @ A2 )
=> ( ( A2 != bot_bo498966703094740906er_nat )
=> ? [X2: filter_nat] :
( ( member_filter_nat @ X2 @ A2 )
& ! [Xa2: filter_nat] :
( ( member_filter_nat @ Xa2 @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8781_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8782_finite__has__minimal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X2: rat] :
( ( member_rat @ X2 @ A2 )
& ! [Xa2: rat] :
( ( member_rat @ Xa2 @ A2 )
=> ( ( ord_less_eq_rat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8783_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8784_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_8785_finite__has__maximal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A2 )
=> ( ( ord_less_eq_o @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8786_finite__has__maximal,axiom,
! [A2: set_filter_nat] :
( ( finite2119507909894593271er_nat @ A2 )
=> ( ( A2 != bot_bo498966703094740906er_nat )
=> ? [X2: filter_nat] :
( ( member_filter_nat @ X2 @ A2 )
& ! [Xa2: filter_nat] :
( ( member_filter_nat @ Xa2 @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8787_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8788_finite__has__maximal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X2: rat] :
( ( member_rat @ X2 @ A2 )
& ! [Xa2: rat] :
( ( member_rat @ Xa2 @ A2 )
=> ( ( ord_less_eq_rat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8789_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8790_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A2 )
=> ( ( ord_less_eq_int @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_8791_finite_Ocases,axiom,
! [A: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A )
=> ( ( A != bot_bo228742789529271731at_nat )
=> ~ ! [A8: set_Pr4329608150637261639at_nat] :
( ? [A6: produc3843707927480180839at_nat] :
( A
= ( insert9069300056098147895at_nat @ A6 @ A8 ) )
=> ~ ( finite4343798906461161616at_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_8792_finite_Ocases,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( A != bot_bo2099793752762293965at_nat )
=> ~ ! [A8: set_Pr1261947904930325089at_nat] :
( ? [A6: product_prod_nat_nat] :
( A
= ( insert8211810215607154385at_nat @ A6 @ A8 ) )
=> ~ ( finite6177210948735845034at_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_8793_finite_Ocases,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ~ ! [A8: set_o] :
( ? [A6: $o] :
( A
= ( insert_o @ A6 @ A8 ) )
=> ~ ( finite_finite_o @ A8 ) ) ) ) ).
% finite.cases
thf(fact_8794_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_8795_finite_Ocases,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ~ ! [A8: set_int] :
( ? [A6: int] :
( A
= ( insert_int @ A6 @ A8 ) )
=> ~ ( finite_finite_int @ A8 ) ) ) ) ).
% finite.cases
thf(fact_8796_finite_Osimps,axiom,
( finite4343798906461161616at_nat
= ( ^ [A3: set_Pr4329608150637261639at_nat] :
( ( A3 = bot_bo228742789529271731at_nat )
| ? [A4: set_Pr4329608150637261639at_nat,B3: produc3843707927480180839at_nat] :
( ( A3
= ( insert9069300056098147895at_nat @ B3 @ A4 ) )
& ( finite4343798906461161616at_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_8797_finite_Osimps,axiom,
( finite6177210948735845034at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat] :
( ( A3 = bot_bo2099793752762293965at_nat )
| ? [A4: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
( ( A3
= ( insert8211810215607154385at_nat @ B3 @ A4 ) )
& ( finite6177210948735845034at_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_8798_finite_Osimps,axiom,
( finite_finite_o
= ( ^ [A3: set_o] :
( ( A3 = bot_bot_set_o )
| ? [A4: set_o,B3: $o] :
( ( A3
= ( insert_o @ B3 @ A4 ) )
& ( finite_finite_o @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_8799_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A4: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_8800_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A3: set_int] :
( ( A3 = bot_bot_set_int )
| ? [A4: set_int,B3: int] :
( ( A3
= ( insert_int @ B3 @ A4 ) )
& ( finite_finite_int @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_8801_finite__induct,axiom,
! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [X2: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F5 )
=> ( ~ ( member8757157785044589968at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert9069300056098147895at_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8802_finite__induct,axiom,
! [F4: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ F4 )
=> ( ( P @ bot_bo1488462491386950373nt_int )
=> ( ! [X2: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ F5 )
=> ( ~ ( member2340774599025711042nt_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8897473484851387113nt_int @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8803_finite__induct,axiom,
! [F4: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X2: set_nat,F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ~ ( member_set_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_set_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8804_finite__induct,axiom,
! [F4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F4 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [X2: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F5 )
=> ( ~ ( member6576561426505652726_nat_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert5175938949040314269_nat_o @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8805_finite__induct,axiom,
! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F5 )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8211810215607154385at_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8806_finite__induct,axiom,
! [F4: set_o,P: set_o > $o] :
( ( finite_finite_o @ F4 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,F5: set_o] :
( ( finite_finite_o @ F5 )
=> ( ~ ( member_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_o @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8807_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8808_finite__induct,axiom,
! [F4: set_int,P: set_int > $o] :
( ( finite_finite_int @ F4 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F5: set_int] :
( ( finite_finite_int @ F5 )
=> ( ~ ( member_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_int @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_8809_finite__ne__induct,axiom,
! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( F4 != bot_bo228742789529271731at_nat )
=> ( ! [X2: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
=> ( ! [X2: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F5 )
=> ( ( F5 != bot_bo228742789529271731at_nat )
=> ( ~ ( member8757157785044589968at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert9069300056098147895at_nat @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8810_finite__ne__induct,axiom,
! [F4: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ F4 )
=> ( ( F4 != bot_bo1488462491386950373nt_int )
=> ( ! [X2: set_Pr958786334691620121nt_int] : ( P @ ( insert8897473484851387113nt_int @ X2 @ bot_bo1488462491386950373nt_int ) )
=> ( ! [X2: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ F5 )
=> ( ( F5 != bot_bo1488462491386950373nt_int )
=> ( ~ ( member2340774599025711042nt_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8897473484851387113nt_int @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8811_finite__ne__induct,axiom,
! [F4: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( F4 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] : ( P @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
=> ( ! [X2: set_nat,F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( F5 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_set_nat @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8812_finite__ne__induct,axiom,
! [F4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F4 )
=> ( ( F4 != bot_bo7824918357723582233_nat_o )
=> ( ! [X2: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) )
=> ( ! [X2: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F5 )
=> ( ( F5 != bot_bo7824918357723582233_nat_o )
=> ( ~ ( member6576561426505652726_nat_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert5175938949040314269_nat_o @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8813_finite__ne__induct,axiom,
! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( F4 != bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
=> ( ! [X2: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F5 )
=> ( ( F5 != bot_bo2099793752762293965at_nat )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8211810215607154385at_nat @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8814_finite__ne__induct,axiom,
! [F4: set_o,P: set_o > $o] :
( ( finite_finite_o @ F4 )
=> ( ( F4 != bot_bot_set_o )
=> ( ! [X2: $o] : ( P @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ! [X2: $o,F5: set_o] :
( ( finite_finite_o @ F5 )
=> ( ( F5 != bot_bot_set_o )
=> ( ~ ( member_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_o @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8815_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( F5 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8816_finite__ne__induct,axiom,
! [F4: set_int,P: set_int > $o] :
( ( finite_finite_int @ F4 )
=> ( ( F4 != bot_bot_set_int )
=> ( ! [X2: int] : ( P @ ( insert_int @ X2 @ bot_bot_set_int ) )
=> ( ! [X2: int,F5: set_int] :
( ( finite_finite_int @ F5 )
=> ( ( F5 != bot_bot_set_int )
=> ( ~ ( member_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_int @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_8817_infinite__finite__induct,axiom,
! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
( ! [A8: set_Pr4329608150637261639at_nat] :
( ~ ( finite4343798906461161616at_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [X2: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F5 )
=> ( ~ ( member8757157785044589968at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert9069300056098147895at_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8818_infinite__finite__induct,axiom,
! [P: set_se6260736226359567993nt_int > $o,A2: set_se6260736226359567993nt_int] :
( ! [A8: set_se6260736226359567993nt_int] :
( ~ ( finite8744585540193469122nt_int @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo1488462491386950373nt_int )
=> ( ! [X2: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ F5 )
=> ( ~ ( member2340774599025711042nt_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8897473484851387113nt_int @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8819_infinite__finite__induct,axiom,
! [P: set_set_nat > $o,A2: set_set_nat] :
( ! [A8: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X2: set_nat,F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ~ ( member_set_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_set_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8820_infinite__finite__induct,axiom,
! [P: set_Pr4532377907799695533_nat_o > $o,A2: set_Pr4532377907799695533_nat_o] :
( ! [A8: set_Pr4532377907799695533_nat_o] :
( ~ ( finite3252695134891459830_nat_o @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [X2: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F5 )
=> ( ~ ( member6576561426505652726_nat_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert5175938949040314269_nat_o @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8821_infinite__finite__induct,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
( ! [A8: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [X2: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F5 )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8211810215607154385at_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8822_infinite__finite__induct,axiom,
! [P: set_o > $o,A2: set_o] :
( ! [A8: set_o] :
( ~ ( finite_finite_o @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,F5: set_o] :
( ( finite_finite_o @ F5 )
=> ( ~ ( member_o @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_o @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8823_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8824_infinite__finite__induct,axiom,
! [P: set_int > $o,A2: set_int] :
( ! [A8: set_int] :
( ~ ( finite_finite_int @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F5: set_int] :
( ( finite_finite_int @ F5 )
=> ( ~ ( member_int @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_int @ X2 @ F5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_8825_finite__prod,axiom,
( ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_8826_finite__prod,axiom,
( ( finite7561617386449315510_o_rat @ top_to642698169864860165_o_rat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_prod
thf(fact_8827_finite__prod,axiom,
( ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_8828_finite__prod,axiom,
( ( finite6891994925333515530_o_int @ top_to4532601362410810969_o_int )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_8829_finite__prod,axiom,
( ( finite1617675504726861388_rat_o @ top_to7320853134386005659_rat_o )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_8830_finite__prod,axiom,
( ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_prod
thf(fact_8831_finite__prod,axiom,
( ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_8832_finite__prod,axiom,
( ( finite1682805285291782414at_int @ top_to7002410609688830997at_int )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_8833_finite__prod,axiom,
( ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_8834_finite__prod,axiom,
( ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_prod
thf(fact_8835_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_8836_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_rat @ top_top_set_rat )
=> ( finite7561617386449315510_o_rat @ top_to642698169864860165_o_rat ) ) ) ).
% finite_Prod_UNIV
thf(fact_8837_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_8838_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite6891994925333515530_o_int @ top_to4532601362410810969_o_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_8839_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite1617675504726861388_rat_o @ top_to7320853134386005659_rat_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_8840_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_rat @ top_top_set_rat )
=> ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat ) ) ) ).
% finite_Prod_UNIV
thf(fact_8841_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_8842_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite1682805285291782414at_int @ top_to7002410609688830997at_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_8843_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_8844_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_rat @ top_top_set_rat )
=> ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat ) ) ) ).
% finite_Prod_UNIV
thf(fact_8845_Finite__Set_Ofinite__set,axiom,
( ( finite9047747110432174090at_nat @ top_to7629004291339433233at_nat )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).
% Finite_Set.finite_set
thf(fact_8846_Finite__Set_Ofinite__set,axiom,
( ( finite7047420756378620717st_nat @ top_to1619013496918823676st_nat )
= ( finite8100373058378681591st_nat @ top_top_set_list_nat ) ) ).
% Finite_Set.finite_set
thf(fact_8847_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_o @ top_top_set_set_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% Finite_Set.finite_set
thf(fact_8848_Finite__Set_Ofinite__set,axiom,
( ( finite6867581373910428453et_rat @ top_top_set_set_rat )
= ( finite_finite_rat @ top_top_set_rat ) ) ).
% Finite_Set.finite_set
thf(fact_8849_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_8850_Finite__Set_Ofinite__set,axiom,
( ( finite6197958912794628473et_int @ top_top_set_set_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% Finite_Set.finite_set
thf(fact_8851_finite__subset__induct_H,axiom,
! [F4: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [A6: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F5 )
=> ( ( member8757157785044589968at_nat @ A6 @ A2 )
=> ( ( ord_le1268244103169919719at_nat @ F5 @ A2 )
=> ( ~ ( member8757157785044589968at_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert9069300056098147895at_nat @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8852_finite__subset__induct_H,axiom,
! [F4: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ F4 )
=> ( ( ord_le483042692224249369nt_int @ F4 @ A2 )
=> ( ( P @ bot_bo1488462491386950373nt_int )
=> ( ! [A6: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ F5 )
=> ( ( member2340774599025711042nt_int @ A6 @ A2 )
=> ( ( ord_le483042692224249369nt_int @ F5 @ A2 )
=> ( ~ ( member2340774599025711042nt_int @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8897473484851387113nt_int @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8853_finite__subset__induct_H,axiom,
! [F4: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_set_nat @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8854_finite__subset__induct_H,axiom,
! [F4: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F4 )
=> ( ( ord_le2965882846123202637_nat_o @ F4 @ A2 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A6: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F5 )
=> ( ( member6576561426505652726_nat_o @ A6 @ A2 )
=> ( ( ord_le2965882846123202637_nat_o @ F5 @ A2 )
=> ( ~ ( member6576561426505652726_nat_o @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert5175938949040314269_nat_o @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8855_finite__subset__induct_H,axiom,
! [F4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A6: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F5 )
=> ( ( member8440522571783428010at_nat @ A6 @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ F5 @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8211810215607154385at_nat @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8856_finite__subset__induct_H,axiom,
! [F4: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F4 )
=> ( ( ord_less_eq_set_o @ F4 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A6: $o,F5: set_o] :
( ( finite_finite_o @ F5 )
=> ( ( member_o @ A6 @ A2 )
=> ( ( ord_less_eq_set_o @ F5 @ A2 )
=> ( ~ ( member_o @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_o @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8857_finite__subset__induct_H,axiom,
! [F4: set_int,A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F4 )
=> ( ( ord_less_eq_set_int @ F4 @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A6: int,F5: set_int] :
( ( finite_finite_int @ F5 )
=> ( ( member_int @ A6 @ A2 )
=> ( ( ord_less_eq_set_int @ F5 @ A2 )
=> ( ~ ( member_int @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_int @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8858_finite__subset__induct_H,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F5 @ A2 )
=> ( ~ ( member_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A6 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_8859_finite__subset__induct,axiom,
! [F4: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ F4 )
=> ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [A6: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ F5 )
=> ( ( member8757157785044589968at_nat @ A6 @ A2 )
=> ( ~ ( member8757157785044589968at_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert9069300056098147895at_nat @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8860_finite__subset__induct,axiom,
! [F4: set_se6260736226359567993nt_int,A2: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ F4 )
=> ( ( ord_le483042692224249369nt_int @ F4 @ A2 )
=> ( ( P @ bot_bo1488462491386950373nt_int )
=> ( ! [A6: set_Pr958786334691620121nt_int,F5: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ F5 )
=> ( ( member2340774599025711042nt_int @ A6 @ A2 )
=> ( ~ ( member2340774599025711042nt_int @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8897473484851387113nt_int @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8861_finite__subset__induct,axiom,
! [F4: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_set_nat @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8862_finite__subset__induct,axiom,
! [F4: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F4 )
=> ( ( ord_le2965882846123202637_nat_o @ F4 @ A2 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A6: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F5 )
=> ( ( member6576561426505652726_nat_o @ A6 @ A2 )
=> ( ~ ( member6576561426505652726_nat_o @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert5175938949040314269_nat_o @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8863_finite__subset__induct,axiom,
! [F4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A6: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ F5 )
=> ( ( member8440522571783428010at_nat @ A6 @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert8211810215607154385at_nat @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8864_finite__subset__induct,axiom,
! [F4: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F4 )
=> ( ( ord_less_eq_set_o @ F4 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A6: $o,F5: set_o] :
( ( finite_finite_o @ F5 )
=> ( ( member_o @ A6 @ A2 )
=> ( ~ ( member_o @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_o @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8865_finite__subset__induct,axiom,
! [F4: set_int,A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F4 )
=> ( ( ord_less_eq_set_int @ F4 @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A6: int,F5: set_int] :
( ( finite_finite_int @ F5 )
=> ( ( member_int @ A6 @ A2 )
=> ( ~ ( member_int @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_int @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8866_finite__subset__induct,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A6 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_8867_infinite__remove,axiom,
! [S: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
( ~ ( finite4343798906461161616at_nat @ S )
=> ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% infinite_remove
thf(fact_8868_infinite__remove,axiom,
! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ~ ( finite6177210948735845034at_nat @ S )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% infinite_remove
thf(fact_8869_infinite__remove,axiom,
! [S: set_o,A: $o] :
( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% infinite_remove
thf(fact_8870_infinite__remove,axiom,
! [S: set_int,A: int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% infinite_remove
thf(fact_8871_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_8872_infinite__coinduct,axiom,
! [X7: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
( ( X7 @ A2 )
=> ( ! [A8: set_Pr4329608150637261639at_nat] :
( ( X7 @ A8 )
=> ? [X5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X5 @ A8 )
& ( ( X7 @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) ) )
| ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
=> ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_8873_infinite__coinduct,axiom,
! [X7: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
( ( X7 @ A2 )
=> ( ! [A8: set_Pr1261947904930325089at_nat] :
( ( X7 @ A8 )
=> ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A8 )
& ( ( X7 @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) )
| ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_8874_infinite__coinduct,axiom,
! [X7: set_o > $o,A2: set_o] :
( ( X7 @ A2 )
=> ( ! [A8: set_o] :
( ( X7 @ A8 )
=> ? [X5: $o] :
( ( member_o @ X5 @ A8 )
& ( ( X7 @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) )
| ~ ( finite_finite_o @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) )
=> ~ ( finite_finite_o @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_8875_infinite__coinduct,axiom,
! [X7: set_int > $o,A2: set_int] :
( ( X7 @ A2 )
=> ( ! [A8: set_int] :
( ( X7 @ A8 )
=> ? [X5: int] :
( ( member_int @ X5 @ A8 )
& ( ( X7 @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) )
| ~ ( finite_finite_int @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) ) ) )
=> ~ ( finite_finite_int @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_8876_infinite__coinduct,axiom,
! [X7: set_nat > $o,A2: set_nat] :
( ( X7 @ A2 )
=> ( ! [A8: set_nat] :
( ( X7 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X7 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_8877_finite__empty__induct,axiom,
! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A8 )
=> ( ( member8757157785044589968at_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ A6 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
=> ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_8878_finite__empty__induct,axiom,
! [A2: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_Pr958786334691620121nt_int,A8: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ A8 )
=> ( ( member2340774599025711042nt_int @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_2612819937483484256nt_int @ A8 @ ( insert8897473484851387113nt_int @ A6 @ bot_bo1488462491386950373nt_int ) ) ) ) ) )
=> ( P @ bot_bo1488462491386950373nt_int ) ) ) ) ).
% finite_empty_induct
thf(fact_8879_finite__empty__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_nat,A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( member_set_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ A6 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_8880_finite__empty__induct,axiom,
! [A2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: produc3658429121746597890et_nat > $o,A8: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A8 )
=> ( ( member6576561426505652726_nat_o @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A8 @ ( insert5175938949040314269_nat_o @ A6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
=> ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% finite_empty_induct
thf(fact_8881_finite__empty__induct,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A8 )
=> ( ( member8440522571783428010at_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ A6 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
=> ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_8882_finite__empty__induct,axiom,
! [A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: $o,A8: set_o] :
( ( finite_finite_o @ A8 )
=> ( ( member_o @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ) )
=> ( P @ bot_bot_set_o ) ) ) ) ).
% finite_empty_induct
thf(fact_8883_finite__empty__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: int,A8: set_int] :
( ( finite_finite_int @ A8 )
=> ( ( member_int @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ A6 @ bot_bot_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_8884_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_8885_finite__remove__induct,axiom,
! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ B2 )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [A8: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A8 )
=> ( ( A8 != bot_bo228742789529271731at_nat )
=> ( ( ord_le1268244103169919719at_nat @ A8 @ B2 )
=> ( ! [X5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X5 @ A8 )
=> ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8886_finite__remove__induct,axiom,
! [B2: set_se6260736226359567993nt_int,P: set_se6260736226359567993nt_int > $o] :
( ( finite8744585540193469122nt_int @ B2 )
=> ( ( P @ bot_bo1488462491386950373nt_int )
=> ( ! [A8: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ A8 )
=> ( ( A8 != bot_bo1488462491386950373nt_int )
=> ( ( ord_le483042692224249369nt_int @ A8 @ B2 )
=> ( ! [X5: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X5 @ A8 )
=> ( P @ ( minus_2612819937483484256nt_int @ A8 @ ( insert8897473484851387113nt_int @ X5 @ bot_bo1488462491386950373nt_int ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8887_finite__remove__induct,axiom,
! [B2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( A8 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A8 )
=> ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8888_finite__remove__induct,axiom,
! [B2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ B2 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A8: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A8 )
=> ( ( A8 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ A8 @ B2 )
=> ( ! [X5: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X5 @ A8 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A8 @ ( insert5175938949040314269_nat_o @ X5 @ bot_bo7824918357723582233_nat_o ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8889_finite__remove__induct,axiom,
! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ B2 )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [A8: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A8 )
=> ( ( A8 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A8 @ B2 )
=> ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A8 )
=> ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8890_finite__remove__induct,axiom,
! [B2: set_o,P: set_o > $o] :
( ( finite_finite_o @ B2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A8: set_o] :
( ( finite_finite_o @ A8 )
=> ( ( A8 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A8 @ B2 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A8 )
=> ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8891_finite__remove__induct,axiom,
! [B2: set_int,P: set_int > $o] :
( ( finite_finite_int @ B2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A8: set_int] :
( ( finite_finite_int @ A8 )
=> ( ( A8 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A8 @ B2 )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A8 )
=> ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8892_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_8893_remove__induct,axiom,
! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
( ( P @ bot_bo228742789529271731at_nat )
=> ( ( ~ ( finite4343798906461161616at_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_Pr4329608150637261639at_nat] :
( ( finite4343798906461161616at_nat @ A8 )
=> ( ( A8 != bot_bo228742789529271731at_nat )
=> ( ( ord_le1268244103169919719at_nat @ A8 @ B2 )
=> ( ! [X5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X5 @ A8 )
=> ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8894_remove__induct,axiom,
! [P: set_se6260736226359567993nt_int > $o,B2: set_se6260736226359567993nt_int] :
( ( P @ bot_bo1488462491386950373nt_int )
=> ( ( ~ ( finite8744585540193469122nt_int @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_se6260736226359567993nt_int] :
( ( finite8744585540193469122nt_int @ A8 )
=> ( ( A8 != bot_bo1488462491386950373nt_int )
=> ( ( ord_le483042692224249369nt_int @ A8 @ B2 )
=> ( ! [X5: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X5 @ A8 )
=> ( P @ ( minus_2612819937483484256nt_int @ A8 @ ( insert8897473484851387113nt_int @ X5 @ bot_bo1488462491386950373nt_int ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8895_remove__induct,axiom,
! [P: set_set_nat > $o,B2: set_set_nat] :
( ( P @ bot_bot_set_set_nat )
=> ( ( ~ ( finite1152437895449049373et_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_set_nat] :
( ( finite1152437895449049373et_nat @ A8 )
=> ( ( A8 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
=> ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A8 )
=> ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8896_remove__induct,axiom,
! [P: set_Pr4532377907799695533_nat_o > $o,B2: set_Pr4532377907799695533_nat_o] :
( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ( ~ ( finite3252695134891459830_nat_o @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A8 )
=> ( ( A8 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ A8 @ B2 )
=> ( ! [X5: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X5 @ A8 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A8 @ ( insert5175938949040314269_nat_o @ X5 @ bot_bo7824918357723582233_nat_o ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8897_remove__induct,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
( ( P @ bot_bo2099793752762293965at_nat )
=> ( ( ~ ( finite6177210948735845034at_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A8 )
=> ( ( A8 != bot_bo2099793752762293965at_nat )
=> ( ( ord_le3146513528884898305at_nat @ A8 @ B2 )
=> ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A8 )
=> ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8898_remove__induct,axiom,
! [P: set_o > $o,B2: set_o] :
( ( P @ bot_bot_set_o )
=> ( ( ~ ( finite_finite_o @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_o] :
( ( finite_finite_o @ A8 )
=> ( ( A8 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A8 @ B2 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A8 )
=> ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8899_remove__induct,axiom,
! [P: set_int > $o,B2: set_int] :
( ( P @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_int] :
( ( finite_finite_int @ A8 )
=> ( ( A8 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A8 @ B2 )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A8 )
=> ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8900_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_8901_finite__induct__select,axiom,
! [S: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
( ( finite4343798906461161616at_nat @ S )
=> ( ( P @ bot_bo228742789529271731at_nat )
=> ( ! [T5: set_Pr4329608150637261639at_nat] :
( ( ord_le2604355607129572851at_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X5 @ ( minus_3314409938677909166at_nat @ S @ T5 ) )
& ( P @ ( insert9069300056098147895at_nat @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_8902_finite__induct__select,axiom,
! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ S )
=> ( ( P @ bot_bo2099793752762293965at_nat )
=> ( ! [T5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ ( minus_1356011639430497352at_nat @ S @ T5 ) )
& ( P @ ( insert8211810215607154385at_nat @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_8903_finite__induct__select,axiom,
! [S: set_o,P: set_o > $o] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [T5: set_o] :
( ( ord_less_set_o @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: $o] :
( ( member_o @ X5 @ ( minus_minus_set_o @ S @ T5 ) )
& ( P @ ( insert_o @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_8904_finite__induct__select,axiom,
! [S: set_int,P: set_int > $o] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [T5: set_int] :
( ( ord_less_set_int @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: int] :
( ( member_int @ X5 @ ( minus_minus_set_int @ S @ T5 ) )
& ( P @ ( insert_int @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_8905_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T5 ) )
& ( P @ ( insert_nat @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_8906_finite__option__UNIV,axiom,
( ( finite6732403688824079472at_nat @ top_to3251141154256563319at_nat )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).
% finite_option_UNIV
thf(fact_8907_finite__option__UNIV,axiom,
( ( finite7857960318081769799st_nat @ top_to1113147461230312982st_nat )
= ( finite8100373058378681591st_nat @ top_top_set_list_nat ) ) ).
% finite_option_UNIV
thf(fact_8908_finite__option__UNIV,axiom,
( ( finite4093902646404507527tion_o @ top_top_set_option_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% finite_option_UNIV
thf(fact_8909_finite__option__UNIV,axiom,
( ( finite2014924581280026175on_rat @ top_to2540212048668676366on_rat )
= ( finite_finite_rat @ top_top_set_rat ) ) ).
% finite_option_UNIV
thf(fact_8910_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_8911_finite__option__UNIV,axiom,
( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% finite_option_UNIV
thf(fact_8912_diff__preserves__multiset,axiom,
! [M4: set_Pr958786334691620121nt_int > nat,N4: set_Pr958786334691620121nt_int > nat] :
( ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8913_diff__preserves__multiset,axiom,
! [M4: set_nat > nat,N4: set_nat > nat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8914_diff__preserves__multiset,axiom,
! [M4: ( produc3658429121746597890et_nat > $o ) > nat,N4: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8915_diff__preserves__multiset,axiom,
! [M4: nat > nat,N4: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8916_diff__preserves__multiset,axiom,
! [M4: int > nat,N4: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8917_diff__preserves__multiset,axiom,
! [M4: product_prod_nat_nat > nat,N4: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M4 @ X3 ) @ ( N4 @ X3 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_8918_add__mset__in__multiset,axiom,
! [M4: set_Pr958786334691620121nt_int > nat,A: set_Pr958786334691620121nt_int] :
( ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8919_add__mset__in__multiset,axiom,
! [M4: set_nat > nat,A: set_nat] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8920_add__mset__in__multiset,axiom,
! [M4: ( produc3658429121746597890et_nat > $o ) > nat,A: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8921_add__mset__in__multiset,axiom,
! [M4: nat > nat,A: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8922_add__mset__in__multiset,axiom,
! [M4: int > nat,A: int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8923_add__mset__in__multiset,axiom,
! [M4: product_prod_nat_nat > nat,A: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A ) @ ( suc @ ( M4 @ X3 ) ) @ ( M4 @ X3 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_8924_finite__linorder__min__induct,axiom,
! [A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B6: $o,A8: set_o] :
( ( finite_finite_o @ A8 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A8 )
=> ( ord_less_o @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_o @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_8925_finite__linorder__min__induct,axiom,
! [A2: set_literal,P: set_literal > $o] :
( ( finite5847741373460823677iteral @ A2 )
=> ( ( P @ bot_bot_set_literal )
=> ( ! [B6: literal,A8: set_literal] :
( ( finite5847741373460823677iteral @ A8 )
=> ( ! [X5: literal] :
( ( member_literal @ X5 @ A8 )
=> ( ord_less_literal @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_literal @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_8926_finite__linorder__min__induct,axiom,
! [A2: set_rat,P: set_rat > $o] :
( ( finite_finite_rat @ A2 )
=> ( ( P @ bot_bot_set_rat )
=> ( ! [B6: rat,A8: set_rat] :
( ( finite_finite_rat @ A8 )
=> ( ! [X5: rat] :
( ( member_rat @ X5 @ A8 )
=> ( ord_less_rat @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_rat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_8927_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_8928_finite__linorder__min__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B6: int,A8: set_int] :
( ( finite_finite_int @ A8 )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A8 )
=> ( ord_less_int @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_int @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_8929_finite__linorder__max__induct,axiom,
! [A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B6: $o,A8: set_o] :
( ( finite_finite_o @ A8 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A8 )
=> ( ord_less_o @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_o @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_8930_finite__linorder__max__induct,axiom,
! [A2: set_literal,P: set_literal > $o] :
( ( finite5847741373460823677iteral @ A2 )
=> ( ( P @ bot_bot_set_literal )
=> ( ! [B6: literal,A8: set_literal] :
( ( finite5847741373460823677iteral @ A8 )
=> ( ! [X5: literal] :
( ( member_literal @ X5 @ A8 )
=> ( ord_less_literal @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_literal @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_8931_finite__linorder__max__induct,axiom,
! [A2: set_rat,P: set_rat > $o] :
( ( finite_finite_rat @ A2 )
=> ( ( P @ bot_bot_set_rat )
=> ( ! [B6: rat,A8: set_rat] :
( ( finite_finite_rat @ A8 )
=> ( ! [X5: rat] :
( ( member_rat @ X5 @ A8 )
=> ( ord_less_rat @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_rat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_8932_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_8933_finite__linorder__max__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B6: int,A8: set_int] :
( ( finite_finite_int @ A8 )
=> ( ! [X5: int] :
( ( member_int @ X5 @ A8 )
=> ( ord_less_int @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_int @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_8934_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F2: $o > rat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,S6: set_o] :
( ( finite_finite_o @ S6 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S6 )
=> ( ord_less_eq_rat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_o @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8935_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S6 )
=> ( ord_less_eq_rat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_nat @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8936_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F2: int > rat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S6: set_int] :
( ( finite_finite_int @ S6 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S6 )
=> ( ord_less_eq_rat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_int @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8937_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F2: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,S6: set_o] :
( ( finite_finite_o @ S6 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S6 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_o @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8938_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S6 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_nat @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8939_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F2: int > nat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S6: set_int] :
( ( finite_finite_int @ S6 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S6 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_int @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8940_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F2: $o > int] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,S6: set_o] :
( ( finite_finite_o @ S6 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S6 )
=> ( ord_less_eq_int @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_o @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8941_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S6 )
=> ( ord_less_eq_int @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_nat @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8942_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F2: int > int] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S6: set_int] :
( ( finite_finite_int @ S6 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S6 )
=> ( ord_less_eq_int @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_int @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8943_finite__ranking__induct,axiom,
! [S: set_set_nat,P: set_set_nat > $o,F2: set_nat > rat] :
( ( finite1152437895449049373et_nat @ S )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X2: set_nat,S6: set_set_nat] :
( ( finite1152437895449049373et_nat @ S6 )
=> ( ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ S6 )
=> ( ord_less_eq_rat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_set_nat @ X2 @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_8944_infinite__growing,axiom,
! [X7: set_o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X7 )
=> ? [Xa2: $o] :
( ( member_o @ Xa2 @ X7 )
& ( ord_less_o @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_o @ X7 ) ) ) ).
% infinite_growing
thf(fact_8945_infinite__growing,axiom,
! [X7: set_literal] :
( ( X7 != bot_bot_set_literal )
=> ( ! [X2: literal] :
( ( member_literal @ X2 @ X7 )
=> ? [Xa2: literal] :
( ( member_literal @ Xa2 @ X7 )
& ( ord_less_literal @ X2 @ Xa2 ) ) )
=> ~ ( finite5847741373460823677iteral @ X7 ) ) ) ).
% infinite_growing
thf(fact_8946_infinite__growing,axiom,
! [X7: set_rat] :
( ( X7 != bot_bot_set_rat )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ X7 )
=> ? [Xa2: rat] :
( ( member_rat @ Xa2 @ X7 )
& ( ord_less_rat @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_rat @ X7 ) ) ) ).
% infinite_growing
thf(fact_8947_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X7 )
& ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_8948_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ X7 )
& ( ord_less_int @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_8949_ex__min__if__finite,axiom,
! [S: set_o] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ S )
& ~ ? [Xa2: $o] :
( ( member_o @ Xa2 @ S )
& ( ord_less_o @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_8950_ex__min__if__finite,axiom,
! [S: set_literal] :
( ( finite5847741373460823677iteral @ S )
=> ( ( S != bot_bot_set_literal )
=> ? [X2: literal] :
( ( member_literal @ X2 @ S )
& ~ ? [Xa2: literal] :
( ( member_literal @ Xa2 @ S )
& ( ord_less_literal @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_8951_ex__min__if__finite,axiom,
! [S: set_rat] :
( ( finite_finite_rat @ S )
=> ( ( S != bot_bot_set_rat )
=> ? [X2: rat] :
( ( member_rat @ X2 @ S )
& ~ ? [Xa2: rat] :
( ( member_rat @ Xa2 @ S )
& ( ord_less_rat @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_8952_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_8953_ex__min__if__finite,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ S )
& ~ ? [Xa2: int] :
( ( member_int @ Xa2 @ S )
& ( ord_less_int @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_8954_filter__preserves__multiset,axiom,
! [M4: set_Pr958786334691620121nt_int > nat,P: set_Pr958786334691620121nt_int > $o] :
( ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite8744585540193469122nt_int
@ ( collec5210948495886036740nt_int
@ ^ [X3: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8955_filter__preserves__multiset,axiom,
! [M4: set_nat > nat,P: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8956_filter__preserves__multiset,axiom,
! [M4: ( produc3658429121746597890et_nat > $o ) > nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite3252695134891459830_nat_o
@ ( collec939566748876313656_nat_o
@ ^ [X3: produc3658429121746597890et_nat > $o] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8957_filter__preserves__multiset,axiom,
! [M4: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8958_filter__preserves__multiset,axiom,
! [M4: int > nat,P: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8959_filter__preserves__multiset,axiom,
! [M4: product_prod_nat_nat > nat,P: product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( M4 @ X3 ) ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M4 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_8960_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K4: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K4 )
= ( sgn_sgn_int @ L2 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L2 @ K4 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_8961_sum__diff1_H__aux,axiom,
! [F4: set_o,I4: set_o,F2: $o > code_integer,I: $o] :
( ( finite_finite_o @ F4 )
=> ( ( ord_less_eq_set_o
@ ( collect_o
@ ^ [I3: $o] :
( ( member_o @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_z3403309356797280102nteger ) ) )
@ F4 )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups1402912129352969042nteger @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( minus_8373710615458151222nteger @ ( groups1402912129352969042nteger @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups1402912129352969042nteger @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( groups1402912129352969042nteger @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8962_sum__diff1_H__aux,axiom,
! [F4: set_int,I4: set_int,F2: int > code_integer,I: int] :
( ( finite_finite_int @ F4 )
=> ( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_z3403309356797280102nteger ) ) )
@ F4 )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups926780983652909934nteger @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( minus_8373710615458151222nteger @ ( groups926780983652909934nteger @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups926780983652909934nteger @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( groups926780983652909934nteger @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8963_sum__diff1_H__aux,axiom,
! [F4: set_o,I4: set_o,F2: $o > rat,I: $o] :
( ( finite_finite_o @ F4 )
=> ( ( ord_less_eq_set_o
@ ( collect_o
@ ^ [I3: $o] :
( ( member_o @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_rat ) ) )
@ F4 )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups3921277224699582669_o_rat @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( minus_minus_rat @ ( groups3921277224699582669_o_rat @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups3921277224699582669_o_rat @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( groups3921277224699582669_o_rat @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8964_sum__diff1_H__aux,axiom,
! [F4: set_int,I4: set_int,F2: int > rat,I: int] :
( ( finite_finite_int @ F4 )
=> ( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_rat ) ) )
@ F4 )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups2350640619554545897nt_rat @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( minus_minus_rat @ ( groups2350640619554545897nt_rat @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups2350640619554545897nt_rat @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( groups2350640619554545897nt_rat @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8965_sum__diff1_H__aux,axiom,
! [F4: set_o,I4: set_o,F2: $o > int,I: $o] :
( ( finite_finite_o @ F4 )
=> ( ( ord_less_eq_set_o
@ ( collect_o
@ ^ [I3: $o] :
( ( member_o @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_int ) ) )
@ F4 )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups4553916814277028129_o_int @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( minus_minus_int @ ( groups4553916814277028129_o_int @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups4553916814277028129_o_int @ F2 @ ( minus_minus_set_o @ I4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
= ( groups4553916814277028129_o_int @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8966_sum__diff1_H__aux,axiom,
! [F4: set_int,I4: set_int,F2: int > int,I: int] :
( ( finite_finite_int @ F4 )
=> ( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_int ) ) )
@ F4 )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups2983280209131991357nt_int @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( minus_minus_int @ ( groups2983280209131991357nt_int @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups2983280209131991357nt_int @ F2 @ ( minus_minus_set_int @ I4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
= ( groups2983280209131991357nt_int @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8967_sum__diff1_H__aux,axiom,
! [F4: set_nat,I4: set_nat,F2: nat > code_integer,I: nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_z3403309356797280102nteger ) ) )
@ F4 )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups555127423416065298nteger @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( minus_8373710615458151222nteger @ ( groups555127423416065298nteger @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups555127423416065298nteger @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( groups555127423416065298nteger @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8968_sum__diff1_H__aux,axiom,
! [F4: set_nat,I4: set_nat,F2: nat > rat,I: nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_rat ) ) )
@ F4 )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups1351286907653491341at_rat @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( minus_minus_rat @ ( groups1351286907653491341at_rat @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups1351286907653491341at_rat @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( groups1351286907653491341at_rat @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8969_sum__diff1_H__aux,axiom,
! [F4: set_nat,I4: set_nat,F2: nat > int,I: nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_zero_int ) ) )
@ F4 )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups1983926497230936801at_int @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( minus_minus_int @ ( groups1983926497230936801at_int @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups1983926497230936801at_int @ F2 @ ( minus_minus_set_nat @ I4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
= ( groups1983926497230936801at_int @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8970_sum__diff1_H__aux,axiom,
! [F4: set_set_nat,I4: set_set_nat,F2: set_nat > code_integer,I: set_nat] :
( ( finite1152437895449049373et_nat @ F4 )
=> ( ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
& ( ( F2 @ I3 )
!= zero_z3403309356797280102nteger ) ) )
@ F4 )
=> ( ( ( member_set_nat @ I @ I4 )
=> ( ( groups9221925307898054856nteger @ F2 @ ( minus_2163939370556025621et_nat @ I4 @ ( insert_set_nat @ I @ bot_bot_set_set_nat ) ) )
= ( minus_8373710615458151222nteger @ ( groups9221925307898054856nteger @ F2 @ I4 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member_set_nat @ I @ I4 )
=> ( ( groups9221925307898054856nteger @ F2 @ ( minus_2163939370556025621et_nat @ I4 @ ( insert_set_nat @ I @ bot_bot_set_set_nat ) ) )
= ( groups9221925307898054856nteger @ F2 @ I4 ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_8971_even__sum__iff,axiom,
! [A2: set_o,F2: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups8507830703676809646_o_nat @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_o
@ ( collect_o
@ ^ [A3: $o] :
( ( member_o @ A3 @ A2 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8972_even__sum__iff,axiom,
! [A2: set_int,F2: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups4541462559716669496nt_nat @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8973_even__sum__iff,axiom,
! [A2: set_o,F2: $o > int] :
( ( finite_finite_o @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups8505340233167759370_o_int @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_o
@ ( collect_o
@ ^ [A3: $o] :
( ( member_o @ A3 @ A2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8974_even__sum__iff,axiom,
! [A2: set_nat,F2: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8975_even__sum__iff,axiom,
! [A2: set_o,F2: $o > code_integer] :
( ( finite_finite_o @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups4406642042086082107nteger @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_o
@ ( collect_o
@ ^ [A3: $o] :
( ( member_o @ A3 @ A2 )
& ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8976_even__sum__iff,axiom,
! [A2: set_nat,F2: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8977_even__sum__iff,axiom,
! [A2: set_int,F2: int > code_integer] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7873554091576472773nteger @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8978_even__sum__iff,axiom,
! [A2: set_o,F2: $o > code_natural] :
( ( finite_finite_o @ A2 )
=> ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups3230237193842799622atural @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_o
@ ( collect_o
@ ^ [A3: $o] :
( ( member_o @ A3 @ A2 )
& ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8979_even__sum__iff,axiom,
! [A2: set_nat,F2: nat > code_natural] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8980_even__sum__iff,axiom,
! [A2: set_int,F2: int > code_natural] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6697149243333190288atural @ F2 @ A2 ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( finite_card_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_8981_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_8982_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_8983_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_8984_card__eq__UNIV,axiom,
! [S: set_o] :
( ( ( finite_card_o @ S )
= ( finite_card_o @ top_top_set_o ) )
= ( S = top_top_set_o ) ) ).
% card_eq_UNIV
thf(fact_8985_card__eq__UNIV2,axiom,
! [S: set_o] :
( ( ( finite_card_o @ top_top_set_o )
= ( finite_card_o @ S ) )
= ( S = top_top_set_o ) ) ).
% card_eq_UNIV2
thf(fact_8986_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_8987_card__UNIV__bool,axiom,
( ( finite_card_o @ top_top_set_o )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card_UNIV_bool
thf(fact_8988_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_8989_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_8990_card_Oempty,axiom,
( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_8991_card_Oempty,axiom,
( ( finite_card_o @ bot_bot_set_o )
= zero_zero_nat ) ).
% card.empty
thf(fact_8992_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_8993_card_Oempty,axiom,
( ( finite_card_int @ bot_bot_set_int )
= zero_zero_nat ) ).
% card.empty
thf(fact_8994_card__ge__UNIV,axiom,
! [S: set_o] :
( ( ord_less_eq_nat @ ( finite_card_o @ top_top_set_o ) @ ( finite_card_o @ S ) )
= ( S = top_top_set_o ) ) ).
% card_ge_UNIV
thf(fact_8995_sum_Oempty_H,axiom,
! [P4: $o > code_integer] :
( ( groups1402912129352969042nteger @ P4 @ bot_bot_set_o )
= zero_z3403309356797280102nteger ) ).
% sum.empty'
thf(fact_8996_sum_Oempty_H,axiom,
! [P4: $o > rat] :
( ( groups3921277224699582669_o_rat @ P4 @ bot_bot_set_o )
= zero_zero_rat ) ).
% sum.empty'
thf(fact_8997_sum_Oempty_H,axiom,
! [P4: $o > nat] :
( ( groups4556407284786078405_o_nat @ P4 @ bot_bot_set_o )
= zero_zero_nat ) ).
% sum.empty'
thf(fact_8998_sum_Oempty_H,axiom,
! [P4: $o > int] :
( ( groups4553916814277028129_o_int @ P4 @ bot_bot_set_o )
= zero_zero_int ) ).
% sum.empty'
thf(fact_8999_sum_Oempty_H,axiom,
! [P4: nat > code_integer] :
( ( groups555127423416065298nteger @ P4 @ bot_bot_set_nat )
= zero_z3403309356797280102nteger ) ).
% sum.empty'
thf(fact_9000_sum_Oempty_H,axiom,
! [P4: nat > rat] :
( ( groups1351286907653491341at_rat @ P4 @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty'
thf(fact_9001_sum_Oempty_H,axiom,
! [P4: nat > nat] :
( ( groups1986416967739987077at_nat @ P4 @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty'
thf(fact_9002_sum_Oempty_H,axiom,
! [P4: nat > int] :
( ( groups1983926497230936801at_int @ P4 @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty'
thf(fact_9003_sum_Oempty_H,axiom,
! [P4: int > code_integer] :
( ( groups926780983652909934nteger @ P4 @ bot_bot_set_int )
= zero_z3403309356797280102nteger ) ).
% sum.empty'
thf(fact_9004_sum_Oempty_H,axiom,
! [P4: int > rat] :
( ( groups2350640619554545897nt_rat @ P4 @ bot_bot_set_int )
= zero_zero_rat ) ).
% sum.empty'
thf(fact_9005_prod__constant,axiom,
! [Y: int,A2: set_list_nat] :
( ( groups2905156660866384563at_int
@ ^ [X3: list_nat] : Y
@ A2 )
= ( power_power_int @ Y @ ( finite_card_list_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9006_prod__constant,axiom,
! [Y: int,A2: set_o] :
( ( groups3502327434004483295_o_int
@ ^ [X3: $o] : Y
@ A2 )
= ( power_power_int @ Y @ ( finite_card_o @ A2 ) ) ) ).
% prod_constant
thf(fact_9007_prod__constant,axiom,
! [Y: int,A2: set_set_nat] :
( ( groups4246057289670975065at_int
@ ^ [X3: set_nat] : Y
@ A2 )
= ( power_power_int @ Y @ ( finite_card_set_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9008_prod__constant,axiom,
! [Y: nat,A2: set_list_nat] :
( ( groups2907647131375434839at_nat
@ ^ [X3: list_nat] : Y
@ A2 )
= ( power_power_nat @ Y @ ( finite_card_list_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9009_prod__constant,axiom,
! [Y: nat,A2: set_o] :
( ( groups3504817904513533571_o_nat
@ ^ [X3: $o] : Y
@ A2 )
= ( power_power_nat @ Y @ ( finite_card_o @ A2 ) ) ) ).
% prod_constant
thf(fact_9010_prod__constant,axiom,
! [Y: nat,A2: set_set_nat] :
( ( groups4248547760180025341at_nat
@ ^ [X3: set_nat] : Y
@ A2 )
= ( power_power_nat @ Y @ ( finite_card_set_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9011_prod__constant,axiom,
! [Y: nat,A2: set_int] :
( ( groups1707563613775114915nt_nat
@ ^ [X3: int] : Y
@ A2 )
= ( power_power_nat @ Y @ ( finite_card_int @ A2 ) ) ) ).
% prod_constant
thf(fact_9012_prod__constant,axiom,
! [Y: nat,A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [X3: nat] : Y
@ A2 )
= ( power_power_nat @ Y @ ( finite_card_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9013_prod__constant,axiom,
! [Y: int,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [X3: nat] : Y
@ A2 )
= ( power_power_int @ Y @ ( finite_card_nat @ A2 ) ) ) ).
% prod_constant
thf(fact_9014_prod__constant,axiom,
! [Y: int,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [X3: int] : Y
@ A2 )
= ( power_power_int @ Y @ ( finite_card_int @ A2 ) ) ) ).
% prod_constant
thf(fact_9015_card__0__eq,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( ( finite_card_list_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_list_nat ) ) ) ).
% card_0_eq
thf(fact_9016_card__0__eq,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( finite_card_set_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_nat ) ) ) ).
% card_0_eq
thf(fact_9017_card__0__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( ( finite711546835091564841at_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ) ).
% card_0_eq
thf(fact_9018_card__0__eq,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( ( finite_card_o @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_o ) ) ) ).
% card_0_eq
thf(fact_9019_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_9020_card__0__eq,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( ( finite_card_int @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_9021_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_9022_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_9023_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_9024_sum__constant,axiom,
! [Y: code_integer,A2: set_o] :
( ( groups4406642042086082107nteger
@ ^ [X3: $o] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9025_sum__constant,axiom,
! [Y: code_integer,A2: set_nat] :
( ( groups7501900531339628137nteger
@ ^ [X3: nat] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9026_sum__constant,axiom,
! [Y: code_integer,A2: set_int] :
( ( groups7873554091576472773nteger
@ ^ [X3: int] : Y
@ A2 )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9027_sum__constant,axiom,
! [Y: int,A2: set_o] :
( ( groups8505340233167759370_o_int
@ ^ [X3: $o] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9028_sum__constant,axiom,
! [Y: int,A2: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X3: nat] : Y
@ A2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9029_sum__constant,axiom,
! [Y: nat,A2: set_o] :
( ( groups8507830703676809646_o_nat
@ ^ [X3: $o] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9030_sum__constant,axiom,
! [Y: nat,A2: set_int] :
( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : Y
@ A2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9031_sum__constant,axiom,
! [Y: rat,A2: set_o] :
( ( groups7872700643590313910_o_rat
@ ^ [X3: $o] : Y
@ A2 )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9032_sum__constant,axiom,
! [Y: rat,A2: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [X3: nat] : Y
@ A2 )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9033_sum__constant,axiom,
! [Y: rat,A2: set_int] :
( ( groups3906332499630173760nt_rat
@ ^ [X3: int] : Y
@ A2 )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A2 ) ) @ Y ) ) ).
% sum_constant
thf(fact_9034_card__Diff__insert,axiom,
! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ A2 )
=> ( ~ ( member8757157785044589968at_nat @ A @ B2 )
=> ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9035_card__Diff__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ A @ B2 )
=> ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9036_card__Diff__insert,axiom,
! [A: set_Pr958786334691620121nt_int,A2: set_se6260736226359567993nt_int,B2: set_se6260736226359567993nt_int] :
( ( member2340774599025711042nt_int @ A @ A2 )
=> ( ~ ( member2340774599025711042nt_int @ A @ B2 )
=> ( ( finite4053189226111446337nt_int @ ( minus_2612819937483484256nt_int @ A2 @ ( insert8897473484851387113nt_int @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite4053189226111446337nt_int @ ( minus_2612819937483484256nt_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9037_card__Diff__insert,axiom,
! [A: produc3658429121746597890et_nat > $o,A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A @ A2 )
=> ( ~ ( member6576561426505652726_nat_o @ A @ B2 )
=> ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9038_card__Diff__insert,axiom,
! [A: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ~ ( member_list_nat @ A @ B2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9039_card__Diff__insert,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( member_o @ A @ A2 )
=> ( ~ ( member_o @ A @ B2 )
=> ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9040_card__Diff__insert,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ A @ B2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9041_card__Diff__insert,axiom,
! [A: int,A2: set_int,B2: set_int] :
( ( member_int @ A @ A2 )
=> ( ~ ( member_int @ A @ B2 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9042_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_9043_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_9044_sum_Oinsert_H,axiom,
! [I4: set_o,P4: $o > code_integer,I: $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups1402912129352969042nteger @ P4 @ ( insert_o @ I @ I4 ) )
= ( groups1402912129352969042nteger @ P4 @ I4 ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups1402912129352969042nteger @ P4 @ ( insert_o @ I @ I4 ) )
= ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups1402912129352969042nteger @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9045_sum_Oinsert_H,axiom,
! [I4: set_nat,P4: nat > code_integer,I: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups555127423416065298nteger @ P4 @ ( insert_nat @ I @ I4 ) )
= ( groups555127423416065298nteger @ P4 @ I4 ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups555127423416065298nteger @ P4 @ ( insert_nat @ I @ I4 ) )
= ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups555127423416065298nteger @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9046_sum_Oinsert_H,axiom,
! [I4: set_int,P4: int > code_integer,I: int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups926780983652909934nteger @ P4 @ ( insert_int @ I @ I4 ) )
= ( groups926780983652909934nteger @ P4 @ I4 ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups926780983652909934nteger @ P4 @ ( insert_int @ I @ I4 ) )
= ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups926780983652909934nteger @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9047_sum_Oinsert_H,axiom,
! [I4: set_o,P4: $o > rat,I: $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_rat ) ) ) )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups3921277224699582669_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
= ( groups3921277224699582669_o_rat @ P4 @ I4 ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups3921277224699582669_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
= ( plus_plus_rat @ ( P4 @ I ) @ ( groups3921277224699582669_o_rat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9048_sum_Oinsert_H,axiom,
! [I4: set_nat,P4: nat > rat,I: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_rat ) ) ) )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
= ( groups1351286907653491341at_rat @ P4 @ I4 ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
= ( plus_plus_rat @ ( P4 @ I ) @ ( groups1351286907653491341at_rat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9049_sum_Oinsert_H,axiom,
! [I4: set_int,P4: int > rat,I: int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_rat ) ) ) )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups2350640619554545897nt_rat @ P4 @ ( insert_int @ I @ I4 ) )
= ( groups2350640619554545897nt_rat @ P4 @ I4 ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups2350640619554545897nt_rat @ P4 @ ( insert_int @ I @ I4 ) )
= ( plus_plus_rat @ ( P4 @ I ) @ ( groups2350640619554545897nt_rat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9050_sum_Oinsert_H,axiom,
! [I4: set_o,P4: $o > nat,I: $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_nat ) ) ) )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o @ I @ I4 ) )
= ( groups4556407284786078405_o_nat @ P4 @ I4 ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o @ I @ I4 ) )
= ( plus_plus_nat @ ( P4 @ I ) @ ( groups4556407284786078405_o_nat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9051_sum_Oinsert_H,axiom,
! [I4: set_nat,P4: nat > nat,I: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_nat ) ) ) )
=> ( ( ( member_nat @ I @ I4 )
=> ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat @ I @ I4 ) )
= ( groups1986416967739987077at_nat @ P4 @ I4 ) ) )
& ( ~ ( member_nat @ I @ I4 )
=> ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat @ I @ I4 ) )
= ( plus_plus_nat @ ( P4 @ I ) @ ( groups1986416967739987077at_nat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9052_sum_Oinsert_H,axiom,
! [I4: set_int,P4: int > nat,I: int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_nat ) ) ) )
=> ( ( ( member_int @ I @ I4 )
=> ( ( groups2985770679641041633nt_nat @ P4 @ ( insert_int @ I @ I4 ) )
= ( groups2985770679641041633nt_nat @ P4 @ I4 ) ) )
& ( ~ ( member_int @ I @ I4 )
=> ( ( groups2985770679641041633nt_nat @ P4 @ ( insert_int @ I @ I4 ) )
= ( plus_plus_nat @ ( P4 @ I ) @ ( groups2985770679641041633nt_nat @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9053_sum_Oinsert_H,axiom,
! [I4: set_o,P4: $o > int,I: $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ I4 )
& ( ( P4 @ X3 )
!= zero_zero_int ) ) ) )
=> ( ( ( member_o @ I @ I4 )
=> ( ( groups4553916814277028129_o_int @ P4 @ ( insert_o @ I @ I4 ) )
= ( groups4553916814277028129_o_int @ P4 @ I4 ) ) )
& ( ~ ( member_o @ I @ I4 )
=> ( ( groups4553916814277028129_o_int @ P4 @ ( insert_o @ I @ I4 ) )
= ( plus_plus_int @ ( P4 @ I ) @ ( groups4553916814277028129_o_int @ P4 @ I4 ) ) ) ) ) ) ).
% sum.insert'
thf(fact_9054_card__doubleton__eq__2__iff,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9055_card__doubleton__eq__2__iff,axiom,
! [A: list_nat,B: list_nat] :
( ( ( finite_card_list_nat @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9056_card__doubleton__eq__2__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( finite_card_set_nat @ ( insert_set_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9057_card__doubleton__eq__2__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9058_card__doubleton__eq__2__iff,axiom,
! [A: $o,B: $o] :
( ( ( finite_card_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9059_card__doubleton__eq__2__iff,axiom,
! [A: nat,B: nat] :
( ( ( finite_card_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9060_card__doubleton__eq__2__iff,axiom,
! [A: int,B: int] :
( ( ( finite_card_int @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( A != B ) ) ).
% card_doubleton_eq_2_iff
thf(fact_9061_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_9062_sum__of__bool__eq,axiom,
! [A2: set_o,P: $o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ A2 )
=> ( ( groups7872700643590313910_o_rat
@ ^ [X3: $o] : ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) )
@ A2 )
= ( semiri681578069525770553at_rat @ ( finite_card_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9063_sum__of__bool__eq,axiom,
! [A2: set_int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [X3: int] : ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) )
@ A2 )
= ( semiri681578069525770553at_rat @ ( finite_card_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9064_sum__of__bool__eq,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [X3: nat] : ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) )
@ A2 )
= ( semiri681578069525770553at_rat @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9065_sum__of__bool__eq,axiom,
! [A2: set_o,P: $o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ A2 )
=> ( ( groups8505340233167759370_o_int
@ ^ [X3: $o] : ( zero_n2684676970156552555ol_int @ ( P @ X3 ) )
@ A2 )
= ( semiri1314217659103216013at_int @ ( finite_card_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9066_sum__of__bool__eq,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( zero_n2684676970156552555ol_int @ ( P @ X3 ) )
@ A2 )
= ( semiri1314217659103216013at_int @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9067_sum__of__bool__eq,axiom,
! [A2: set_o,P: $o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_o @ A2 )
=> ( ( groups8507830703676809646_o_nat
@ ^ [X3: $o] : ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) )
@ A2 )
= ( semiri1316708129612266289at_nat @ ( finite_card_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9068_sum__of__bool__eq,axiom,
! [A2: set_int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) )
@ A2 )
= ( semiri1316708129612266289at_nat @ ( finite_card_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9069_sum__of__bool__eq,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( zero_n2687167440665602831ol_nat @ ( P @ X3 ) )
@ A2 )
= ( semiri1316708129612266289at_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9070_sum__of__bool__eq,axiom,
! [A2: set_int,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( zero_n2684676970156552555ol_int @ ( P @ X3 ) )
@ A2 )
= ( semiri1314217659103216013at_int @ ( finite_card_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9071_sum__of__bool__eq,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ A2 )
=> ( ( groups3760926236672600436at_rat
@ ^ [X3: list_nat] : ( zero_n2052037380579107095ol_rat @ ( P @ X3 ) )
@ A2 )
= ( semiri681578069525770553at_rat @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A2 @ ( collect_list_nat @ P ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_9072_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_9073_n__subsets,axiom,
! [A2: set_Pr958786334691620121nt_int,K: nat] :
( ( finite2998713641127702882nt_int @ A2 )
=> ( ( finite4053189226111446337nt_int
@ ( collec5210948495886036740nt_int
@ ^ [B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B4 @ A2 )
& ( ( finite6756421564338198497nt_int @ B4 )
= K ) ) ) )
= ( binomial @ ( finite6756421564338198497nt_int @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9074_n__subsets,axiom,
! [A2: set_list_nat,K: nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite2364142230527598318st_nat
@ ( collect_set_list_nat
@ ^ [B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B4 @ A2 )
& ( ( finite_card_list_nat @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card_list_nat @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9075_n__subsets,axiom,
! [A2: set_o,K: nat] :
( ( finite_finite_o @ A2 )
=> ( ( finite_card_set_o
@ ( collect_set_o
@ ^ [B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A2 )
& ( ( finite_card_o @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card_o @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9076_n__subsets,axiom,
! [A2: set_set_nat,K: nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1149291290879098388et_nat
@ ( collect_set_set_nat
@ ^ [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A2 )
& ( ( finite_card_set_nat @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card_set_nat @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9077_n__subsets,axiom,
! [A2: set_int,K: nat] :
( ( finite_finite_int @ A2 )
=> ( ( finite_card_set_int
@ ( collect_set_int
@ ^ [B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A2 )
& ( ( finite_card_int @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card_int @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9078_n__subsets,axiom,
! [A2: set_Pr1261947904930325089at_nat,K: nat] :
( ( finite6177210948735845034at_nat @ A2 )
=> ( ( finite4356350796350151305at_nat
@ ( collec5514110066124741708at_nat
@ ^ [B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ B4 @ A2 )
& ( ( finite711546835091564841at_nat @ B4 )
= K ) ) ) )
= ( binomial @ ( finite711546835091564841at_nat @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9079_n__subsets,axiom,
! [A2: set_nat,K: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_set_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
& ( ( finite_card_nat @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card_nat @ A2 ) @ K ) ) ) ).
% n_subsets
thf(fact_9080_sum_Onon__neutral_H,axiom,
! [G2: nat > code_integer,I4: set_nat] :
( ( groups555127423416065298nteger @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
= ( groups555127423416065298nteger @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9081_sum_Onon__neutral_H,axiom,
! [G2: int > code_integer,I4: set_int] :
( ( groups926780983652909934nteger @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
= ( groups926780983652909934nteger @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9082_sum_Onon__neutral_H,axiom,
! [G2: nat > rat,I4: set_nat] :
( ( groups1351286907653491341at_rat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_rat ) ) ) )
= ( groups1351286907653491341at_rat @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9083_sum_Onon__neutral_H,axiom,
! [G2: int > rat,I4: set_int] :
( ( groups2350640619554545897nt_rat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_rat ) ) ) )
= ( groups2350640619554545897nt_rat @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9084_sum_Onon__neutral_H,axiom,
! [G2: nat > nat,I4: set_nat] :
( ( groups1986416967739987077at_nat @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_nat ) ) ) )
= ( groups1986416967739987077at_nat @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9085_sum_Onon__neutral_H,axiom,
! [G2: int > nat,I4: set_int] :
( ( groups2985770679641041633nt_nat @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_nat ) ) ) )
= ( groups2985770679641041633nt_nat @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9086_sum_Onon__neutral_H,axiom,
! [G2: nat > int,I4: set_nat] :
( ( groups1983926497230936801at_int @ G2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_int ) ) ) )
= ( groups1983926497230936801at_int @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9087_sum_Onon__neutral_H,axiom,
! [G2: int > int,I4: set_int] :
( ( groups2983280209131991357nt_int @ G2
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_int ) ) ) )
= ( groups2983280209131991357nt_int @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9088_sum_Onon__neutral_H,axiom,
! [G2: set_nat > code_integer,I4: set_set_nat] :
( ( groups9221925307898054856nteger @ G2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_z3403309356797280102nteger ) ) ) )
= ( groups9221925307898054856nteger @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9089_sum_Onon__neutral_H,axiom,
! [G2: set_nat > rat,I4: set_set_nat] :
( ( groups713094340420062787at_rat @ G2
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ I4 )
& ( ( G2 @ X3 )
!= zero_zero_rat ) ) ) )
= ( groups713094340420062787at_rat @ G2 @ I4 ) ) ).
% sum.non_neutral'
thf(fact_9090_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ top_top_set_set_nat ) )
=> ( A2 = top_top_set_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9091_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( ( finite711546835091564841at_nat @ A2 )
= ( finite711546835091564841at_nat @ top_to4669805908274784177at_nat ) )
=> ( A2 = top_to4669805908274784177at_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9092_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ top_top_set_list_nat )
=> ( ( ( finite_card_list_nat @ A2 )
= ( finite_card_list_nat @ top_top_set_list_nat ) )
=> ( A2 = top_top_set_list_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9093_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_o] :
( ( finite_finite_o @ top_top_set_o )
=> ( ( ( finite_card_o @ A2 )
= ( finite_card_o @ top_top_set_o ) )
=> ( A2 = top_top_set_o ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9094_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( ( finite_card_rat @ A2 )
= ( finite_card_rat @ top_top_set_rat ) )
=> ( A2 = top_top_set_rat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9095_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A2 = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9096_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A2: set_int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( ( finite_card_int @ A2 )
= ( finite_card_int @ top_top_set_int ) )
=> ( A2 = top_top_set_int ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_9097_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_9098_sum__multicount__gen,axiom,
! [S3: set_o,T3: set_nat,R: $o > nat > $o,K: nat > nat] :
( ( finite_finite_o @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I3: $o] :
( ( member_o @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I3: $o] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9099_sum__multicount__gen,axiom,
! [S3: set_int,T3: set_nat,R: int > nat > $o,K: nat > nat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I3: int] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9100_sum__multicount__gen,axiom,
! [S3: set_nat,T3: set_o,R: nat > $o > $o,K: $o > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_o @ T3 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( finite_card_o
@ ( collect_o
@ ^ [J3: $o] :
( ( member_o @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups8507830703676809646_o_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9101_sum__multicount__gen,axiom,
! [S3: set_nat,T3: set_int,R: nat > int > $o,K: int > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T3 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( finite_card_int
@ ( collect_int
@ ^ [J3: int] :
( ( member_int @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups4541462559716669496nt_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9102_sum__multicount__gen,axiom,
! [S3: set_nat,T3: set_nat,R: nat > nat > $o,K: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9103_sum__multicount__gen,axiom,
! [S3: set_list_nat,T3: set_nat,R: list_nat > nat > $o,K: nat > nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [I3: list_nat] :
( ( member_list_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat
@ ^ [I3: list_nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9104_sum__multicount__gen,axiom,
! [S3: set_set_nat,T3: set_nat,R: set_nat > nat > $o,K: nat > nat] :
( ( finite1152437895449049373et_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite_card_set_nat
@ ( collect_set_nat
@ ^ [I3: set_nat] :
( ( member_set_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [I3: set_nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9105_sum__multicount__gen,axiom,
! [S3: set_nat,T3: set_list_nat,R: nat > list_nat > $o,K: list_nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite8100373058378681591st_nat @ T3 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ T3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( finite_card_list_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups4396056296759096172at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9106_sum__multicount__gen,axiom,
! [S3: set_nat,T3: set_set_nat,R: nat > set_nat > $o,K: set_nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite1152437895449049373et_nat @ T3 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ T3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I3: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [J3: set_nat] :
( ( member_set_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups8294997508430121362at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9107_sum__multicount__gen,axiom,
! [S3: set_Pr1261947904930325089at_nat,T3: set_nat,R: product_prod_nat_nat > nat > $o,K: nat > nat] :
( ( finite6177210948735845034at_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ( ( finite711546835091564841at_nat
@ ( collec3392354462482085612at_nat
@ ^ [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ S3 )
& ( R @ I3 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups977919841031483927at_nat
@ ^ [I3: product_prod_nat_nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T3 )
& ( R @ I3 @ J3 ) ) ) )
@ S3 )
= ( groups3542108847815614940at_nat @ K @ T3 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_9108_card__less,axiom,
! [M4: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M4 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_9109_card__less__Suc,axiom,
! [M4: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M4 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M4 )
& ( ord_less_nat @ K4 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M4 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_9110_card__less__Suc2,axiom,
! [M4: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M4 )
& ( ord_less_nat @ K4 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M4 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_9111_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_9112_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_9113_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_9114_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_9115_nat__mod__as__int,axiom,
( modulo_modulo_nat
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_9116_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_9117_card__sum__le__nat__sum,axiom,
! [S: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_9118_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_9119_nat__mult__distrib__neg,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z8 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_9120_diff__nat__eq__if,axiom,
! [Z8: int,Z: int] :
( ( ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_9121_card__UNIV__char,axiom,
( ( finite_card_char @ top_top_set_char )
= ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_9122_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K4 )
= ( sgn_sgn_Code_integer @ L2 ) )
@ ( code_divmod_abs @ K4 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R4: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S2 ) ) )
@ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9123_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_9124_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_9125_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_9126_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_9127_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_9128_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N2: nat,K4: int] : ( bit_se1409905431419307370or_int @ K4 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_9129_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M2: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9130_or__not__numerals_I1_J,axiom,
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(1)
thf(fact_9131_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_9132_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% or_not_numerals(4)
thf(fact_9133_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_9134_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(7)
thf(fact_9135_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_9136_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_9137_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_9138_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K4: int,L2: int] :
( if_int
@ ( ( K4
= ( uminus_uminus_int @ one_one_int ) )
| ( L2
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K4 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_9139_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_9140_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_9141_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9142_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_9143_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_9144_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_9145_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_9146_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
thf(fact_9147_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9148_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_9149_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_9150_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_9151_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9152_max__nat_Osemilattice__neutr__axioms,axiom,
semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.semilattice_neutr_axioms
thf(fact_9153_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.comm_monoid_axioms
thf(fact_9154_max__nat_Omonoid__axioms,axiom,
monoid_nat @ ord_max_nat @ zero_zero_nat ).
% max_nat.monoid_axioms
thf(fact_9155_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_9156_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9157_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
@ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9158_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
= ( set_or4266950643985792945nteger @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_9159_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% Code_Numeral.negative_def
thf(fact_9160_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_9161_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_9162_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_9163_Inf__int__def,axiom,
( complete_Inf_Inf_int
= ( ^ [X4: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X4 ) ) ) ) ) ).
% Inf_int_def
thf(fact_9164_Inf__nat__def1,axiom,
! [K5: set_nat] :
( ( K5 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_9165_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_9166_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_9167_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_9168_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9169_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image_4470545334726330049nteger
@ ^ [X3: code_integer] : ( plus_p5714425477246183910nteger @ X3 @ L )
@ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
= ( set_or8404916559141939852nteger @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_9170_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_9171_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9172_UNIV__char__of__nat,axiom,
( top_top_set_char
= ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_9173_range__nat__of__char,axiom,
( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_9174_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_9175_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_9176_binomial__def,axiom,
( binomial
= ( ^ [N2: nat,K4: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [K6: set_nat] :
( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
& ( ( finite_card_nat @ K6 )
= K4 ) ) ) ) ) ) ).
% binomial_def
thf(fact_9177_times__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_9178_rat__minus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_9179_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_9180_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9181_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9182_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_9183_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral_rat @ K ) )
= ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% quotient_of_number(3)
thf(fact_9184_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% quotient_of_number(4)
thf(fact_9185_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_9186_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_9187_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% quotient_of_number(5)
thf(fact_9188_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q5: rat,R4: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R4 ) ) ) ) ).
% diff_rat_def
thf(fact_9189_quotient__of__div,axiom,
! [R3: rat,N: int,D2: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair_int_int @ N @ D2 ) )
=> ( R3
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).
% quotient_of_div
thf(fact_9190_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X2: nat,Y2: nat] :
( Z
!= ( abs_Integ @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ).
% eq_Abs_Integ
thf(fact_9191_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_9192_quotient__of__denom__pos,axiom,
! [R3: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_9193_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9194_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9195_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
= ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_9196_rat__floor__code,axiom,
( archim3151403230148437115or_rat
= ( ^ [P6: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_9197_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_9198_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_9199_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P6: rat,Q5: rat] :
( produc4947309494688390418_int_o
@ ^ [A3: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C4 @ B3 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_9200_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P6: rat,Q5: rat] :
( produc4947309494688390418_int_o
@ ^ [A3: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D4: int] : ( ord_less_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C4 @ B3 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_9201_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_9202_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ N3 @ M6 )
=> ( ( bit_se1146084159140164899it_int @ K @ M6 )
= ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_9203_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9204_less__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_9205_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9206_plus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_9207_minus__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_9208_rat__divide__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C4 @ B3 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_9209_rat__times__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_9210_rat__plus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_9211_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_9212_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_9213_abs__rat__def,axiom,
( abs_abs_rat
= ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% abs_rat_def
thf(fact_9214_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_9215_char__of__integer__code,axiom,
( char_of_integer
= ( ^ [K4: code_integer] :
( produc4188289175737317920o_char
@ ^ [Q0: code_integer,B0: $o] :
( produc4188289175737317920o_char
@ ^ [Q1: code_integer,B1: $o] :
( produc4188289175737317920o_char
@ ^ [Q22: code_integer,B22: $o] :
( produc4188289175737317920o_char
@ ^ [Q32: code_integer,B32: $o] :
( produc4188289175737317920o_char
@ ^ [Q42: code_integer,B42: $o] :
( produc4188289175737317920o_char
@ ^ [Q52: code_integer,B52: $o] :
( produc4188289175737317920o_char
@ ^ [Q62: code_integer,B62: $o] :
( produc4188289175737317920o_char
@ ^ [Uu: code_integer] : ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 )
@ ( code_bit_cut_integer @ Q62 ) )
@ ( code_bit_cut_integer @ Q52 ) )
@ ( code_bit_cut_integer @ Q42 ) )
@ ( code_bit_cut_integer @ Q32 ) )
@ ( code_bit_cut_integer @ Q22 ) )
@ ( code_bit_cut_integer @ Q1 ) )
@ ( code_bit_cut_integer @ Q0 ) )
@ ( code_bit_cut_integer @ K4 ) ) ) ) ).
% char_of_integer_code
thf(fact_9216_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_9217_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X3: rat] :
( the_int
@ ^ [Z3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9218_Frct__code__post_I9_J,axiom,
! [Q4: product_prod_int_int] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ ( frct @ Q4 ) ) )
= ( frct @ Q4 ) ) ).
% Frct_code_post(9)
thf(fact_9219_Frct__code__post_I1_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
= zero_zero_rat ) ).
% Frct_code_post(1)
thf(fact_9220_Frct__code__post_I2_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
= zero_zero_rat ) ).
% Frct_code_post(2)
thf(fact_9221_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
% Frct_code_post(3)
thf(fact_9222_Frct__code__post_I7_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(7)
thf(fact_9223_Frct__code__post_I8_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(8)
thf(fact_9224_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
= ( numeral_numeral_rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_9225_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
= ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_9226_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y3: nat,Z3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V2 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9227_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X3: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y3: nat,Z3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V2 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9228_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) ) ) ) ).
% uminus_int_def
thf(fact_9229_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M2: nat,N2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_9230_less__eq,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M @ N ) ) ).
% less_eq
thf(fact_9231_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_9232_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9233_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_9234_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9235_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9236_Gcd__remove0__nat,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ( ( gcd_Gcd_nat @ M4 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_9237_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int
@ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q5 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9238_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_9239_minus__rat__cancel,axiom,
! [A: int,B: int] :
( ( fract @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( fract @ A @ B ) ) ).
% minus_rat_cancel
thf(fact_9240_minus__rat,axiom,
! [A: int,B: int] :
( ( uminus_uminus_rat @ ( fract @ A @ B ) )
= ( fract @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_rat
thf(fact_9241_Gcd__nat__eq__one,axiom,
! [N4: set_nat] :
( ( member_nat @ one_one_nat @ N4 )
=> ( ( gcd_Gcd_nat @ N4 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_9242_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% eq_rat(2)
thf(fact_9243_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
= ( semiri681578069525770553at_rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_9244_quotient__of__eq,axiom,
! [A: int,B: int,P4: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A @ B ) ) ) ).
% quotient_of_eq
thf(fact_9245_One__rat__def,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
% One_rat_def
thf(fact_9246_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ one_one_int )
= ( ring_1_of_int_rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_9247_normalize__eq,axiom,
! [A: int,B: int,P4: int,Q4: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A @ B ) ) ) ).
% normalize_eq
thf(fact_9248_Zero__rat__def,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% Zero_rat_def
thf(fact_9249_rat__number__expand_I3_J,axiom,
( numeral_numeral_rat
= ( ^ [K4: num] : ( fract @ ( numeral_numeral_int @ K4 ) @ one_one_int ) ) ) ).
% rat_number_expand(3)
thf(fact_9250_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
= ( numeral_numeral_rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_9251_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% quotient_of_Fract
thf(fact_9252_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% one_less_Fract_iff
thf(fact_9253_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_int @ A @ B ) ) ) ).
% Fract_less_one_iff
thf(fact_9254_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% rat_number_collapse(5)
thf(fact_9255_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9256_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% one_le_Fract_iff
thf(fact_9257_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% Fract_le_one_iff
thf(fact_9258_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_9259_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
= ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% rat_number_expand(5)
thf(fact_9260_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9261_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9262_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9263_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9264_Gcd__int__eq,axiom,
! [N4: set_nat] :
( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N4 ) )
= ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N4 ) ) ) ).
% Gcd_int_eq
thf(fact_9265_Gcd__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
= ( gcd_Gcd_int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_9266_Gcd__nat__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Gcd_nat
@ ( image_int_nat
@ ^ [K4: int] : ( nat2 @ ( abs_abs_int @ K4 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_9267_Gcd__int__def,axiom,
( gcd_Gcd_int
= ( ^ [K6: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K6 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_9268_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_9269_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_9270_Gcd__eq__Max,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M4 )
=> ( ( gcd_Gcd_nat @ M4 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M2: nat] :
( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) )
@ M4 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_9271_take__bit__num__simps_I7_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_9272_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X3: nat] :
( collect_nat
@ ^ [N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_9273_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_9274_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_9275_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9276_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M3: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M3 ) ) ).
% prod_decode_aux.cases
thf(fact_9277_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X4: set_nat] : ( if_nat @ ( X4 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% Sup_nat_def
thf(fact_9278_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_9279_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K4 @ N2 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_9280_Max__divisors__self__int,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
= ( abs_abs_int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_9281_take__bit__num__simps_I6_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_9282_surj__int__encode,axiom,
( ( image_int_nat @ nat_int_encode @ top_top_set_int )
= top_top_set_nat ) ).
% surj_int_encode
thf(fact_9283_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_9284_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9285_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K4 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K4 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M2 @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_9286_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( produc478579273971653890on_num
@ ^ [A3: nat,X3: num] :
( case_nat_option_num @ none_num
@ ^ [O: nat] :
( case_num_option_num @ ( some_num @ one )
@ ^ [P6: num] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X3 )
@ A3 )
@ ( product_Pair_nat_num @ N2 @ M2 ) ) ) ) ).
% take_bit_num_code
thf(fact_9287_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_9288_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_9289_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_9290_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_9291_surj__int__decode,axiom,
( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
= top_top_set_int ) ).
% surj_int_decode
thf(fact_9292_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( order_mono_nat_nat
@ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_9293_and__num_Oelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_9294_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_9295_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_int_def
thf(fact_9296_less__eq__int__def,axiom,
( ord_less_eq_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_9297_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_9298_uminus__integer__def,axiom,
( uminus1351360451143612070nteger
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ uminus_uminus_int ) ) ).
% uminus_integer_def
thf(fact_9299_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9300_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_9301_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
@ M ) ) ).
% min_Suc1
thf(fact_9302_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
@ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_9303_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_9304_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N ) ) @ one_one_Code_integer ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_9305_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fu6891787308814931657nteger @ id_num @ ( map_fu8638147718074629079nteger @ id_num @ code_integer_of_int )
@ ^ [M2: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_9306_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_9307_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_9308_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( bit0 @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M3 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit0 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M3 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M3: num] :
( ( Xa
= ( bit1 @ M3 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_9309_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ ( bit1 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ ( bit0 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit1 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_9310_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) )
= ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_9311_range__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
= S ) ) ).
% range_enumerate
thf(fact_9312_Inf__nat__def,axiom,
( complete_Inf_Inf_nat
= ( ^ [X4: set_nat] :
( ord_Least_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ X4 ) ) ) ) ).
% Inf_nat_def
thf(fact_9313_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_9314_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_9315_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_9316_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_9317_bezw_Osimps,axiom,
( bezw
= ( ^ [X3: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9318_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_9319_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_9320_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_9321_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_9322_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_9323_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_9324_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P6 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_9325_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
% gcd_1_int
thf(fact_9326_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_neg1_int
thf(fact_9327_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% gcd_neg2_int
thf(fact_9328_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_9329_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
= ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_9330_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 )
@ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 ) ) ).
% dup.rsp
thf(fact_9331_sub_Orsp,axiom,
( bNF_re8402795839162346335um_int
@ ^ [Y4: num,Z4: num] : ( Y4 = Z4 )
@ ( bNF_re1822329894187522285nt_int
@ ^ [Y4: num,Z4: num] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
@ ^ [M2: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
@ ^ [M2: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% sub.rsp
thf(fact_9332_uminus__integer_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ uminus_uminus_int
@ uminus_uminus_int ) ).
% uminus_integer.rsp
thf(fact_9333_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_9334_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( gcd_gcd_int @ M @ N )
= ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D4: int] :
( ( dvd_dvd_int @ D4 @ M )
& ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_9335_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_9336_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_9337_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
% gcd_1_nat
thf(fact_9338_Gcd__in,axiom,
! [A2: set_nat] :
( ! [A6: nat,B6: nat] :
( ( member_nat @ A6 @ A2 )
=> ( ( member_nat @ B6 @ A2 )
=> ( member_nat @ ( gcd_gcd_nat @ A6 @ B6 ) @ A2 ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% Gcd_in
thf(fact_9339_gcd__nat_Osemilattice__neutr__axioms,axiom,
semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_9340_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.comm_monoid_axioms
thf(fact_9341_gcd__nat_Omonoid__axioms,axiom,
monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).
% gcd_nat.monoid_axioms
thf(fact_9342_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9343_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% one_rat.transfer
thf(fact_9344_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D4: nat] :
( ( dvd_dvd_nat @ D4 @ M )
& ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_9345_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% zero_rat.transfer
thf(fact_9346_Fract_Otransfer,axiom,
( bNF_re3461391660133120880nt_rat
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ( bNF_re2214769303045360666nt_rat
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ pcr_rat )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_9347_uminus__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
@ uminus_uminus_rat ) ).
% uminus_rat.transfer
thf(fact_9348_times__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
@ times_times_rat ) ).
% times_rat.transfer
thf(fact_9349_inverse__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
@ inverse_inverse_rat ) ).
% inverse_rat.transfer
thf(fact_9350_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9351_positive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_9352_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_9353_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_9354_positive__minus,axiom,
! [X: rat] :
( ~ ( positive @ X )
=> ( ( X != zero_zero_rat )
=> ( positive @ ( uminus_uminus_rat @ X ) ) ) ) ).
% positive_minus
thf(fact_9355_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ pcr_int
@ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_9356_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_9357_nat_Otransfer,axiom,
( bNF_re4555766996558763186at_nat @ pcr_int
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ nat2 ) ).
% nat.transfer
thf(fact_9358_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_9359_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_9360_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_9361_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_9362_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_9363_positive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% positive_def
thf(fact_9364_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_9365_inverse__rat__def,axiom,
( inverse_inverse_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_9366_times__rat__def,axiom,
( times_times_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_9367_uminus__rat__def,axiom,
( uminus_uminus_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% uminus_rat_def
thf(fact_9368_one__rat__def,axiom,
( one_one_rat
= ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% one_rat_def
thf(fact_9369_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_9370_zero__rat__def,axiom,
( zero_zero_rat
= ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% zero_rat_def
thf(fact_9371_Fract__def,axiom,
( fract
= ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ) ).
% Fract_def
thf(fact_9372_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_9373_inverse__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_9374_times__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_9375_one__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% one_rat.rsp
thf(fact_9376_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% zero_rat.rsp
thf(fact_9377_Fract_Orsp,axiom,
( bNF_re157797125943740599nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ( bNF_re6250860962936578807nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ratrel )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
@ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_9378_ratrel__def,axiom,
( ratrel
= ( ^ [X3: product_prod_int_int,Y3: product_prod_int_int] :
( ( ( product_snd_int_int @ X3 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y3 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) )
= ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_9379_times__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_9380_uminus__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
@ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ).
% uminus_rat.rsp
thf(fact_9381_positive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
@ ^ [X3: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).
% positive.rsp
thf(fact_9382_inverse__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
@ ^ [X3: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X3 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_9383_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
@ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_9384_uminus__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_9385_cr__rat__def,axiom,
( cr_rat
= ( ^ [X3: product_prod_int_int,Y3: rat] :
( ( ratrel @ X3 @ X3 )
& ( ( abs_Rat @ X3 )
= Y3 ) ) ) ) ).
% cr_rat_def
thf(fact_9386_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_9387_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
= ( ( plus_plus_nat @ X @ V )
= ( plus_plus_nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_9388_zero__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% zero_int.rsp
thf(fact_9389_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) ) ) ).
% uminus_int.rsp
thf(fact_9390_nat_Orsp,axiom,
( bNF_re8246922863344978751at_nat @ intrel
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% nat.rsp
thf(fact_9391_one__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% one_int.rsp
thf(fact_9392_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] :
( ( plus_plus_nat @ X3 @ V2 )
= ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).
% intrel_def
thf(fact_9393_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_9394_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X3: nat,Y3: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_9395_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_9396_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X3: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_9397_quotient__of__def,axiom,
( quotient_of
= ( ^ [X3: rat] :
( the_Pr4378521158711661632nt_int
@ ^ [Pair: product_prod_int_int] :
( ( X3
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_9398_signed__take__bit__eq__concat__bit,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K4: int] : ( bit_concat_bit @ N2 @ K4 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K4 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_9399_normalize__stable,axiom,
! [Q4: int,P4: int] :
( ( ord_less_int @ zero_zero_int @ Q4 )
=> ( ( algebr932160517623751201me_int @ P4 @ Q4 )
=> ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
= ( product_Pair_int_int @ P4 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_9400_quotient__of__coprime,axiom,
! [R3: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% quotient_of_coprime
thf(fact_9401_normalize__coprime,axiom,
! [R3: product_prod_int_int,P4: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair_int_int @ P4 @ Q4 ) )
=> ( algebr932160517623751201me_int @ P4 @ Q4 ) ) ).
% normalize_coprime
thf(fact_9402_coprime__common__divisor__int,axiom,
! [A: int,B: int,X: int] :
( ( algebr932160517623751201me_int @ A @ B )
=> ( ( dvd_dvd_int @ X @ A )
=> ( ( dvd_dvd_int @ X @ B )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_9403_coprime__common__divisor__nat,axiom,
! [A: nat,B: nat,X: nat] :
( ( algebr934650988132801477me_nat @ A @ B )
=> ( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( X = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_9404_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_9405_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_9406_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_9407_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9408_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_9409_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9410_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9411_length__upto,axiom,
! [I: int,J: int] :
( ( size_size_list_int @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_9412_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9413_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9414_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_9415_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9416_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_9417_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9418_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_9419_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_9420_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_9421_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_9422_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9423_sort__upto,axiom,
! [I: int,J: int] :
( ( linord1735203802627413978nt_int
@ ^ [X3: int] : X3
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_9424_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X3: nat] : X3
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_9425_map__add__upt_H,axiom,
! [Ofs: nat,A: nat,B: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ Ofs )
@ ( upt @ A @ B ) )
= ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_9426_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_9427_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_9428_upt__filter__extend,axiom,
! [U: nat,U3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ U @ U3 )
=> ( ! [I2: nat] :
( ( ( ord_less_eq_nat @ U @ I2 )
& ( ord_less_nat @ I2 @ U3 ) )
=> ~ ( P @ I2 ) )
=> ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
= ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_9429_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_9430_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_9431_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N4 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N4 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_9432_card__length__sum__list,axiom,
! [M: nat,N4: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N4 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).
% card_length_sum_list
thf(fact_9433_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ K @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_9434_rat__denum,axiom,
? [F3: nat > rat] :
( ( image_nat_rat @ F3 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_9435_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_9436_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_9437_Sup__int__def,axiom,
( complete_Sup_Sup_int
= ( ^ [X4: set_int] :
( the_int
@ ^ [X3: int] :
( ( member_int @ X3 @ X4 )
& ! [Y3: int] :
( ( member_int @ Y3 @ X4 )
=> ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_9438_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_9439_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N2: nat] :
( produc6207742614233964070at_rat
@ ^ [A3: nat,B3: nat] : ( fract @ ( nat_int_decode @ A3 ) @ ( nat_int_decode @ B3 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_9440_surj__prod__decode,axiom,
( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% surj_prod_decode
thf(fact_9441_list__decode_Oelims,axiom,
! [X: nat,Y: list_nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( produc2761476792215241774st_nat
@ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_9442_surj__list__decode,axiom,
( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
= top_top_set_list_nat ) ).
% surj_list_decode
thf(fact_9443_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( produc2761476792215241774st_nat
@ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_9444_list__decode_Opelims,axiom,
! [X: nat,Y: list_nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( accp_nat @ nat_list_decode_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( produc2761476792215241774st_nat
@ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N3 ) ) )
=> ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_9445_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp_nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
=> ( P @ zero_zero_nat ) )
=> ( ! [N3: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X5: nat,Y5: nat] :
( ( ( product_Pair_nat_nat @ X5 @ Y5 )
= ( nat_prod_decode @ N3 ) )
=> ( P @ Y5 ) )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_9446_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( produc2761476792215241774st_nat
@ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_9447_bij__int__decode,axiom,
bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% bij_int_decode
thf(fact_9448_bij__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).
% bij_enumerate
thf(fact_9449_bij__int__encode,axiom,
bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% bij_int_encode
thf(fact_9450_bij__prod__decode,axiom,
bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).
% bij_prod_decode
thf(fact_9451_bij__list__decode,axiom,
bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).
% bij_list_decode
thf(fact_9452_Rep__unit,axiom,
! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% Rep_unit
thf(fact_9453_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ! [X2: product_unit] : ( P @ ( product_Rep_unit @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_9454_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ~ ! [X2: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X2 ) ) ) ) ).
% Rep_unit_cases
thf(fact_9455_type__definition__unit,axiom,
type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% type_definition_unit
thf(fact_9456_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_9457_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( ( product_Abs_unit @ X )
= ( product_Abs_unit @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_9458_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X
= ( product_Abs_unit @ Y2 ) )
=> ~ ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% Abs_unit_cases
thf(fact_9459_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( P @ ( product_Abs_unit @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_9460_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_We3818239936649020644el_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_9461_pair__lessI2,axiom,
! [A: nat,B: nat,S3: nat,T3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S3 @ T3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_9462_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z ) ) ).
% pair_less_iff1
thf(fact_9463_butlast__upt,axiom,
! [M: nat,N: nat] :
( ( butlast_nat @ ( upt @ M @ N ) )
= ( upt @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% butlast_upt
thf(fact_9464_pair__lessI1,axiom,
! [A: nat,B: nat,S3: nat,T3: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_9465_smin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_9466_smin__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] :
( ( X7 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_9467_smax__insertI,axiom,
! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ Y6 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y6 ) @ fun_max_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X7 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_9468_pair__leqI2,axiom,
! [A: nat,B: nat,S3: nat,T3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S3 @ T3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_9469_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_9470_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat
@ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_9471_le__sequentially,axiom,
! [F4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
= ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_9472_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).
% pair_leq_def
thf(fact_9473_smax__emptyI,axiom,
! [Y6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ Y6 )
=> ( ( Y6 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_9474_filterlim__atMost__at__top,axiom,
filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% filterlim_atMost_at_top
thf(fact_9475_filterlim__lessThan__at__top,axiom,
filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% filterlim_lessThan_at_top
thf(fact_9476_pair__leqI1,axiom,
! [A: nat,B: nat,S3: nat,T3: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_9477_wmin__insertI,axiom,
! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_9478_wmax__insertI,axiom,
! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ YS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_9479_max__weak__def,axiom,
( fun_max_weak
= ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% max_weak_def
thf(fact_9480_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
thf(fact_9481_eventually__False__sequentially,axiom,
~ ( eventually_nat
@ ^ [N2: nat] : $false
@ at_top_nat ) ).
% eventually_False_sequentially
thf(fact_9482_max__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_9483_min__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_9484_wmax__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ X7 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_9485_wmin__emptyI,axiom,
! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_9486_min__weak__def,axiom,
( fun_min_weak
= ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% min_weak_def
thf(fact_9487_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_9488_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_9489_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_9490_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9491_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
= ( set_or2715278749043346189nteger @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_9492_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_9493_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
= one_one_int ) ).
% abs_division_segment
thf(fact_9494_division__segment__nat__def,axiom,
( euclid3398187327856392827nt_nat
= ( ^ [N2: nat] : one_one_nat ) ) ).
% division_segment_nat_def
thf(fact_9495_division__segment__int__def,axiom,
( euclid3395696857347342551nt_int
= ( ^ [K4: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% division_segment_int_def
thf(fact_9496_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
= ( ord_less_nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_9497_total__less__than,axiom,
total_on_nat @ top_top_set_nat @ less_than ).
% total_less_than
thf(fact_9498_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_2888998067076097458on_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_9499_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_2888998067076097458on_nat
@ ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_9500_ctwo__Cnotzero,axiom,
( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
& ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% ctwo_Cnotzero
thf(fact_9501_ctwo__def,axiom,
( bNF_Cardinal_ctwo
= ( bNF_Ca3172636760949973492d_of_o @ top_top_set_o ) ) ).
% ctwo_def
thf(fact_9502_ctwo__ordLess__natLeq,axiom,
member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).
% ctwo_ordLess_natLeq
thf(fact_9503_natLeq__card__order,axiom,
bNF_Ca1281551314933786834on_nat @ top_top_set_nat @ bNF_Ca8665028551170535155natLeq ).
% natLeq_card_order
thf(fact_9504_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_9505_Field__natLeq,axiom,
( ( field_nat @ bNF_Ca8665028551170535155natLeq )
= top_top_set_nat ) ).
% Field_natLeq
thf(fact_9506_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% natLeq_def
thf(fact_9507_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_9508_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
@ ^ [Uu: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ N )
& ( ord_less_nat @ Y3 @ N )
& ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_9509_card__of__nat,axiom,
member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).
% card_of_nat
thf(fact_9510_card__of__Field__natLeq,axiom,
member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).
% card_of_Field_natLeq
thf(fact_9511_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_9512_eventually__prod__sequentially,axiom,
! [P: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N6: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ N6 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_9513_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_9514_int_Oid__abs__transfer,axiom,
( bNF_re7934895593101944656at_int
@ ( basic_5328504652464829177at_nat
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
@ pcr_int
@ ^ [X3: product_prod_nat_nat] : X3
@ abs_Integ ) ).
% int.id_abs_transfer
thf(fact_9515_smsI,axiom,
! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B2 ) ) @ fun_max_strict )
=> ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B2 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_9516_wmsI,axiom,
! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B2 ) ) @ fun_max_strict )
| ( ( A2 = zero_z1048942125864253310at_nat )
& ( B2 = zero_z1048942125864253310at_nat ) ) )
=> ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B2 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_9517_ms__weak__def,axiom,
( ms_weak
= ( sup_su3035147773818789531at_nat @ ms_strict @ id_mul2649389997224486051at_nat ) ) ).
% ms_weak_def
thf(fact_9518_ms__reduction__pair,axiom,
fun_re7357418987779152907at_nat @ ( produc5245064249948416855at_nat @ ms_strict @ ms_weak ) ).
% ms_reduction_pair
thf(fact_9519_ms__strictI,axiom,
! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
( ( pw_leq @ Z5 @ Z9 )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B2 ) ) @ fun_max_strict )
=> ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B2 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_9520_ms__weakI1,axiom,
! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
( ( pw_leq @ Z5 @ Z9 )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B2 ) ) @ fun_max_strict )
=> ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B2 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_9521_pw__leq__lstep,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( pw_leq @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) ) ) ).
% pw_leq_lstep
thf(fact_9522_pw__leq_Ocases,axiom,
! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
( ( pw_leq @ A1 @ A22 )
=> ( ( ( A1 = zero_z1048942125864253310at_nat )
=> ( A22 != zero_z1048942125864253310at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,X8: multis2468970476368604999at_nat] :
( ( A1
= ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X2 @ zero_z1048942125864253310at_nat ) @ X8 ) )
=> ! [Y7: multis2468970476368604999at_nat] :
( ( A22
= ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y2 @ zero_z1048942125864253310at_nat ) @ Y7 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X8 @ Y7 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_9523_pw__leq_Osimps,axiom,
( pw_leq
= ( ^ [A12: multis2468970476368604999at_nat,A23: multis2468970476368604999at_nat] :
( ( ( A12 = zero_z1048942125864253310at_nat )
& ( A23 = zero_z1048942125864253310at_nat ) )
| ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X4: multis2468970476368604999at_nat,Y8: multis2468970476368604999at_nat] :
( ( A12
= ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X3 @ zero_z1048942125864253310at_nat ) @ X4 ) )
& ( A23
= ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y3 @ zero_z1048942125864253310at_nat ) @ Y8 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
& ( pw_leq @ X4 @ Y8 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_9524_pw__leq__step,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
=> ( ( pw_leq @ X7 @ Y6 )
=> ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X7 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y6 ) ) ) ) ).
% pw_leq_step
thf(fact_9525_ms__weakI2,axiom,
! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat] :
( ( pw_leq @ Z5 @ Z9 )
=> ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z9 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_9526_pw__leq__split,axiom,
! [X7: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
( ( pw_leq @ X7 @ Y6 )
=> ? [A8: multis2468970476368604999at_nat,B9: multis2468970476368604999at_nat,Z10: multis2468970476368604999at_nat] :
( ( X7
= ( plus_p7104986032573967614at_nat @ A8 @ Z10 ) )
& ( Y6
= ( plus_p7104986032573967614at_nat @ B9 @ Z10 ) )
& ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A8 ) @ ( set_ms8126754132646256062at_nat @ B9 ) ) @ fun_max_strict )
| ( ( B9 = zero_z1048942125864253310at_nat )
& ( A8 = zero_z1048942125864253310at_nat ) ) ) ) ) ).
% pw_leq_split
thf(fact_9527_rat_Odomain,axiom,
( ( domain9213661015745956397nt_rat @ pcr_rat )
= ( ^ [X3: product_prod_int_int] :
? [Y3: product_prod_int_int] :
( ( basic_4387203522000727145nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ X3
@ Y3 )
& ( ratrel @ Y3 @ Y3 ) ) ) ) ).
% rat.domain
thf(fact_9528_rat_Odomain__par,axiom,
! [DR1: int > $o,DR2: int > $o,P22: product_prod_int_int > $o] :
( ( ( domainp_int_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= DR1 )
=> ( ( ( domainp_int_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= DR2 )
=> ( ( bNF_re8699439704749558557nt_o_o
@ ( basic_4387203522000727145nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ P22
@ ^ [X3: product_prod_int_int] : ( ratrel @ X3 @ X3 ) )
=> ( ( domain9213661015745956397nt_rat @ pcr_rat )
= ( inf_in3604695632404883862_int_o @ ( basic_1567116559311922317nt_int @ DR1 @ DR2 ) @ P22 ) ) ) ) ) ).
% rat.domain_par
thf(fact_9529_rat_Odomain__par__left__total,axiom,
! [P7: product_prod_int_int > $o] :
( ( left_t3131394472396969446nt_int
@ ( basic_4387203522000727145nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) ) )
=> ( ( bNF_re8699439704749558557nt_o_o
@ ( basic_4387203522000727145nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ P7
@ ^ [X3: product_prod_int_int] : ( ratrel @ X3 @ X3 ) )
=> ( ( domain9213661015745956397nt_rat @ pcr_rat )
= P7 ) ) ) ).
% rat.domain_par_left_total
thf(fact_9530_Abs__rat__inject,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( member2340774599025711042nt_int @ Y
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( ( abs_rat @ X )
= ( abs_rat @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_rat_inject
thf(fact_9531_Abs__rat__induct,axiom,
! [P: rat > $o,X: rat] :
( ! [Y2: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y2
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ( P @ ( abs_rat @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_rat_induct
thf(fact_9532_Abs__rat__cases,axiom,
! [X: rat] :
~ ! [Y2: set_Pr958786334691620121nt_int] :
( ( X
= ( abs_rat @ Y2 ) )
=> ~ ( member2340774599025711042nt_int @ Y2
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) ) ) ).
% Abs_rat_cases
thf(fact_9533_type__definition__rat,axiom,
( type_d8554052265237484179nt_int @ rep_rat @ abs_rat
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) ) ).
% type_definition_rat
thf(fact_9534_Abs__rat__inverse,axiom,
! [Y: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ( rep_rat @ ( abs_rat @ Y ) )
= Y ) ) ).
% Abs_rat_inverse
thf(fact_9535_Rep__rat__induct,axiom,
! [Y: set_Pr958786334691620121nt_int,P: set_Pr958786334691620121nt_int > $o] :
( ( member2340774599025711042nt_int @ Y
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ( ! [X2: rat] : ( P @ ( rep_rat @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_rat_induct
thf(fact_9536_Rep__rat__cases,axiom,
! [Y: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Y
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) )
=> ~ ! [X2: rat] :
( Y
!= ( rep_rat @ X2 ) ) ) ).
% Rep_rat_cases
thf(fact_9537_Rep__rat,axiom,
! [X: rat] :
( member2340774599025711042nt_int @ ( rep_rat @ X )
@ ( collec5210948495886036740nt_int
@ ^ [C4: set_Pr958786334691620121nt_int] :
? [X3: product_prod_int_int] :
( ( ratrel @ X3 @ X3 )
& ( C4
= ( collec213857154873943460nt_int @ ( ratrel @ X3 ) ) ) ) ) ) ).
% Rep_rat
thf(fact_9538_cr__int__def,axiom,
( cr_int
= ( ^ [X3: product_prod_nat_nat] :
( ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ( abs_Integ @ X3 ) ) ) ) ).
% cr_int_def
thf(fact_9539_in__range_Osimps,axiom,
! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ As )
=> ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).
% in_range.simps
thf(fact_9540_in__range_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( in_range @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ As4 )
=> ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_9541_in__range_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( in_range @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ~ ! [X5: nat] :
( ( member_nat @ X5 @ As4 )
=> ( ord_less_nat @ X5 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_9542_in__range_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As4 ) )
=> ( Y
= ( ~ ! [X3: nat] :
( ( member_nat @ X3 @ As4 )
=> ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_9543_surj__prod__encode,axiom,
( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
= top_top_set_nat ) ).
% surj_prod_encode
thf(fact_9544_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_9545_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_9546_bij__prod__encode,axiom,
bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% bij_prod_encode
thf(fact_9547_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_9548_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M2: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N2 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_9549_surj__list__encode,axiom,
( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
= top_top_set_nat ) ).
% surj_list_encode
thf(fact_9550_bij__list__encode,axiom,
bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% bij_list_encode
thf(fact_9551_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_9552_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_9553_String_Oless__literal__def,axiom,
( ord_less_literal
= ( map_fu2661873803724901240eral_o @ explode @ ( map_fu6698795530259886495ar_o_o @ explode @ id_o )
@ ( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) ) ) ) ).
% String.less_literal_def
thf(fact_9554_less__literal_Orep__eq,axiom,
( ord_less_literal
= ( ^ [X3: literal,Xa4: literal] :
( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) )
@ ( explode @ X3 )
@ ( explode @ Xa4 ) ) ) ) ).
% less_literal.rep_eq
thf(fact_9555_String_Oless__eq__literal__def,axiom,
( ord_less_eq_literal
= ( map_fu2661873803724901240eral_o @ explode @ ( map_fu6698795530259886495ar_o_o @ explode @ id_o )
@ ( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) ) ) ) ).
% String.less_eq_literal_def
thf(fact_9556_less__literal_Otransfer,axiom,
( bNF_re7831395832754058691eral_o @ pcr_literal
@ ( bNF_re8730477052009823892al_o_o @ pcr_literal
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) )
@ ord_less_literal ) ).
% less_literal.transfer
thf(fact_9557_literal_Orep__transfer,axiom,
( bNF_re4991257583289732696t_char @ pcr_literal
@ ( list_all2_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 ) )
@ ^ [X3: list_char] : X3
@ explode ) ).
% literal.rep_transfer
thf(fact_9558_less__eq__literal_Otransfer,axiom,
( bNF_re7831395832754058691eral_o @ pcr_literal
@ ( bNF_re8730477052009823892al_o_o @ pcr_literal
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) )
@ ord_less_eq_literal ) ).
% less_eq_literal.transfer
thf(fact_9559_less__eq__literal_Orep__eq,axiom,
( ord_less_eq_literal
= ( ^ [X3: literal,Xa4: literal] :
( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) )
@ ( explode @ X3 )
@ ( explode @ Xa4 ) ) ) ) ).
% less_eq_literal.rep_eq
thf(fact_9560_String_Ocr__literal__def,axiom,
( cr_literal
= ( ^ [X3: list_char,Y3: literal] :
( X3
= ( explode @ Y3 ) ) ) ) ).
% String.cr_literal_def
thf(fact_9561_MOST__SucD,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ cofinite_nat )
=> ( eventually_nat @ P @ cofinite_nat ) ) ).
% MOST_SucD
thf(fact_9562_MOST__SucI,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ cofinite_nat )
=> ( eventually_nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ cofinite_nat ) ) ).
% MOST_SucI
thf(fact_9563_MOST__Suc__iff,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
@ cofinite_nat )
= ( eventually_nat @ P @ cofinite_nat ) ) ).
% MOST_Suc_iff
thf(fact_9564_MOST__nat,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ cofinite_nat )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat
thf(fact_9565_MOST__ge__nat,axiom,
! [M: nat] : ( eventually_nat @ ( ord_less_eq_nat @ M ) @ cofinite_nat ) ).
% MOST_ge_nat
thf(fact_9566_MOST__nat__le,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ cofinite_nat )
= ( ? [M2: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat_le
thf(fact_9567_less__literal_Orsp,axiom,
( bNF_re7775871201573978401char_o
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_re6668474094832959905ar_o_o
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) )
@ ( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) ) ) ).
% less_literal.rsp
thf(fact_9568_plus__literal_Orsp,axiom,
( bNF_re7399511318543894245t_char
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_re1407854171455731685t_char
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
@ append_char
@ append_char ) ).
% plus_literal.rsp
thf(fact_9569_asciis__of__literal_Orsp,axiom,
( bNF_re901197953848981245nteger
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y4: list_Code_integer,Z4: list_Code_integer] : ( Y4 = Z4 )
@ ( map_ch5843194146513361140nteger @ comm_s4049598766089966537nteger )
@ ( map_ch5843194146513361140nteger @ comm_s4049598766089966537nteger ) ) ).
% asciis_of_literal.rsp
thf(fact_9570_size__literal_Orsp,axiom,
( bNF_re76688982720252355at_nat
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ size_size_list_char
@ size_size_list_char ) ).
% size_literal.rsp
thf(fact_9571_Literal_Orsp,axiom,
( bNF_re5598663031114558885t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re7347849368244578525t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re7972843584066424825t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re796896441190010845t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re4583855572895197261t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re441219624009962717t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re6157522605288126113t_char
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re1407854171455731685t_char
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ).
% Literal.rsp
thf(fact_9572_literal_Oexplode,axiom,
! [X: literal] :
( member_list_char @ ( explode @ X )
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.explode
thf(fact_9573_literal_Oexplode__cases,axiom,
! [Y: list_char] :
( ( member_list_char @ Y
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ~ ! [X2: literal] :
( Y
!= ( explode @ X2 ) ) ) ).
% literal.explode_cases
thf(fact_9574_literal_Oexplode__induct,axiom,
! [Y: list_char,P: list_char > $o] :
( ( member_list_char @ Y
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ! [X2: literal] : ( P @ ( explode @ X2 ) )
=> ( P @ Y ) ) ) ).
% literal.explode_induct
thf(fact_9575_zero__literal_Orsp,axiom,
( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ nil_char
@ nil_char ) ).
% zero_literal.rsp
thf(fact_9576_literal_Odomain__eq,axiom,
( ( domain7888411570377768909iteral @ pcr_literal )
= ( ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.domain_eq
thf(fact_9577_literal_Odomain,axiom,
( ( domain7888411570377768909iteral @ pcr_literal )
= ( ^ [X3: list_char] :
? [Y3: list_char] :
( ( list_all2_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 )
@ X3
@ Y3 )
& ! [Z3: char] :
( ( member_char @ Z3 @ ( set_char2 @ Y3 ) )
=> ~ ( digit7 @ Z3 ) ) ) ) ) ).
% literal.domain
thf(fact_9578_less__eq__literal_Orsp,axiom,
( bNF_re7775871201573978401char_o
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( bNF_re6668474094832959905ar_o_o
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 ) )
@ ( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) )
@ ( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) ) ) ) ).
% less_eq_literal.rsp
thf(fact_9579_less__eq__literal_Oabs__eq,axiom,
! [Xa: list_char,X: list_char] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( ord_less_eq_literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp_eq_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) )
@ Xa
@ X ) ) ) ) ).
% less_eq_literal.abs_eq
thf(fact_9580_less__literal_Oabs__eq,axiom,
! [Xa: list_char,X: list_char] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( ord_less_literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp_char
@ ^ [C4: char,D4: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C4 ) @ ( comm_s629917340098488124ar_nat @ D4 ) )
@ Xa
@ X ) ) ) ) ).
% less_literal.abs_eq
thf(fact_9581_literal_OAbs__literal__cases,axiom,
! [X: literal] :
~ ! [Y2: list_char] :
( ( X
= ( abs_literal @ Y2 ) )
=> ~ ( member_list_char @ Y2
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ) ).
% literal.Abs_literal_cases
thf(fact_9582_literal_OAbs__literal__induct,axiom,
! [P: literal > $o,X: literal] :
( ! [Y2: list_char] :
( ( member_list_char @ Y2
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( P @ ( abs_literal @ Y2 ) ) )
=> ( P @ X ) ) ).
% literal.Abs_literal_induct
thf(fact_9583_literal_OAbs__literal__inject,axiom,
! [X: list_char,Y: list_char] :
( ( member_list_char @ X
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( member_list_char @ Y
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( ( abs_literal @ X )
= ( abs_literal @ Y ) )
= ( X = Y ) ) ) ) ).
% literal.Abs_literal_inject
thf(fact_9584_literal_Otype__definition__literal,axiom,
( type_d4752411451802217481t_char @ explode @ abs_literal
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) ) ).
% literal.type_definition_literal
thf(fact_9585_literal_OAbs__literal__inverse,axiom,
! [Y: list_char] :
( ( member_list_char @ Y
@ ( collect_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) ) )
=> ( ( explode @ ( abs_literal @ Y ) )
= Y ) ) ).
% literal.Abs_literal_inverse
thf(fact_9586_plus__literal_Oabs__eq,axiom,
! [Xa: list_char,X: list_char] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( plus_plus_literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( abs_literal @ ( append_char @ Xa @ X ) ) ) ) ) ).
% plus_literal.abs_eq
thf(fact_9587_String_OQuotient__literal,axiom,
( quotie6109894551476798619iteral
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ abs_literal
@ explode
@ cr_literal ) ).
% String.Quotient_literal
thf(fact_9588_asciis__of__literal_Oabs__eq,axiom,
! [X: list_char] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( asciis_of_literal @ ( abs_literal @ X ) )
= ( map_ch5843194146513361140nteger @ comm_s4049598766089966537nteger @ X ) ) ) ).
% asciis_of_literal.abs_eq
thf(fact_9589_size__literal_Oabs__eq,axiom,
! [X: list_char] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( size_size_literal @ ( abs_literal @ X ) )
= ( size_size_list_char @ X ) ) ) ).
% size_literal.abs_eq
thf(fact_9590_literal_Odomain__par__left__total,axiom,
! [P7: list_char > $o] :
( ( left_t8440299596084406506t_char
@ ( list_all2_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 ) ) )
=> ( ( bNF_re6668474094832959905ar_o_o
@ ( list_all2_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ P7
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
=> ( ( domain7888411570377768909iteral @ pcr_literal )
= P7 ) ) ) ).
% literal.domain_par_left_total
thf(fact_9591_literal_Odomain__par,axiom,
! [DR: char > $o,P22: list_char > $o] :
( ( ( domainp_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 ) )
= DR )
=> ( ( bNF_re6668474094832959905ar_o_o
@ ( list_all2_char_char
@ ^ [Y4: char,Z4: char] : ( Y4 = Z4 ) )
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ P22
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
=> ( ( domain7888411570377768909iteral @ pcr_literal )
= ( inf_inf_list_char_o @ ( list_all_char @ DR ) @ P22 ) ) ) ) ).
% literal.domain_par
thf(fact_9592_implode_Orsp,axiom,
( bNF_re1407854171455731685t_char
@ ^ [Y4: list_char,Z4: list_char] : ( Y4 = Z4 )
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( map_char_char @ ascii_of )
@ ( map_char_char @ ascii_of ) ) ).
% implode.rsp
thf(fact_9593_literal__of__asciis_Orsp,axiom,
( bNF_re5828396963709814525t_char
@ ^ [Y4: list_Code_integer,Z4: list_Code_integer] : ( Y4 = Z4 )
@ ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) ) )
@ ( map_Co843364623960961780r_char @ ( comp_c4562247934908617188nteger @ ascii_of @ unique1708501630192214702nteger ) )
@ ( map_Co843364623960961780r_char @ ( comp_c4562247934908617188nteger @ ascii_of @ unique1708501630192214702nteger ) ) ) ).
% literal_of_asciis.rsp
thf(fact_9594_Literal_Oabs__eq,axiom,
! [X: list_char,Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xa: $o] :
( ( bNF_eq_onp_list_char
@ ^ [Cs: list_char] :
! [X3: char] :
( ( member_char @ X3 @ ( set_char2 @ Cs ) )
=> ~ ( digit7 @ X3 ) )
@ X
@ X )
=> ( ( literal2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ ( abs_literal @ X ) )
= ( abs_literal @ ( cons_char @ ( char2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ $false ) @ X ) ) ) ) ).
% Literal.abs_eq
thf(fact_9595_String_OLiteral__def,axiom,
( literal2
= ( map_fu1104370778922912482iteral @ id_o @ ( map_fu7309004089335677210iteral @ id_o @ ( map_fu1929762555066093878iteral @ id_o @ ( map_fu5333924603067766298iteral @ id_o @ ( map_fu1888220742600445322iteral @ id_o @ ( map_fu5700776415251466522iteral @ id_o @ ( map_fu4627020735845646302iteral @ id_o @ ( map_fu7398138089906338514iteral @ explode @ abs_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ) ).
% String.Literal_def
thf(fact_9596_Literal_Otransfer,axiom,
( bNF_re8691941649098027561iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re5719858693005739489iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re136309367175754621iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re2365650225897248225iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re5778109699181701841iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re1595102025518187489iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re6148748351543924773iteral
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ( bNF_re5453467372598631581iteral @ pcr_literal @ pcr_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ literal2 ) ).
% Literal.transfer
thf(fact_9597_cr__integer__def,axiom,
( code_cr_integer
= ( ^ [X3: int,Y3: code_integer] :
( X3
= ( code_int_of_integer @ Y3 ) ) ) ) ).
% cr_integer_def
thf(fact_9598_int__encode__def,axiom,
( nat_int_encode
= ( ^ [I3: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I3 ) @ ( sum_Inl_nat_nat @ ( nat2 @ I3 ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I3 ) @ one_one_int ) ) ) ) ) ) ) ).
% int_encode_def
thf(fact_9599_surj__sum__encode,axiom,
( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
= top_top_set_nat ) ).
% surj_sum_encode
thf(fact_9600_bij__sum__encode,axiom,
bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).
% bij_sum_encode
thf(fact_9601_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ^ [B3: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).
% sum_encode_def
thf(fact_9602_surj__sum__decode,axiom,
( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% surj_sum_decode
thf(fact_9603_bij__sum__decode,axiom,
bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).
% bij_sum_decode
thf(fact_9604_nth__item_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp_nat @ nth_item_rel @ A0 )
=> ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
=> ( P @ zero_zero_nat ) )
=> ( ! [N3: nat] :
( ( accp_nat @ nth_item_rel @ ( suc @ N3 ) )
=> ( ! [A9: nat,Aa: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl_nat_nat @ A9 ) )
=> ( ( ( nat_sum_decode @ A9 )
= ( sum_Inl_nat_nat @ Aa ) )
=> ( P @ Aa ) ) )
=> ( ! [A9: nat,B10: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl_nat_nat @ A9 ) )
=> ( ( ( nat_sum_decode @ A9 )
= ( sum_Inr_nat_nat @ B10 ) )
=> ( P @ B10 ) ) )
=> ( ! [B10: nat,Ba: nat,X5: nat,Y5: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr_nat_nat @ B10 ) )
=> ( ( ( nat_sum_decode @ B10 )
= ( sum_Inr_nat_nat @ Ba ) )
=> ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
= ( nat_prod_decode @ Ba ) )
=> ( P @ X5 ) ) ) )
=> ( ! [B10: nat,Ba: nat,X5: nat,Y5: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr_nat_nat @ B10 ) )
=> ( ( ( nat_sum_decode @ B10 )
= ( sum_Inr_nat_nat @ Ba ) )
=> ( ( ( product_Pair_nat_nat @ X5 @ Y5 )
= ( nat_prod_decode @ Ba ) )
=> ( P @ Y5 ) ) ) )
=> ( P @ ( suc @ N3 ) ) ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_9605_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N2: nat] :
( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
@ ^ [B3: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B3 ) ) @ one_one_int )
@ ( nat_sum_decode @ N2 ) ) ) ) ).
% int_decode_def
thf(fact_9606_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_9607_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% cone_def
thf(fact_9608_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_9609_uminus__unit__def,axiom,
( uminus2952777764628376836t_unit
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_9610_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_9611_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_9612_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_9613_natural__zero__minus__one,axiom,
( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
= zero_z2226904508553997617atural ) ).
% natural_zero_minus_one
thf(fact_9614_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa @ X ) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa @ X ) )
=> ( Y
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_9615_log_Osimps,axiom,
( log
= ( ^ [B3: code_natural,I3: code_natural] :
( if_Code_natural
@ ( ( ord_le1926595141338095240atural @ B3 @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ I3 @ B3 ) )
@ one_one_Code_natural
@ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B3 @ ( divide5121882707175180666atural @ I3 @ B3 ) ) ) ) ) ) ).
% log.simps
thf(fact_9616_Random_Orange__def,axiom,
( range
= ( ^ [K4: code_natural] :
( produc5538323210962509403atural
@ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
@ ^ [L2: code_natural] :
( produc5538323210962509403atural @ next
@ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V2 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ one_one_Code_natural )
@ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V2 @ K4 ) ) ) ) ) ).
% Random.range_def
thf(fact_9617_next_Osimps,axiom,
! [V: code_natural,W: code_natural] :
( ( next @ ( produc3574140220909816553atural @ V @ W ) )
= ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_9618_log_Ocases,axiom,
! [X: produc7822875418678951345atural] :
~ ! [B6: code_natural,I2: code_natural] :
( X
!= ( produc3574140220909816553atural @ B6 @ I2 ) ) ).
% log.cases
thf(fact_9619_full__exhaustive__natural_H_Ocases,axiom,
! [X: produc989692990947075319atural] :
~ ! [F3: produc4972180933644002618e_term > option6357759511663192854e_term,D: code_natural,I2: code_natural] :
( X
!= ( produc3831813291587773865atural @ F3 @ ( produc3574140220909816553atural @ D @ I2 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_9620_exhaustive__natural_H_Ocases,axiom,
! [X: produc8731074985263844745atural] :
~ ! [F3: code_natural > option6357759511663192854e_term,D: code_natural,I2: code_natural] :
( X
!= ( produc2252593628808123835atural @ F3 @ ( produc3574140220909816553atural @ D @ I2 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_9621_split__seed__def,axiom,
( split_seed
= ( ^ [S2: produc7822875418678951345atural] :
( produc8282080750456430313atural
@ ^ [V2: code_natural,W3: code_natural] :
( produc8282080750456430313atural
@ ^ [V3: code_natural,W4: code_natural] : ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W4 ) @ ( produc3574140220909816553atural @ V3 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W3 ) ) )
@ ( produc6591912806276919810atural @ ( next @ S2 ) ) )
@ S2 ) ) ) ).
% split_seed_def
thf(fact_9622_inc__shift__def,axiom,
( inc_shift
= ( ^ [V2: code_natural,K4: code_natural] : ( if_Code_natural @ ( V2 = K4 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K4 @ one_one_Code_natural ) ) ) ) ).
% inc_shift_def
thf(fact_9623_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa @ X ) )
=> ( Y = one_one_Code_natural ) )
& ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
| ( ord_le5570908160329646204atural @ Xa @ X ) )
=> ( Y
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_9624_one__natural__def,axiom,
( one_one_Code_natural
= ( code_natural_of_nat @ one_one_nat ) ) ).
% one_natural_def
thf(fact_9625_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y2: nat] :
( ( X
= ( code_natural_of_nat @ Y2 ) )
=> ~ ( member_nat @ Y2 @ top_top_set_nat ) ) ).
% natural_of_nat_cases
thf(fact_9626_natural__of__nat__induct,axiom,
! [P: code_natural > $o,X: code_natural] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ top_top_set_nat )
=> ( P @ ( code_natural_of_nat @ Y2 ) ) )
=> ( P @ X ) ) ).
% natural_of_nat_induct
thf(fact_9627_natural__of__nat__inject,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( ( ( code_natural_of_nat @ X )
= ( code_natural_of_nat @ Y ) )
= ( X = Y ) ) ) ) ).
% natural_of_nat_inject
thf(fact_9628_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ one_one_Code_natural )
= one_one_nat ) ).
% one_natural.rep_eq
thf(fact_9629_type__definition__natural,axiom,
type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).
% type_definition_natural
thf(fact_9630_natural__of__nat__inverse,axiom,
! [Y: nat] :
( ( member_nat @ Y @ top_top_set_nat )
=> ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
= Y ) ) ).
% natural_of_nat_inverse
thf(fact_9631_nat__of__natural__induct,axiom,
! [Y: nat,P: nat > $o] :
( ( member_nat @ Y @ top_top_set_nat )
=> ( ! [X2: code_natural] : ( P @ ( code_nat_of_natural @ X2 ) )
=> ( P @ Y ) ) ) ).
% nat_of_natural_induct
thf(fact_9632_nat__of__natural__cases,axiom,
! [Y: nat] :
( ( member_nat @ Y @ top_top_set_nat )
=> ~ ! [X2: code_natural] :
( Y
!= ( code_nat_of_natural @ X2 ) ) ) ).
% nat_of_natural_cases
thf(fact_9633_nat__of__natural,axiom,
! [X: code_natural] : ( member_nat @ ( code_nat_of_natural @ X ) @ top_top_set_nat ) ).
% nat_of_natural
thf(fact_9634_cr__natural__def,axiom,
( code_cr_natural
= ( ^ [X3: nat,Y3: code_natural] :
( X3
= ( code_nat_of_natural @ Y3 ) ) ) ) ).
% cr_natural_def
thf(fact_9635_INFM__nat,axiom,
! [P: nat > $o] :
( ( frequently_nat @ P @ cofinite_nat )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat
thf(fact_9636_INFM__nat__le,axiom,
! [P: nat > $o] :
( ( frequently_nat @ P @ cofinite_nat )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat_le
thf(fact_9637_INFM__nat__inductI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [I2: nat] :
( ( P @ I2 )
=> ? [J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
& ( P @ J4 )
& ( Q2 @ J4 ) ) )
=> ( frequently_nat @ Q2 @ cofinite_nat ) ) ) ).
% INFM_nat_inductI
thf(fact_9638_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ one_one_nat )
= one_one_Code_integer ) ).
% integer_of_nat_1
thf(fact_9639_Lcm__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K5 ) )
= ( gcd_Lcm_int @ K5 ) ) ).
% Lcm_abs_eq
thf(fact_9640_Lcm__int__eq,axiom,
! [N4: set_nat] :
( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N4 ) )
= ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N4 ) ) ) ).
% Lcm_int_eq
thf(fact_9641_Lcm__nat__abs__eq,axiom,
! [K5: set_int] :
( ( gcd_Lcm_nat
@ ( image_int_nat
@ ^ [K4: int] : ( nat2 @ ( abs_abs_int @ K4 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Lcm_int @ K5 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_9642_Lcm__nat__empty,axiom,
( ( gcd_Lcm_nat @ bot_bot_set_nat )
= one_one_nat ) ).
% Lcm_nat_empty
thf(fact_9643_Gcd__nat__def,axiom,
( gcd_Gcd_nat
= ( ^ [M7: set_nat] :
( gcd_Lcm_nat
@ ( collect_nat
@ ^ [D4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ M7 )
=> ( dvd_dvd_nat @ D4 @ X3 ) ) ) ) ) ) ).
% Gcd_nat_def
thf(fact_9644_Lcm__int__def,axiom,
( gcd_Lcm_int
= ( ^ [K6: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K6 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_9645_Lcm__eq__Max__nat,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M4 )
=> ( ! [M3: nat,N3: nat] :
( ( member_nat @ M3 @ M4 )
=> ( ( member_nat @ N3 @ M4 )
=> ( member_nat @ ( gcd_lcm_nat @ M3 @ N3 ) @ M4 ) ) )
=> ( ( gcd_Lcm_nat @ M4 )
= ( lattic8265883725875713057ax_nat @ M4 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_9646_Lcm__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_9647_Lcm__in__lcm__closed__set__nat,axiom,
! [M4: set_nat] :
( ( finite_finite_nat @ M4 )
=> ( ( M4 != bot_bot_set_nat )
=> ( ! [M3: nat,N3: nat] :
( ( member_nat @ M3 @ M4 )
=> ( ( member_nat @ N3 @ M4 )
=> ( member_nat @ ( gcd_lcm_nat @ M3 @ N3 ) @ M4 ) ) )
=> ( member_nat @ ( gcd_Lcm_nat @ M4 ) @ M4 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_9648_Lcm__nat__def,axiom,
( gcd_Lcm_nat
= ( ^ [M7: set_nat] : ( if_nat @ ( finite_finite_nat @ M7 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M7 ) @ zero_zero_nat ) ) ) ).
% Lcm_nat_def
thf(fact_9649_lcm__1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) )
& ( ( N = one_one_int )
| ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% lcm_1_iff_int
thf(fact_9650_lcm__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_neg1_int
thf(fact_9651_lcm__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_lcm_int @ X @ Y ) ) ).
% lcm_neg2_int
thf(fact_9652_lcm__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_lcm_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_lcm_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_lcm_int @ X @ Y ) ) ) ) ) ) ).
% lcm_cases_int
thf(fact_9653_Lcm__int__set__eq__fold,axiom,
! [Xs: list_int] :
( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
= ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).
% Lcm_int_set_eq_fold
thf(fact_9654_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
= one_one_nat ) ).
% unit_factor_simps(2)
thf(fact_9655_unit__factor__nat__def,axiom,
( unit_f2748546683901255202or_nat
= ( ^ [N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% unit_factor_nat_def
thf(fact_9656_unit__pred__cases,axiom,
! [P: pred_Product_unit > $o,Q2: pred_Product_unit] :
( ( P @ bot_bo2717538794563056311t_unit )
=> ( ( P @ ( single_Product_unit @ product_Unity ) )
=> ( P @ Q2 ) ) ) ).
% unit_pred_cases
thf(fact_9657_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P2: produc7822875418678951345atural > produc4675096598859438275atural,S2: produc7822875418678951345atural] :
( produc6665183775751917029atural
@ ^ [P8: pred_Product_unit,S7: produc7822875418678951345atural] : ( if_Pro3444522238938527101atural @ ( eval_Product_unit @ P8 @ product_Unity ) @ ( produc5069803637994805237atural @ bot_bo2717538794563056311t_unit @ S7 ) @ ( produc5069803637994805237atural @ ( single_Product_unit @ product_Unity ) @ S7 ) )
@ ( P2 @ S2 ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_9658_not__pred__eq,axiom,
( not_pred
= ( ^ [P2: pred_Product_unit] : ( if_pred_Product_unit @ ( eval_Product_unit @ P2 @ product_Unity ) @ bot_bo2717538794563056311t_unit @ ( single_Product_unit @ product_Unity ) ) ) ) ).
% not_pred_eq
thf(fact_9659_if__pred__eq,axiom,
( if_pred
= ( ^ [B3: $o] : ( if_pred_Product_unit @ B3 @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) ).
% if_pred_eq
thf(fact_9660_not__predE,axiom,
! [P: $o,X: product_unit] :
( ( eval_Product_unit
@ ( not_pred
@ ( pred_Product_unit2
@ ^ [U2: product_unit] : P ) )
@ X )
=> ~ P ) ).
% not_predE
thf(fact_9661_not__predI,axiom,
! [P: $o] :
( ~ P
=> ( eval_Product_unit
@ ( not_pred
@ ( pred_Product_unit2
@ ^ [U2: product_unit] : P ) )
@ product_Unity ) ) ).
% not_predI
thf(fact_9662_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_9663_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one )
= one_one_nat ) ).
% nat_of_num_code(1)
thf(fact_9664_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_9665_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
= N ) ).
% Suc_natural_minus_one
thf(fact_9666_gcd__nat_Osemilattice__order__axioms,axiom,
( semila1248733672344298208er_nat @ gcd_gcd_nat @ dvd_dvd_nat
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_order_axioms
thf(fact_9667_random__aux__set_Ocases,axiom,
! [X: produc7822875418678951345atural] :
( ! [J2: code_natural] :
( X
!= ( produc3574140220909816553atural @ zero_z2226904508553997617atural @ J2 ) )
=> ~ ! [I2: code_natural,J2: code_natural] :
( X
!= ( produc3574140220909816553atural @ ( code_Suc @ I2 ) @ J2 ) ) ) ).
% random_aux_set.cases
thf(fact_9668_or__num_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_un6697907153464112080or_num @ X @ Xa )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ X @ Xa ) )
=> ( ( ( X = one )
=> ( ( Xa = one )
=> ( ( Y = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit0 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one )
=> ( ( Y
= ( bit1 @ M3 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
thf(fact_9669_less__nat__rel,axiom,
( ord_less_nat
= ( transi2163837189807498211lp_nat
@ ^ [M2: nat,N2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% less_nat_rel
thf(fact_9670_sub_Otransfer,axiom,
( bNF_re7876454716742015248nteger
@ ^ [Y4: num,Z4: num] : ( Y4 = Z4 )
@ ( bNF_re6501075790457514782nteger
@ ^ [Y4: num,Z4: num] : ( Y4 = Z4 )
@ code_pcr_integer )
@ ^ [M2: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_9671_uminus__integer_Otransfer,axiom,
bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ uminus_uminus_int @ uminus1351360451143612070nteger ).
% uminus_integer.transfer
thf(fact_9672_integer_Oid__abs__transfer,axiom,
( bNF_re982302072995117890nteger
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ code_pcr_integer
@ ^ [X3: int] : X3
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_9673_integer_Orep__transfer,axiom,
( bNF_re3804157879324367682nt_int @ code_pcr_integer
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [X3: int] : X3
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_9674_dup_Otransfer,axiom,
( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
@ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 )
@ code_dup ) ).
% dup.transfer
thf(fact_9675_one__integer_Otransfer,axiom,
code_pcr_integer @ one_one_int @ one_one_Code_integer ).
% one_integer.transfer
thf(fact_9676_gcd__nat_Oordering__axioms,axiom,
( ordering_nat @ dvd_dvd_nat
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.ordering_axioms
thf(fact_9677_Abs__int__cases,axiom,
! [X: int] :
~ ! [Y2: set_Pr1261947904930325089at_nat] :
( ( X
= ( abs_int @ Y2 ) )
=> ~ ( member2643936169264416010at_nat @ Y2
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) ) ) ).
% Abs_int_cases
thf(fact_9678_Abs__int__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y2
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ( P @ ( abs_int @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_int_induct
thf(fact_9679_Abs__int__inject,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( member2643936169264416010at_nat @ Y
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( ( abs_int @ X )
= ( abs_int @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_int_inject
thf(fact_9680_Abs__int__inverse,axiom,
! [Y: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ( ( rep_int @ ( abs_int @ Y ) )
= Y ) ) ).
% Abs_int_inverse
thf(fact_9681_type__definition__int,axiom,
( type_d8752208705193531015at_nat @ rep_int @ abs_int
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) ) ).
% type_definition_int
thf(fact_9682_Rep__int__induct,axiom,
! [Y: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( member2643936169264416010at_nat @ Y
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ( ! [X2: int] : ( P @ ( rep_int @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_int_induct
thf(fact_9683_Rep__int__cases,axiom,
! [Y: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) )
=> ~ ! [X2: int] :
( Y
!= ( rep_int @ X2 ) ) ) ).
% Rep_int_cases
thf(fact_9684_Rep__int,axiom,
! [X: int] :
( member2643936169264416010at_nat @ ( rep_int @ X )
@ ( collec5514110066124741708at_nat
@ ^ [C4: set_Pr1261947904930325089at_nat] :
? [X3: product_prod_nat_nat] :
( ( intrel @ X3 @ X3 )
& ( C4
= ( collec3392354462482085612at_nat @ ( intrel @ X3 ) ) ) ) ) ) ).
% Rep_int
thf(fact_9685_Int_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% Int.sub_code(9)
thf(fact_9686_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus_int @ ( dup @ ( sub @ M @ N ) ) @ one_one_int ) ) ).
% Int.sub_code(8)
thf(fact_9687_gcd__nat_Opreordering__axioms,axiom,
( preordering_nat @ dvd_dvd_nat
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.preordering_axioms
% Helper facts (37)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
! [X: assn,Y: assn] :
( ( if_assn @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
! [X: assn,Y: assn] :
( ( if_assn @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y: option_num] :
( ( if_option_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,
! [X: pred_Product_unit,Y: pred_Product_unit] :
( ( if_pred_Product_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,
! [X: pred_Product_unit,Y: pred_Product_unit] :
( ( if_pred_Product_unit @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( if_Sum_sum_nat_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
( ( if_Sum_sum_nat_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
( ( if_mul8430962117462786573at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
( ( if_mul8430962117462786573at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
! [X: produc4675096598859438275atural,Y: produc4675096598859438275atural] :
( ( if_Pro3444522238938527101atural @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
! [X: produc4675096598859438275atural,Y: produc4675096598859438275atural] :
( ( if_Pro3444522238938527101atural @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( x
= ( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ x @ H2 )
& ( rep_assn @ y @ H2 ) ) ) )
& ( x != y ) )
= ( ( x
= ( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ x @ H2 )
& ( rep_assn @ y @ H2 ) ) ) )
& ( y
!= ( abs_assn
@ ^ [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ y @ H2 )
& ( rep_assn @ x @ H2 ) ) ) ) ) ) ).
%------------------------------------------------------------------------------