TPTP Problem File: SLH0847^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0009_Dtree/prob_00793_032290__15085810_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1476 ( 546 unt; 203 typ; 0 def)
% Number of atoms : 3697 (1163 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10976 ( 416 ~; 74 |; 231 &;8570 @)
% ( 0 <=>;1685 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 1262 (1262 >; 0 *; 0 +; 0 <<)
% Number of symbols : 183 ( 180 usr; 15 con; 0-5 aty)
% Number of variables : 4072 ( 449 ^;3478 !; 145 ?;4072 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:06:19.004
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr2458342521480944603at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_Pr5582243495563764594_nat_o: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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set_Pr1261947904930325089at_nat: $tType ).
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set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
fset_P9214369701362650254od_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
set_Product_prod_a_b: $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,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
product_prod_a_b: $tType ).
thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
fset_nat: $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__FSet__Ofset_Itf__a_J,type,
fset_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $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__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (180)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Nat__Onat,type,
bNF_Ca1968104039914474786nt_nat: set_Pr958786334691620121nt_int > ( int > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Real__Oreal,type,
bNF_Ca2154799978004375294t_real: set_Pr958786334691620121nt_int > ( int > real ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Ca968750328013420230at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Real__Oreal,type,
bNF_Ca9191250440166129314t_real: set_Pr1261947904930325089at_nat > ( nat > real ) > $o ).
thf(sy_c_FSet_Ofcard_001t__Nat__Onat,type,
fcard_nat: fset_nat > nat ).
thf(sy_c_FSet_Ofcard_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet_Ofimage_001t__Nat__Onat_001t__Nat__Onat,type,
fimage_nat_nat: ( nat > nat ) > fset_nat > fset_nat ).
thf(sy_c_FSet_Ofimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_FSet_Ofimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_FSet_Ofimage_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
fimage8245688998986472637nt_int: ( product_prod_int_int > product_prod_int_int ) > fset_P5367158941141162143nt_int > fset_P5367158941141162143nt_int ).
thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
fimage2200814269739838981at_nat: ( product_prod_int_int > product_prod_nat_nat ) > fset_P5367158941141162143nt_int > fset_P5670320511379867111at_nat ).
thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
fimage1850745034776345876at_nat: ( product_prod_nat_nat > nat ) > fset_P5670320511379867111at_nat > fset_nat ).
thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
fimage6539872720770606011_b_nat: ( product_prod_a_b > nat ) > fset_P9214369701362650254od_a_b > fset_nat ).
thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet_Ofimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a,type,
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thf(sy_c_FSet_Ofinsert_001t__Nat__Onat,type,
finsert_nat: nat > fset_nat > fset_nat ).
thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
fmember_nat: nat > fset_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
fmembe7738088520663304047od_a_b: product_prod_a_b > fset_P9214369701362650254od_a_b > $o ).
thf(sy_c_FSet_Ofmember_001tf__a,type,
fmember_a: a > fset_a > $o ).
thf(sy_c_FSet_Ofset_Ofset_001t__Nat__Onat,type,
fset_nat2: fset_nat > set_nat ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
fset_P322984321831570088nt_int: fset_P5367158941141162143nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
fset_P3501481629439712240at_nat: fset_P5670320511379867111at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_FSet_Ofset_Ofset_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
fset_P2369346149119917079od_a_b: fset_P9214369701362650254od_a_b > set_Product_prod_a_b ).
thf(sy_c_FSet_Ofset_Ofset_001tf__a,type,
fset_a2: fset_a > set_a ).
thf(sy_c_FSet_Osize__fset_001t__Nat__Onat,type,
size_fset_nat: ( nat > nat ) > fset_nat > nat ).
thf(sy_c_FSet_Osize__fset_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
size_f6359142247592877058od_a_b: ( product_prod_a_b > nat ) > fset_P9214369701362650254od_a_b > nat ).
thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
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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_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Int__Oint,type,
groups7002347999527655834_b_int: ( product_prod_a_b > int ) > set_Product_prod_a_b > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
groups7004838470036706110_b_nat: ( product_prod_a_b > nat ) > set_Product_prod_a_b > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Real__Oreal,type,
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thf(sy_c_If_001t__FSet__Ofset_It__Nat__Onat_J,type,
if_fset_nat: $o > fset_nat > fset_nat > fset_nat ).
thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
if_Product_prod_a_b: $o > product_prod_a_b > product_prod_a_b > product_prod_a_b ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: product_prod_nat_nat > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > product_prod_nat_nat ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
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__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__FSet__Ofset_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_Itf__a_Mtf__b_J_M_Eo_J,type,
ord_le8027066870050541877_a_b_o: ( product_prod_a_b > $o ) > ( product_prod_a_b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__b_M_Eo_J_J,type,
ord_less_eq_a_b_o: ( a > b > $o ) > ( a > b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Nat__Onat_J,type,
ord_less_eq_fset_nat: fset_nat > fset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
ord_le7868007465744610350od_a_b: fset_P9214369701362650254od_a_b > fset_P9214369701362650254od_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
ord_le817736998455962536od_a_b: set_Product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__b,type,
product_Pair_a_b: a > b > product_prod_a_b ).
thf(sy_c_Product__Type_Ointernal__case__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc8005341501107743676_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Ointernal__case__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
produc4780622933104268256_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Ointernal__case__prod_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,
produc1854806715440696265at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Ointernal__case__prod_001tf__a_001tf__b_001t__Nat__Onat,type,
produc2938035986357027795_b_nat: ( a > b > nat ) > product_prod_a_b > nat ).
thf(sy_c_Product__Type_Ointernal__case__prod_001tf__a_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
produc3797153600616317541od_a_b: ( a > b > product_prod_a_b ) > product_prod_a_b > product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
produc8213879946458358998nt_nat: ( int > int > nat ) > product_prod_int_int > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
produc8019077043795531554od_a_b: ( int > int > product_prod_a_b ) > product_prod_int_int > product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
produc5321327879229220482od_a_b: ( int > int > set_Product_prod_a_b ) > product_prod_int_int > set_Product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_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,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
produc5281920923555503578od_a_b: ( nat > nat > product_prod_a_b ) > product_prod_nat_nat > product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
produc6189476227299908564et_nat: ( nat > nat > set_nat ) > product_prod_nat_nat > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
produc3410858885478534458od_a_b: ( nat > nat > set_Product_prod_a_b ) > product_prod_nat_nat > set_Product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001_Eo,type,
produc3537405659489547051_a_b_o: ( a > b > $o ) > product_prod_a_b > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001t__Nat__Onat,type,
produc5088076833887504125_b_nat: ( a > b > nat ) > product_prod_a_b > nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
produc8992199381948149691od_a_b: ( a > b > product_prod_a_b ) > product_prod_a_b > product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
produc2976218243292053171et_nat: ( a > b > set_nat ) > product_prod_a_b > set_nat ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
produc7078681169951578907od_a_b: ( a > b > set_Product_prod_a_b ) > product_prod_a_b > set_Product_prod_a_b ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__b_001tf__a,type,
produc6028431345588019473_a_b_a: ( a > b > a ) > product_prod_a_b > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__b,type,
product_fst_a_b: product_prod_a_b > a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
collec3336397801687681299od_a_b: ( product_prod_a_b > $o ) > set_Product_prod_a_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
image_2667626500211843362nt_int: ( nat > product_prod_int_int ) > set_nat > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
image_372941892535967121od_a_b: ( nat > product_prod_a_b ) > set_nat > set_Product_prod_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
image_2135063354759101220_int_o: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int > set_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
image_5042161079198086560nt_int: ( product_prod_int_int > int ) > set_Pr958786334691620121nt_int > set_int ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
image_5044651549707136836nt_nat: ( product_prod_int_int > nat ) > set_Pr958786334691620121nt_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_5831868185956570253at_nat: ( product_prod_int_int > product_prod_nat_nat ) > set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
image_2370028551316859444od_a_b: ( product_prod_int_int > product_prod_a_b ) > set_Pr958786334691620121nt_int > set_Product_prod_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
image_2972642778337070200_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat > set_Pr5582243495563764594_nat_o ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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,
image_8730593652825689185at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_Pr2458342521480944603at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_5168914502847457605at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
image_4894260360327267052od_a_b: ( product_prod_nat_nat > product_prod_a_b ) > set_Pr1261947904930325089at_nat > set_Product_prod_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
image_5426517477246462771_b_nat: ( product_prod_a_b > nat ) > set_Product_prod_a_b > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
image_3300603549555413765od_a_b: ( product_prod_a_b > product_prod_a_b ) > set_Product_prod_a_b > set_Product_prod_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a,type,
image_2802296252294471259_a_b_a: ( product_prod_a_b > a ) > set_Product_prod_a_b > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Int__Oint,type,
mlex_prod_int: ( int > nat ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Nat__Onat,type,
mlex_prod_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
member2006080708140975699_nat_o: ( product_prod_nat_nat > $o ) > set_Pr5582243495563764594_nat_o > $o ).
thf(sy_c_member_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,
member8885076297122219836at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr2458342521480944603at_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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__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_Itf__a_Mtf__b_J,type,
member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_e____,type,
e: b ).
thf(sy_v_f,type,
f: a > a ).
thf(sy_v_g,type,
g: a > nat ).
thf(sy_v_t____,type,
t: a ).
thf(sy_v_x____,type,
x: product_prod_a_b ).
thf(sy_v_xs,type,
xs: fset_P9214369701362650254od_a_b ).
thf(sy_v_xsa____,type,
xsa: fset_P9214369701362650254od_a_b ).
% Relevant facts (1263)
thf(fact_0_notin__fset,axiom,
! [X: nat,S: fset_nat] :
( ( ~ ( fmember_nat @ X @ S ) )
= ( ~ ( member_nat @ X @ ( fset_nat2 @ S ) ) ) ) ).
% notin_fset
thf(fact_1_notin__fset,axiom,
! [X: product_prod_a_b,S: fset_P9214369701362650254od_a_b] :
( ( ~ ( fmembe7738088520663304047od_a_b @ X @ S ) )
= ( ~ ( member1426531481828664017od_a_b @ X @ ( fset_P2369346149119917079od_a_b @ S ) ) ) ) ).
% notin_fset
thf(fact_2_fcard__image__le,axiom,
! [F: product_prod_a_b > product_prod_a_b,Xs: fset_P9214369701362650254od_a_b] : ( ord_less_eq_nat @ ( fcard_8555586198630727417od_a_b @ ( fimage3179447126440047741od_a_b @ F @ Xs ) ) @ ( fcard_8555586198630727417od_a_b @ Xs ) ) ).
% fcard_image_le
thf(fact_3_insert_Ohyps,axiom,
~ ( fmembe7738088520663304047od_a_b @ x @ xsa ) ).
% insert.hyps
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_Ae_O_Ax_A_061_A_It_M_Ae_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T: a,E: b] :
( x
!= ( product_Pair_a_b @ T @ E ) ) ).
% \<open>\<And>thesis. (\<And>t e. x = (t, e) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_t__def,axiom,
( x
= ( product_Pair_a_b @ t @ e ) ) ).
% t_def
thf(fact_6_insert_Oprems_I2_J,axiom,
( ( fcard_8555586198630727417od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( f @ T2 ) ) )
@ ( finser5399165414501630960od_a_b @ x @ xsa ) ) )
= ( fcard_8555586198630727417od_a_b @ ( finser5399165414501630960od_a_b @ x @ xsa ) ) ) ).
% insert.prems(2)
thf(fact_7_assms_I2_J,axiom,
( ( fcard_8555586198630727417od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( f @ T2 ) ) )
@ xs ) )
= ( fcard_8555586198630727417od_a_b @ xs ) ) ).
% assms(2)
thf(fact_8_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_9_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_10_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_11_case__prod__conv,axiom,
! [F: a > b > product_prod_a_b,A: a,B: b] :
( ( produc8992199381948149691od_a_b @ F @ ( product_Pair_a_b @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_12_case__prod__conv,axiom,
! [F: a > b > nat,A: a,B: b] :
( ( produc5088076833887504125_b_nat @ F @ ( product_Pair_a_b @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_13_fset_Omap__ident,axiom,
! [T3: fset_P9214369701362650254od_a_b] :
( ( fimage3179447126440047741od_a_b
@ ^ [X2: product_prod_a_b] : X2
@ T3 )
= T3 ) ).
% fset.map_ident
thf(fact_14_fimage__ident,axiom,
! [Y: fset_P9214369701362650254od_a_b] :
( ( fimage3179447126440047741od_a_b
@ ^ [X2: product_prod_a_b] : X2
@ Y )
= Y ) ).
% fimage_ident
thf(fact_15_case__prod__Pair__iden,axiom,
! [P: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_16_case__prod__Pair__iden,axiom,
! [P: product_prod_int_int] :
( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_17_case__prod__Pair__iden,axiom,
! [P: product_prod_a_b] :
( ( produc8992199381948149691od_a_b @ product_Pair_a_b @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_18_prod_Oinject,axiom,
! [X1: a,X22: b,Y1: a,Y2: b] :
( ( ( product_Pair_a_b @ X1 @ X22 )
= ( product_Pair_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_20_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A: a,B: b,A2: a,B2: b] :
( ( ( product_Pair_a_b @ A @ B )
= ( product_Pair_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_24_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( Q @ ( produc27273713700761075at_nat @ P2 @ Z ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_25_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
( ( Q @ ( produc8739625826339149834_nat_o @ P2 @ Z ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_26_case__prodE2,axiom,
! [Q: $o > $o,P2: int > int > $o,Z: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P2 @ Z ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_27_case__prodE2,axiom,
! [Q: product_prod_a_b > $o,P2: a > b > product_prod_a_b,Z: product_prod_a_b] :
( ( Q @ ( produc8992199381948149691od_a_b @ P2 @ Z ) )
=> ~ ! [X3: a,Y3: b] :
( ( Z
= ( product_Pair_a_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_28_case__prodE2,axiom,
! [Q: nat > $o,P2: a > b > nat,Z: product_prod_a_b] :
( ( Q @ ( produc5088076833887504125_b_nat @ P2 @ Z ) )
=> ~ ! [X3: a,Y3: b] :
( ( Z
= ( product_Pair_a_b @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_29_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X2: nat,Y4: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_30_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y4: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_31_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X2: int,Y4: int] : ( F @ ( product_Pair_int_int @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_32_case__prod__eta,axiom,
! [F: product_prod_a_b > product_prod_a_b] :
( ( produc8992199381948149691od_a_b
@ ^ [X2: a,Y4: b] : ( F @ ( product_Pair_a_b @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_33_case__prod__eta,axiom,
! [F: product_prod_a_b > nat] :
( ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( F @ ( product_Pair_a_b @ X2 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_34_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_35_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_36_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_37_cond__case__prod__eta,axiom,
! [F: a > b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
( ! [X3: a,Y3: b] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_a_b @ X3 @ Y3 ) ) )
=> ( ( produc8992199381948149691od_a_b @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_38_cond__case__prod__eta,axiom,
! [F: a > b > nat,G: product_prod_a_b > nat] :
( ! [X3: a,Y3: b] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_a_b @ X3 @ Y3 ) ) )
=> ( ( produc5088076833887504125_b_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_39_fset_Omap__cong,axiom,
! [X: fset_P9214369701362650254od_a_b,Ya: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
( ( X = Ya )
=> ( ! [Z2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( fset_P2369346149119917079od_a_b @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( fimage3179447126440047741od_a_b @ F @ X )
= ( fimage3179447126440047741od_a_b @ G @ Ya ) ) ) ) ).
% fset.map_cong
thf(fact_40_fset_Omap__cong0,axiom,
! [X: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
( ! [Z2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( fset_P2369346149119917079od_a_b @ X ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( fimage3179447126440047741od_a_b @ F @ X )
= ( fimage3179447126440047741od_a_b @ G @ X ) ) ) ).
% fset.map_cong0
thf(fact_41_fset_Oinj__map__strong,axiom,
! [X: fset_P9214369701362650254od_a_b,Xa: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,Fa: product_prod_a_b > product_prod_a_b] :
( ! [Z2: product_prod_a_b,Za: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( fset_P2369346149119917079od_a_b @ X ) )
=> ( ( member1426531481828664017od_a_b @ Za @ ( fset_P2369346149119917079od_a_b @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( fimage3179447126440047741od_a_b @ F @ X )
= ( fimage3179447126440047741od_a_b @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fset.inj_map_strong
thf(fact_42_fcard__finsert__if,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( fcard_8555586198630727417od_a_b @ ( finser5399165414501630960od_a_b @ X @ A3 ) )
= ( fcard_8555586198630727417od_a_b @ A3 ) ) )
& ( ~ ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( fcard_8555586198630727417od_a_b @ ( finser5399165414501630960od_a_b @ X @ A3 ) )
= ( suc @ ( fcard_8555586198630727417od_a_b @ A3 ) ) ) ) ) ).
% fcard_finsert_if
thf(fact_43_fcard__finsert__if,axiom,
! [X: nat,A3: fset_nat] :
( ( ( fmember_nat @ X @ A3 )
=> ( ( fcard_nat @ ( finsert_nat @ X @ A3 ) )
= ( fcard_nat @ A3 ) ) )
& ( ~ ( fmember_nat @ X @ A3 )
=> ( ( fcard_nat @ ( finsert_nat @ X @ A3 ) )
= ( suc @ ( fcard_nat @ A3 ) ) ) ) ) ).
% fcard_finsert_if
thf(fact_44_finsert__absorb2,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( finser5399165414501630960od_a_b @ X @ ( finser5399165414501630960od_a_b @ X @ A3 ) )
= ( finser5399165414501630960od_a_b @ X @ A3 ) ) ).
% finsert_absorb2
thf(fact_45_case__prodI2,axiom,
! [P: product_prod_a_b,C: a > b > $o] :
( ! [A4: a,B3: b] :
( ( P
= ( product_Pair_a_b @ A4 @ B3 ) )
=> ( C @ A4 @ B3 ) )
=> ( produc3537405659489547051_a_b_o @ C @ P ) ) ).
% case_prodI2
thf(fact_46_case__prodI2,axiom,
! [P: product_prod_nat_nat,C: nat > nat > $o] :
( ! [A4: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( C @ A4 @ B3 ) )
=> ( produc6081775807080527818_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_47_case__prodI2,axiom,
! [P: product_prod_int_int,C: int > int > $o] :
( ! [A4: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A4 @ B3 ) )
=> ( C @ A4 @ B3 ) )
=> ( produc4947309494688390418_int_o @ C @ P ) ) ).
% case_prodI2
thf(fact_48_case__prodI,axiom,
! [F: a > b > $o,A: a,B: b] :
( ( F @ A @ B )
=> ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ A @ B ) ) ) ).
% case_prodI
thf(fact_49_case__prodI,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( F @ A @ B )
=> ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_50_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_51_mem__case__prodI2,axiom,
! [P: product_prod_a_b,Z: product_prod_a_b,C: a > b > set_Product_prod_a_b] :
( ! [A4: a,B3: b] :
( ( P
= ( product_Pair_a_b @ A4 @ B3 ) )
=> ( member1426531481828664017od_a_b @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc7078681169951578907od_a_b @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_52_mem__case__prodI2,axiom,
! [P: product_prod_a_b,Z: nat,C: a > b > set_nat] :
( ! [A4: a,B3: b] :
( ( P
= ( product_Pair_a_b @ A4 @ B3 ) )
=> ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc2976218243292053171et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_53_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: product_prod_a_b,C: nat > nat > set_Product_prod_a_b] :
( ! [A4: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( member1426531481828664017od_a_b @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc3410858885478534458od_a_b @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_54_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
( ! [A4: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_55_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: product_prod_a_b,C: int > int > set_Product_prod_a_b] :
( ! [A4: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A4 @ B3 ) )
=> ( member1426531481828664017od_a_b @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc5321327879229220482od_a_b @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_56_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z: nat,C: int > int > set_nat] :
( ! [A4: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A4 @ B3 ) )
=> ( member_nat @ Z @ ( C @ A4 @ B3 ) ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_57_mem__case__prodI,axiom,
! [Z: product_prod_a_b,C: a > b > set_Product_prod_a_b,A: a,B: b] :
( ( member1426531481828664017od_a_b @ Z @ ( C @ A @ B ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc7078681169951578907od_a_b @ C @ ( product_Pair_a_b @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_58_mem__case__prodI,axiom,
! [Z: nat,C: a > b > set_nat,A: a,B: b] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc2976218243292053171et_nat @ C @ ( product_Pair_a_b @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_59_mem__case__prodI,axiom,
! [Z: product_prod_a_b,C: nat > nat > set_Product_prod_a_b,A: nat,B: nat] :
( ( member1426531481828664017od_a_b @ Z @ ( C @ A @ B ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc3410858885478534458od_a_b @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_60_mem__case__prodI,axiom,
! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_61_mem__case__prodI,axiom,
! [Z: product_prod_a_b,C: int > int > set_Product_prod_a_b,A: int,B: int] :
( ( member1426531481828664017od_a_b @ Z @ ( C @ A @ B ) )
=> ( member1426531481828664017od_a_b @ Z @ ( produc5321327879229220482od_a_b @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_62_mem__case__prodI,axiom,
! [Z: nat,C: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z @ ( C @ A @ B ) )
=> ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_63_case__prodI2_H,axiom,
! [P: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A4 @ B3 )
= P )
=> ( C @ A4 @ B3 @ X ) )
=> ( produc8739625826339149834_nat_o @ C @ P @ X ) ) ).
% case_prodI2'
thf(fact_64_finsertCI,axiom,
! [A: product_prod_a_b,B4: fset_P9214369701362650254od_a_b,B: product_prod_a_b] :
( ( ~ ( fmembe7738088520663304047od_a_b @ A @ B4 )
=> ( A = B ) )
=> ( fmembe7738088520663304047od_a_b @ A @ ( finser5399165414501630960od_a_b @ B @ B4 ) ) ) ).
% finsertCI
thf(fact_65_finsertCI,axiom,
! [A: nat,B4: fset_nat,B: nat] :
( ( ~ ( fmember_nat @ A @ B4 )
=> ( A = B ) )
=> ( fmember_nat @ A @ ( finsert_nat @ B @ B4 ) ) ) ).
% finsertCI
thf(fact_66_finsert__iff,axiom,
! [A: product_prod_a_b,B: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ ( finser5399165414501630960od_a_b @ B @ A3 ) )
= ( ( A = B )
| ( fmembe7738088520663304047od_a_b @ A @ A3 ) ) ) ).
% finsert_iff
thf(fact_67_finsert__iff,axiom,
! [A: nat,B: nat,A3: fset_nat] :
( ( fmember_nat @ A @ ( finsert_nat @ B @ A3 ) )
= ( ( A = B )
| ( fmember_nat @ A @ A3 ) ) ) ).
% finsert_iff
thf(fact_68_fimage__eqI,axiom,
! [B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b,X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( B
= ( F @ X ) )
=> ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( fmembe7738088520663304047od_a_b @ B @ ( fimage3179447126440047741od_a_b @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_69_fimage__eqI,axiom,
! [B: nat,F: product_prod_a_b > nat,X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( B
= ( F @ X ) )
=> ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( fmember_nat @ B @ ( fimage6539872720770606011_b_nat @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_70_fimage__eqI,axiom,
! [B: product_prod_a_b,F: nat > product_prod_a_b,X: nat,A3: fset_nat] :
( ( B
= ( F @ X ) )
=> ( ( fmember_nat @ X @ A3 )
=> ( fmembe7738088520663304047od_a_b @ B @ ( fimage1486297136060110361od_a_b @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_71_fimage__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A3: fset_nat] :
( ( B
= ( F @ X ) )
=> ( ( fmember_nat @ X @ A3 )
=> ( fmember_nat @ B @ ( fimage_nat_nat @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_72_fimage__finsert,axiom,
! [F: product_prod_a_b > product_prod_a_b,A: product_prod_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( fimage3179447126440047741od_a_b @ F @ ( finser5399165414501630960od_a_b @ A @ B4 ) )
= ( finser5399165414501630960od_a_b @ ( F @ A ) @ ( fimage3179447126440047741od_a_b @ F @ B4 ) ) ) ).
% fimage_finsert
thf(fact_73_finsert__fimage,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( finser5399165414501630960od_a_b @ ( F @ X ) @ ( fimage3179447126440047741od_a_b @ F @ A3 ) )
= ( fimage3179447126440047741od_a_b @ F @ A3 ) ) ) ).
% finsert_fimage
thf(fact_74_finsert__fimage,axiom,
! [X: nat,A3: fset_nat,F: nat > product_prod_a_b] :
( ( fmember_nat @ X @ A3 )
=> ( ( finser5399165414501630960od_a_b @ ( F @ X ) @ ( fimage1486297136060110361od_a_b @ F @ A3 ) )
= ( fimage1486297136060110361od_a_b @ F @ A3 ) ) ) ).
% finsert_fimage
thf(fact_75_finsertE,axiom,
! [A: product_prod_a_b,B: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ ( finser5399165414501630960od_a_b @ B @ A3 ) )
=> ( ( A != B )
=> ( fmembe7738088520663304047od_a_b @ A @ A3 ) ) ) ).
% finsertE
thf(fact_76_finsertE,axiom,
! [A: nat,B: nat,A3: fset_nat] :
( ( fmember_nat @ A @ ( finsert_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( fmember_nat @ A @ A3 ) ) ) ).
% finsertE
thf(fact_77_fset__eqI,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ! [X3: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X3 @ A3 )
= ( fmembe7738088520663304047od_a_b @ X3 @ B4 ) )
=> ( A3 = B4 ) ) ).
% fset_eqI
thf(fact_78_fset__eqI,axiom,
! [A3: fset_nat,B4: fset_nat] :
( ! [X3: nat] :
( ( fmember_nat @ X3 @ A3 )
= ( fmember_nat @ X3 @ B4 ) )
=> ( A3 = B4 ) ) ).
% fset_eqI
thf(fact_79_finsertI1,axiom,
! [A: product_prod_a_b,B4: fset_P9214369701362650254od_a_b] : ( fmembe7738088520663304047od_a_b @ A @ ( finser5399165414501630960od_a_b @ A @ B4 ) ) ).
% finsertI1
thf(fact_80_finsertI1,axiom,
! [A: nat,B4: fset_nat] : ( fmember_nat @ A @ ( finsert_nat @ A @ B4 ) ) ).
% finsertI1
thf(fact_81_finsertI2,axiom,
! [A: product_prod_a_b,B4: fset_P9214369701362650254od_a_b,B: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ B4 )
=> ( fmembe7738088520663304047od_a_b @ A @ ( finser5399165414501630960od_a_b @ B @ B4 ) ) ) ).
% finsertI2
thf(fact_82_finsertI2,axiom,
! [A: nat,B4: fset_nat,B: nat] :
( ( fmember_nat @ A @ B4 )
=> ( fmember_nat @ A @ ( finsert_nat @ B @ B4 ) ) ) ).
% finsertI2
thf(fact_83_fequalityCE,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,C: product_prod_a_b] :
( ( A3 = B4 )
=> ( ( ( fmembe7738088520663304047od_a_b @ C @ A3 )
=> ~ ( fmembe7738088520663304047od_a_b @ C @ B4 ) )
=> ~ ( ~ ( fmembe7738088520663304047od_a_b @ C @ A3 )
=> ( fmembe7738088520663304047od_a_b @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_84_fequalityCE,axiom,
! [A3: fset_nat,B4: fset_nat,C: nat] :
( ( A3 = B4 )
=> ( ( ( fmember_nat @ C @ A3 )
=> ~ ( fmember_nat @ C @ B4 ) )
=> ~ ( ~ ( fmember_nat @ C @ A3 )
=> ( fmember_nat @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_85_set__finsert,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ~ ! [B5: fset_P9214369701362650254od_a_b] :
( ( A3
= ( finser5399165414501630960od_a_b @ X @ B5 ) )
=> ( fmembe7738088520663304047od_a_b @ X @ B5 ) ) ) ).
% set_finsert
thf(fact_86_set__finsert,axiom,
! [X: nat,A3: fset_nat] :
( ( fmember_nat @ X @ A3 )
=> ~ ! [B5: fset_nat] :
( ( A3
= ( finsert_nat @ X @ B5 ) )
=> ( fmember_nat @ X @ B5 ) ) ) ).
% set_finsert
thf(fact_87_eq__fmem__trans,axiom,
! [A: product_prod_a_b,B: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( A = B )
=> ( ( fmembe7738088520663304047od_a_b @ B @ A3 )
=> ( fmembe7738088520663304047od_a_b @ A @ A3 ) ) ) ).
% eq_fmem_trans
thf(fact_88_eq__fmem__trans,axiom,
! [A: nat,B: nat,A3: fset_nat] :
( ( A = B )
=> ( ( fmember_nat @ B @ A3 )
=> ( fmember_nat @ A @ A3 ) ) ) ).
% eq_fmem_trans
thf(fact_89_finsert__ident,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ~ ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ~ ( fmembe7738088520663304047od_a_b @ X @ B4 )
=> ( ( ( finser5399165414501630960od_a_b @ X @ A3 )
= ( finser5399165414501630960od_a_b @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% finsert_ident
thf(fact_90_finsert__ident,axiom,
! [X: nat,A3: fset_nat,B4: fset_nat] :
( ~ ( fmember_nat @ X @ A3 )
=> ( ~ ( fmember_nat @ X @ B4 )
=> ( ( ( finsert_nat @ X @ A3 )
= ( finsert_nat @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% finsert_ident
thf(fact_91_eqfset__imp__iff,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,X: product_prod_a_b] :
( ( A3 = B4 )
=> ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
= ( fmembe7738088520663304047od_a_b @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_92_eqfset__imp__iff,axiom,
! [A3: fset_nat,B4: fset_nat,X: nat] :
( ( A3 = B4 )
=> ( ( fmember_nat @ X @ A3 )
= ( fmember_nat @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_93_finsert__absorb,axiom,
! [A: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ A3 )
=> ( ( finser5399165414501630960od_a_b @ A @ A3 )
= A3 ) ) ).
% finsert_absorb
thf(fact_94_finsert__absorb,axiom,
! [A: nat,A3: fset_nat] :
( ( fmember_nat @ A @ A3 )
=> ( ( finsert_nat @ A @ A3 )
= A3 ) ) ).
% finsert_absorb
thf(fact_95_finsert__eq__iff,axiom,
! [A: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B: product_prod_a_b,B4: fset_P9214369701362650254od_a_b] :
( ~ ( fmembe7738088520663304047od_a_b @ A @ A3 )
=> ( ~ ( fmembe7738088520663304047od_a_b @ B @ B4 )
=> ( ( ( finser5399165414501630960od_a_b @ A @ A3 )
= ( finser5399165414501630960od_a_b @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C2: fset_P9214369701362650254od_a_b] :
( ( A3
= ( finser5399165414501630960od_a_b @ B @ C2 ) )
& ~ ( fmembe7738088520663304047od_a_b @ B @ C2 )
& ( B4
= ( finser5399165414501630960od_a_b @ A @ C2 ) )
& ~ ( fmembe7738088520663304047od_a_b @ A @ C2 ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_96_finsert__eq__iff,axiom,
! [A: nat,A3: fset_nat,B: nat,B4: fset_nat] :
( ~ ( fmember_nat @ A @ A3 )
=> ( ~ ( fmember_nat @ B @ B4 )
=> ( ( ( finsert_nat @ A @ A3 )
= ( finsert_nat @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C2: fset_nat] :
( ( A3
= ( finsert_nat @ B @ C2 ) )
& ~ ( fmember_nat @ B @ C2 )
& ( B4
= ( finsert_nat @ A @ C2 ) )
& ~ ( fmember_nat @ A @ C2 ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_97_if__split__fmem1,axiom,
! [Q: $o,X: product_prod_a_b,Y5: product_prod_a_b,B: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ ( if_Product_prod_a_b @ Q @ X @ Y5 ) @ B )
= ( ( Q
=> ( fmembe7738088520663304047od_a_b @ X @ B ) )
& ( ~ Q
=> ( fmembe7738088520663304047od_a_b @ Y5 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_98_if__split__fmem1,axiom,
! [Q: $o,X: nat,Y5: nat,B: fset_nat] :
( ( fmember_nat @ ( if_nat @ Q @ X @ Y5 ) @ B )
= ( ( Q
=> ( fmember_nat @ X @ B ) )
& ( ~ Q
=> ( fmember_nat @ Y5 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_99_if__split__fmem2,axiom,
! [A: product_prod_a_b,Q: $o,X: fset_P9214369701362650254od_a_b,Y5: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ ( if_fse8748871985657954388od_a_b @ Q @ X @ Y5 ) )
= ( ( Q
=> ( fmembe7738088520663304047od_a_b @ A @ X ) )
& ( ~ Q
=> ( fmembe7738088520663304047od_a_b @ A @ Y5 ) ) ) ) ).
% if_split_fmem2
thf(fact_100_if__split__fmem2,axiom,
! [A: nat,Q: $o,X: fset_nat,Y5: fset_nat] :
( ( fmember_nat @ A @ ( if_fset_nat @ Q @ X @ Y5 ) )
= ( ( Q
=> ( fmember_nat @ A @ X ) )
& ( ~ Q
=> ( fmember_nat @ A @ Y5 ) ) ) ) ).
% if_split_fmem2
thf(fact_101_mem__Collect__eq,axiom,
! [A: product_prod_a_b,P2: product_prod_a_b > $o] :
( ( member1426531481828664017od_a_b @ A @ ( collec3336397801687681299od_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_b] :
( ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A3: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A3: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_109_Collect__cong,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3392354462482085612at_nat @ P2 )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_110_Collect__cong,axiom,
! [P2: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X3: product_prod_int_int] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec213857154873943460nt_int @ P2 )
= ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_cong
thf(fact_111_eqfelem__imp__iff,axiom,
! [X: product_prod_a_b,Y5: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( X = Y5 )
=> ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
= ( fmembe7738088520663304047od_a_b @ Y5 @ A3 ) ) ) ).
% eqfelem_imp_iff
thf(fact_112_eqfelem__imp__iff,axiom,
! [X: nat,Y5: nat,A3: fset_nat] :
( ( X = Y5 )
=> ( ( fmember_nat @ X @ A3 )
= ( fmember_nat @ Y5 @ A3 ) ) ) ).
% eqfelem_imp_iff
thf(fact_113_finsert__commute,axiom,
! [X: product_prod_a_b,Y5: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( finser5399165414501630960od_a_b @ X @ ( finser5399165414501630960od_a_b @ Y5 @ A3 ) )
= ( finser5399165414501630960od_a_b @ Y5 @ ( finser5399165414501630960od_a_b @ X @ A3 ) ) ) ).
% finsert_commute
thf(fact_114_fcard__finsert__le,axiom,
! [A3: fset_P9214369701362650254od_a_b,X: product_prod_a_b] : ( ord_less_eq_nat @ ( fcard_8555586198630727417od_a_b @ A3 ) @ ( fcard_8555586198630727417od_a_b @ ( finser5399165414501630960od_a_b @ X @ A3 ) ) ) ).
% fcard_finsert_le
thf(fact_115_mk__disjoint__finsert,axiom,
! [A: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ A @ A3 )
=> ? [B5: fset_P9214369701362650254od_a_b] :
( ( A3
= ( finser5399165414501630960od_a_b @ A @ B5 ) )
& ~ ( fmembe7738088520663304047od_a_b @ A @ B5 ) ) ) ).
% mk_disjoint_finsert
thf(fact_116_mk__disjoint__finsert,axiom,
! [A: nat,A3: fset_nat] :
( ( fmember_nat @ A @ A3 )
=> ? [B5: fset_nat] :
( ( A3
= ( finsert_nat @ A @ B5 ) )
& ~ ( fmember_nat @ A @ B5 ) ) ) ).
% mk_disjoint_finsert
thf(fact_117_mem__case__prodE,axiom,
! [Z: product_prod_a_b,C: a > b > set_Product_prod_a_b,P: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Z @ ( produc7078681169951578907od_a_b @ C @ P ) )
=> ~ ! [X3: a,Y3: b] :
( ( P
= ( product_Pair_a_b @ X3 @ Y3 ) )
=> ~ ( member1426531481828664017od_a_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_118_mem__case__prodE,axiom,
! [Z: nat,C: a > b > set_nat,P: product_prod_a_b] :
( ( member_nat @ Z @ ( produc2976218243292053171et_nat @ C @ P ) )
=> ~ ! [X3: a,Y3: b] :
( ( P
= ( product_Pair_a_b @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_119_mem__case__prodE,axiom,
! [Z: product_prod_a_b,C: nat > nat > set_Product_prod_a_b,P: product_prod_nat_nat] :
( ( member1426531481828664017od_a_b @ Z @ ( produc3410858885478534458od_a_b @ C @ P ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( member1426531481828664017od_a_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_120_mem__case__prodE,axiom,
! [Z: nat,C: nat > nat > set_nat,P: product_prod_nat_nat] :
( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_121_mem__case__prodE,axiom,
! [Z: product_prod_a_b,C: int > int > set_Product_prod_a_b,P: product_prod_int_int] :
( ( member1426531481828664017od_a_b @ Z @ ( produc5321327879229220482od_a_b @ C @ P ) )
=> ~ ! [X3: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member1426531481828664017od_a_b @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_122_mem__case__prodE,axiom,
! [Z: nat,C: int > int > set_nat,P: product_prod_int_int] :
( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P ) )
=> ~ ! [X3: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_123_case__prodE,axiom,
! [C: a > b > $o,P: product_prod_a_b] :
( ( produc3537405659489547051_a_b_o @ C @ P )
=> ~ ! [X3: a,Y3: b] :
( ( P
= ( product_Pair_a_b @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_124_case__prodE,axiom,
! [C: nat > nat > $o,P: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C @ P )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_125_case__prodE,axiom,
! [C: int > int > $o,P: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P )
=> ~ ! [X3: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_126_case__prodD,axiom,
! [F: a > b > $o,A: a,B: b] :
( ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_127_case__prodD,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_128_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_129_case__prodE_H,axiom,
! [C: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C @ P @ Z )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_130_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_131_fmember__iff__member__fset,axiom,
( fmembe7738088520663304047od_a_b
= ( ^ [X2: product_prod_a_b,A5: fset_P9214369701362650254od_a_b] : ( member1426531481828664017od_a_b @ X2 @ ( fset_P2369346149119917079od_a_b @ A5 ) ) ) ) ).
% fmember_iff_member_fset
thf(fact_132_fmember__iff__member__fset,axiom,
( fmember_nat
= ( ^ [X2: nat,A5: fset_nat] : ( member_nat @ X2 @ ( fset_nat2 @ A5 ) ) ) ) ).
% fmember_iff_member_fset
thf(fact_133_fmember_Orep__eq,axiom,
( fmembe7738088520663304047od_a_b
= ( ^ [X2: product_prod_a_b,Xa2: fset_P9214369701362650254od_a_b] : ( member1426531481828664017od_a_b @ X2 @ ( fset_P2369346149119917079od_a_b @ Xa2 ) ) ) ) ).
% fmember.rep_eq
thf(fact_134_fmember_Orep__eq,axiom,
( fmember_nat
= ( ^ [X2: nat,Xa2: fset_nat] : ( member_nat @ X2 @ ( fset_nat2 @ Xa2 ) ) ) ) ).
% fmember.rep_eq
thf(fact_135_rev__fimage__eqI,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( fmembe7738088520663304047od_a_b @ B @ ( fimage3179447126440047741od_a_b @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_136_rev__fimage__eqI,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B: nat,F: product_prod_a_b > nat] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( fmember_nat @ B @ ( fimage6539872720770606011_b_nat @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_137_rev__fimage__eqI,axiom,
! [X: nat,A3: fset_nat,B: product_prod_a_b,F: nat > product_prod_a_b] :
( ( fmember_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( fmembe7738088520663304047od_a_b @ B @ ( fimage1486297136060110361od_a_b @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_138_rev__fimage__eqI,axiom,
! [X: nat,A3: fset_nat,B: nat,F: nat > nat] :
( ( fmember_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( fmember_nat @ B @ ( fimage_nat_nat @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_139_fimage__cong,axiom,
! [M: fset_P9214369701362650254od_a_b,N: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
( ( M = N )
=> ( ! [X3: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fimage3179447126440047741od_a_b @ F @ M )
= ( fimage3179447126440047741od_a_b @ G @ N ) ) ) ) ).
% fimage_cong
thf(fact_140_fimageI,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( fmembe7738088520663304047od_a_b @ ( F @ X ) @ ( fimage3179447126440047741od_a_b @ F @ A3 ) ) ) ).
% fimageI
thf(fact_141_fimageI,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > nat] :
( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( fmember_nat @ ( F @ X ) @ ( fimage6539872720770606011_b_nat @ F @ A3 ) ) ) ).
% fimageI
thf(fact_142_fimageI,axiom,
! [X: nat,A3: fset_nat,F: nat > product_prod_a_b] :
( ( fmember_nat @ X @ A3 )
=> ( fmembe7738088520663304047od_a_b @ ( F @ X ) @ ( fimage1486297136060110361od_a_b @ F @ A3 ) ) ) ).
% fimageI
thf(fact_143_fimageI,axiom,
! [X: nat,A3: fset_nat,F: nat > nat] :
( ( fmember_nat @ X @ A3 )
=> ( fmember_nat @ ( F @ X ) @ ( fimage_nat_nat @ F @ A3 ) ) ) ).
% fimageI
thf(fact_144_fimageE,axiom,
! [B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fmembe7738088520663304047od_a_b @ B @ ( fimage3179447126440047741od_a_b @ F @ A3 ) )
=> ~ ! [X3: product_prod_a_b] :
( ( B
= ( F @ X3 ) )
=> ~ ( fmembe7738088520663304047od_a_b @ X3 @ A3 ) ) ) ).
% fimageE
thf(fact_145_fimageE,axiom,
! [B: product_prod_a_b,F: nat > product_prod_a_b,A3: fset_nat] :
( ( fmembe7738088520663304047od_a_b @ B @ ( fimage1486297136060110361od_a_b @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( fmember_nat @ X3 @ A3 ) ) ) ).
% fimageE
thf(fact_146_fimageE,axiom,
! [B: nat,F: product_prod_a_b > nat,A3: fset_P9214369701362650254od_a_b] :
( ( fmember_nat @ B @ ( fimage6539872720770606011_b_nat @ F @ A3 ) )
=> ~ ! [X3: product_prod_a_b] :
( ( B
= ( F @ X3 ) )
=> ~ ( fmembe7738088520663304047od_a_b @ X3 @ A3 ) ) ) ).
% fimageE
thf(fact_147_fimageE,axiom,
! [B: nat,F: nat > nat,A3: fset_nat] :
( ( fmember_nat @ B @ ( fimage_nat_nat @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( fmember_nat @ X3 @ A3 ) ) ) ).
% fimageE
thf(fact_148_Pair__inject,axiom,
! [A: a,B: b,A2: a,B2: b] :
( ( ( product_Pair_a_b @ A @ B )
= ( product_Pair_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_149_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_150_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_151_prod__cases,axiom,
! [P2: product_prod_a_b > $o,P: product_prod_a_b] :
( ! [A4: a,B3: b] : ( P2 @ ( product_Pair_a_b @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_152_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_153_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A4: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A4 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_154_surj__pair,axiom,
! [P: product_prod_a_b] :
? [X3: a,Y3: b] :
( P
= ( product_Pair_a_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_155_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_156_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X3: int,Y3: int] :
( P
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_157_old_Oprod_Oexhaust,axiom,
! [Y5: product_prod_a_b] :
~ ! [A4: a,B3: b] :
( Y5
!= ( product_Pair_a_b @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_158_old_Oprod_Oexhaust,axiom,
! [Y5: product_prod_nat_nat] :
~ ! [A4: nat,B3: nat] :
( Y5
!= ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_159_old_Oprod_Oexhaust,axiom,
! [Y5: product_prod_int_int] :
~ ! [A4: int,B3: int] :
( Y5
!= ( product_Pair_int_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_160_fset__cong,axiom,
! [X: fset_P9214369701362650254od_a_b,Y5: fset_P9214369701362650254od_a_b] :
( ( ( fset_P2369346149119917079od_a_b @ X )
= ( fset_P2369346149119917079od_a_b @ Y5 ) )
= ( X = Y5 ) ) ).
% fset_cong
thf(fact_161_fset__cong,axiom,
! [X: fset_nat,Y5: fset_nat] :
( ( ( fset_nat2 @ X )
= ( fset_nat2 @ Y5 ) )
= ( X = Y5 ) ) ).
% fset_cong
thf(fact_162_case__prod__app,axiom,
( produc27273713700761075at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( produc2626176000494625587at_nat
@ ^ [L: nat,R2: nat] : ( F2 @ L @ R2 @ Y4 )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_163_case__prod__app,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( produc6081775807080527818_nat_o
@ ^ [L: nat,R2: nat] : ( F2 @ L @ R2 @ Y4 )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_164_prod_Ocase__distrib,axiom,
! [H: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( H @ ( produc27273713700761075at_nat @ F @ Prod ) )
= ( produc27273713700761075at_nat
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_165_prod_Ocase__distrib,axiom,
! [H: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( H @ ( produc27273713700761075at_nat @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_166_prod_Ocase__distrib,axiom,
! [H: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc27273713700761075at_nat
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_167_prod_Ocase__distrib,axiom,
! [H: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_168_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_169_prod_Ocase__distrib,axiom,
! [H: product_prod_a_b > product_prod_a_b,F: a > b > product_prod_a_b,Prod: product_prod_a_b] :
( ( H @ ( produc8992199381948149691od_a_b @ F @ Prod ) )
= ( produc8992199381948149691od_a_b
@ ^ [X12: a,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_170_prod_Ocase__distrib,axiom,
! [H: product_prod_a_b > nat,F: a > b > product_prod_a_b,Prod: product_prod_a_b] :
( ( H @ ( produc8992199381948149691od_a_b @ F @ Prod ) )
= ( produc5088076833887504125_b_nat
@ ^ [X12: a,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_171_prod_Ocase__distrib,axiom,
! [H: nat > product_prod_a_b,F: a > b > nat,Prod: product_prod_a_b] :
( ( H @ ( produc5088076833887504125_b_nat @ F @ Prod ) )
= ( produc8992199381948149691od_a_b
@ ^ [X12: a,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_172_prod_Ocase__distrib,axiom,
! [H: nat > nat,F: a > b > nat,Prod: product_prod_a_b] :
( ( H @ ( produc5088076833887504125_b_nat @ F @ Prod ) )
= ( produc5088076833887504125_b_nat
@ ^ [X12: a,X23: b] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_173_fimage__fimage,axiom,
! [F: product_prod_a_b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( fimage3179447126440047741od_a_b @ F @ ( fimage3179447126440047741od_a_b @ G @ A3 ) )
= ( fimage3179447126440047741od_a_b
@ ^ [X2: product_prod_a_b] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% fimage_fimage
thf(fact_174_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_175_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_176_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_177_old_Oprod_Ocase,axiom,
! [F: a > b > product_prod_a_b,X1: a,X22: b] :
( ( produc8992199381948149691od_a_b @ F @ ( product_Pair_a_b @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_178_old_Oprod_Ocase,axiom,
! [F: a > b > nat,X1: a,X22: b] :
( ( produc5088076833887504125_b_nat @ F @ ( product_Pair_a_b @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_179_fset_Omap__ident__strong,axiom,
! [T3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b] :
( ! [Z2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( fset_P2369346149119917079od_a_b @ T3 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( fimage3179447126440047741od_a_b @ F @ T3 )
= T3 ) ) ).
% fset.map_ident_strong
thf(fact_180_fset_Omap__ident__strong,axiom,
! [T3: fset_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( fset_nat2 @ T3 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( fimage_nat_nat @ F @ T3 )
= T3 ) ) ).
% fset.map_ident_strong
thf(fact_181_fcard__finsert__disjoint,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ~ ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( fcard_8555586198630727417od_a_b @ ( finser5399165414501630960od_a_b @ X @ A3 ) )
= ( suc @ ( fcard_8555586198630727417od_a_b @ A3 ) ) ) ) ).
% fcard_finsert_disjoint
thf(fact_182_fcard__finsert__disjoint,axiom,
! [X: nat,A3: fset_nat] :
( ~ ( fmember_nat @ X @ A3 )
=> ( ( fcard_nat @ ( finsert_nat @ X @ A3 ) )
= ( suc @ ( fcard_nat @ A3 ) ) ) ) ).
% fcard_finsert_disjoint
thf(fact_183_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_184_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_185_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_186_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_187_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_188_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_189_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat] :
( ! [X3: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc27273713700761075at_nat @ F @ P )
= ( produc27273713700761075at_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_190_split__cong,axiom,
! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [X3: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc8739625826339149834_nat_o @ F @ P )
= ( produc8739625826339149834_nat_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_191_split__cong,axiom,
! [Q2: product_prod_int_int,F: int > int > $o,G: int > int > $o,P: product_prod_int_int] :
( ! [X3: int,Y3: int] :
( ( ( product_Pair_int_int @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc4947309494688390418_int_o @ F @ P )
= ( produc4947309494688390418_int_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_192_split__cong,axiom,
! [Q2: product_prod_a_b,F: a > b > product_prod_a_b,G: a > b > product_prod_a_b,P: product_prod_a_b] :
( ! [X3: a,Y3: b] :
( ( ( product_Pair_a_b @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc8992199381948149691od_a_b @ F @ P )
= ( produc8992199381948149691od_a_b @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_193_split__cong,axiom,
! [Q2: product_prod_a_b,F: a > b > nat,G: a > b > nat,P: product_prod_a_b] :
( ! [X3: a,Y3: b] :
( ( ( product_Pair_a_b @ X3 @ Y3 )
= Q2 )
=> ( ( F @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( produc5088076833887504125_b_nat @ F @ P )
= ( produc5088076833887504125_b_nat @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_194_insert_Oprems_I1_J,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ ( finser5399165414501630960od_a_b @ x @ xsa ) ) ) )
=> ( ( g @ ( f @ X4 ) )
= ( g @ X4 ) ) ) ).
% insert.prems(1)
thf(fact_195_internal__case__prod__def,axiom,
produc1854806715440696265at_nat = produc27273713700761075at_nat ).
% internal_case_prod_def
thf(fact_196_internal__case__prod__def,axiom,
produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).
% internal_case_prod_def
thf(fact_197_internal__case__prod__def,axiom,
produc8005341501107743676_int_o = produc4947309494688390418_int_o ).
% internal_case_prod_def
thf(fact_198_internal__case__prod__def,axiom,
produc3797153600616317541od_a_b = produc8992199381948149691od_a_b ).
% internal_case_prod_def
thf(fact_199_internal__case__prod__def,axiom,
produc2938035986357027795_b_nat = produc5088076833887504125_b_nat ).
% internal_case_prod_def
thf(fact_200_pred__equals__eq2,axiom,
! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
( ( ( ^ [X2: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y4 ) @ R ) )
= ( ^ [X2: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_201_pred__equals__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R ) )
= ( ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_202_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( ( ^ [X2: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ R ) )
= ( ^ [X2: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_203_insert_OIH,axiom,
( ! [X3: a] :
( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ xsa ) ) )
=> ( ( g @ ( f @ X3 ) )
= ( g @ X3 ) ) )
=> ( ( ( fcard_8555586198630727417od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( f @ T2 ) ) )
@ xsa ) )
= ( fcard_8555586198630727417od_a_b @ xsa ) )
=> ( ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( g @ X2 ) )
@ ( fset_P2369346149119917079od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( f @ T2 ) ) )
@ xsa ) ) )
= ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( g @ X2 ) )
@ ( fset_P2369346149119917079od_a_b @ xsa ) ) ) ) ) ).
% insert.IH
thf(fact_204_relChain__def,axiom,
( bNF_Ca968750328013420230at_nat
= ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > nat] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_205_relChain__def,axiom,
( bNF_Ca1968104039914474786nt_nat
= ( ^ [R2: set_Pr958786334691620121nt_int,As: int > nat] :
! [I: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R2 )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_206_relChain__def,axiom,
( bNF_Ca966259857504369954at_int
= ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > int] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
=> ( ord_less_eq_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_207_relChain__def,axiom,
( bNF_Ca1965613569405424510nt_int
= ( ^ [R2: set_Pr958786334691620121nt_int,As: int > int] :
! [I: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R2 )
=> ( ord_less_eq_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_208_relChain__def,axiom,
( bNF_Ca9191250440166129314t_real
= ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > real] :
! [I: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
=> ( ord_less_eq_real @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_209_relChain__def,axiom,
( bNF_Ca2154799978004375294t_real
= ( ^ [R2: set_Pr958786334691620121nt_int,As: int > real] :
! [I: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R2 )
=> ( ord_less_eq_real @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_210_assms_I1_J,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ xs ) ) )
=> ( ( g @ ( f @ X4 ) )
= ( g @ X4 ) ) ) ).
% assms(1)
thf(fact_211_fsubsetI,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ! [X3: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X3 @ A3 )
=> ( fmembe7738088520663304047od_a_b @ X3 @ B4 ) )
=> ( ord_le7868007465744610350od_a_b @ A3 @ B4 ) ) ).
% fsubsetI
thf(fact_212_fsubsetI,axiom,
! [A3: fset_nat,B4: fset_nat] :
( ! [X3: nat] :
( ( fmember_nat @ X3 @ A3 )
=> ( fmember_nat @ X3 @ B4 ) )
=> ( ord_less_eq_fset_nat @ A3 @ B4 ) ) ).
% fsubsetI
thf(fact_213_split__part,axiom,
! [P2: $o,Q: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A6: int,B6: int] :
( P2
& ( Q @ A6 @ B6 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P2
& ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_214_finsert__fsubset,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ ( finser5399165414501630960od_a_b @ X @ A3 ) @ B4 )
= ( ( fmembe7738088520663304047od_a_b @ X @ B4 )
& ( ord_le7868007465744610350od_a_b @ A3 @ B4 ) ) ) ).
% finsert_fsubset
thf(fact_215_finsert__fsubset,axiom,
! [X: nat,A3: fset_nat,B4: fset_nat] :
( ( ord_less_eq_fset_nat @ ( finsert_nat @ X @ A3 ) @ B4 )
= ( ( fmember_nat @ X @ B4 )
& ( ord_less_eq_fset_nat @ A3 @ B4 ) ) ) ).
% finsert_fsubset
thf(fact_216_fimage_Orep__eq,axiom,
! [X: product_prod_a_b > a,Xa: fset_P9214369701362650254od_a_b] :
( ( fset_a2 @ ( fimage1853113884124084179_a_b_a @ X @ Xa ) )
= ( image_2802296252294471259_a_b_a @ X @ ( fset_P2369346149119917079od_a_b @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_217_fimage_Orep__eq,axiom,
! [X: product_prod_a_b > product_prod_a_b,Xa: fset_P9214369701362650254od_a_b] :
( ( fset_P2369346149119917079od_a_b @ ( fimage3179447126440047741od_a_b @ X @ Xa ) )
= ( image_3300603549555413765od_a_b @ X @ ( fset_P2369346149119917079od_a_b @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_218_fimage_Orep__eq,axiom,
! [X: nat > product_prod_a_b,Xa: fset_nat] :
( ( fset_P2369346149119917079od_a_b @ ( fimage1486297136060110361od_a_b @ X @ Xa ) )
= ( image_372941892535967121od_a_b @ X @ ( fset_nat2 @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_219_fimage_Orep__eq,axiom,
! [X: product_prod_a_b > nat,Xa: fset_P9214369701362650254od_a_b] :
( ( fset_nat2 @ ( fimage6539872720770606011_b_nat @ X @ Xa ) )
= ( image_5426517477246462771_b_nat @ X @ ( fset_P2369346149119917079od_a_b @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_220_fimage_Orep__eq,axiom,
! [X: nat > nat,Xa: fset_nat] :
( ( fset_nat2 @ ( fimage_nat_nat @ X @ Xa ) )
= ( image_nat_nat @ X @ ( fset_nat2 @ Xa ) ) ) ).
% fimage.rep_eq
thf(fact_221_fset_Oset__map,axiom,
! [F: product_prod_a_b > a,V: fset_P9214369701362650254od_a_b] :
( ( fset_a2 @ ( fimage1853113884124084179_a_b_a @ F @ V ) )
= ( image_2802296252294471259_a_b_a @ F @ ( fset_P2369346149119917079od_a_b @ V ) ) ) ).
% fset.set_map
thf(fact_222_fset_Oset__map,axiom,
! [F: product_prod_a_b > product_prod_a_b,V: fset_P9214369701362650254od_a_b] :
( ( fset_P2369346149119917079od_a_b @ ( fimage3179447126440047741od_a_b @ F @ V ) )
= ( image_3300603549555413765od_a_b @ F @ ( fset_P2369346149119917079od_a_b @ V ) ) ) ).
% fset.set_map
thf(fact_223_fset_Oset__map,axiom,
! [F: nat > product_prod_a_b,V: fset_nat] :
( ( fset_P2369346149119917079od_a_b @ ( fimage1486297136060110361od_a_b @ F @ V ) )
= ( image_372941892535967121od_a_b @ F @ ( fset_nat2 @ V ) ) ) ).
% fset.set_map
thf(fact_224_fset_Oset__map,axiom,
! [F: product_prod_a_b > nat,V: fset_P9214369701362650254od_a_b] :
( ( fset_nat2 @ ( fimage6539872720770606011_b_nat @ F @ V ) )
= ( image_5426517477246462771_b_nat @ F @ ( fset_P2369346149119917079od_a_b @ V ) ) ) ).
% fset.set_map
thf(fact_225_fset_Oset__map,axiom,
! [F: nat > nat,V: fset_nat] :
( ( fset_nat2 @ ( fimage_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( fset_nat2 @ V ) ) ) ).
% fset.set_map
thf(fact_226_pair__imageI,axiom,
! [A: a,B: b,A3: set_Product_prod_a_b,F: a > b > a] :
( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ A3 )
=> ( member_a @ ( F @ A @ B ) @ ( image_2802296252294471259_a_b_a @ ( produc6028431345588019473_a_b_a @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_227_pair__imageI,axiom,
! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_a_b] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ A @ B ) @ ( image_4894260360327267052od_a_b @ ( produc5281920923555503578od_a_b @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_228_pair__imageI,axiom,
! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_229_pair__imageI,axiom,
! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > product_prod_a_b] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ A @ B ) @ ( image_2370028551316859444od_a_b @ ( produc8019077043795531554od_a_b @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_230_pair__imageI,axiom,
! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_5044651549707136836nt_nat @ ( produc8213879946458358998nt_nat @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_231_pair__imageI,axiom,
! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
=> ( member8885076297122219836at_nat @ ( F @ A @ B ) @ ( image_8730593652825689185at_nat @ ( produc27273713700761075at_nat @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_232_pair__imageI,axiom,
! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
=> ( member2006080708140975699_nat_o @ ( F @ A @ B ) @ ( image_2972642778337070200_nat_o @ ( produc8739625826339149834_nat_o @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_233_pair__imageI,axiom,
! [A: int,B: int,A3: set_Pr958786334691620121nt_int,F: int > int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A3 )
=> ( member_o @ ( F @ A @ B ) @ ( image_2135063354759101220_int_o @ ( produc4947309494688390418_int_o @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_234_pair__imageI,axiom,
! [A: a,B: b,A3: set_Product_prod_a_b,F: a > b > product_prod_a_b] :
( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ A @ B ) @ ( image_3300603549555413765od_a_b @ ( produc8992199381948149691od_a_b @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_235_pair__imageI,axiom,
! [A: a,B: b,A3: set_Product_prod_a_b,F: a > b > nat] :
( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ A3 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_5426517477246462771_b_nat @ ( produc5088076833887504125_b_nat @ F ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_236_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_237_Collect__case__prod__mono,axiom,
! [A3: nat > nat > $o,B4: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A3 @ B4 )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A3 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_238_Collect__case__prod__mono,axiom,
! [A3: int > int > $o,B4: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A3 @ B4 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A3 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_239_pred__subset__eq2,axiom,
! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
( ( ord_less_eq_a_b_o
@ ^ [X2: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y4 ) @ R )
@ ^ [X2: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X2 @ Y4 ) @ S ) )
= ( ord_le817736998455962536od_a_b @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_240_pred__subset__eq2,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
@ ^ [X2: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ S ) )
= ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_241_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X2: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ R )
@ ^ [X2: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ S ) )
= ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_242_less__eq__fset_Orep__eq,axiom,
( ord_le7868007465744610350od_a_b
= ( ^ [X2: fset_P9214369701362650254od_a_b,Xa2: fset_P9214369701362650254od_a_b] : ( ord_le817736998455962536od_a_b @ ( fset_P2369346149119917079od_a_b @ X2 ) @ ( fset_P2369346149119917079od_a_b @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_243_less__eq__fset_Orep__eq,axiom,
( ord_less_eq_fset_nat
= ( ^ [X2: fset_nat,Xa2: fset_nat] : ( ord_less_eq_set_nat @ ( fset_nat2 @ X2 ) @ ( fset_nat2 @ Xa2 ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_244_subrelI,axiom,
! [R3: set_Product_prod_a_b,S2: set_Product_prod_a_b] :
( ! [X3: a,Y3: b] :
( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y3 ) @ R3 )
=> ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le817736998455962536od_a_b @ R3 @ S2 ) ) ).
% subrelI
thf(fact_245_subrelI,axiom,
! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R3 @ S2 ) ) ).
% subrelI
thf(fact_246_subrelI,axiom,
! [R3: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le2843351958646193337nt_int @ R3 @ S2 ) ) ).
% subrelI
thf(fact_247_fst__eqD,axiom,
! [X: nat,Y5: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y5 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_248_fst__eqD,axiom,
! [X: int,Y5: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y5 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_249_fst__eqD,axiom,
! [X: a,Y5: b,A: a] :
( ( ( product_fst_a_b @ ( product_Pair_a_b @ X @ Y5 ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_250_fst__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_251_fst__conv,axiom,
! [X1: int,X22: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_252_fst__conv,axiom,
! [X1: a,X22: b] :
( ( product_fst_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_253_fsubsetD,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,C: product_prod_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ( fmembe7738088520663304047od_a_b @ C @ A3 )
=> ( fmembe7738088520663304047od_a_b @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_254_fsubsetD,axiom,
! [A3: fset_nat,B4: fset_nat,C: nat] :
( ( ord_less_eq_fset_nat @ A3 @ B4 )
=> ( ( fmember_nat @ C @ A3 )
=> ( fmember_nat @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_255_fin__mono,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,X: product_prod_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( fmembe7738088520663304047od_a_b @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_256_fin__mono,axiom,
! [A3: fset_nat,B4: fset_nat,X: nat] :
( ( ord_less_eq_fset_nat @ A3 @ B4 )
=> ( ( fmember_nat @ X @ A3 )
=> ( fmember_nat @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_257_fsubset__finsertI2,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,B: product_prod_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ord_le7868007465744610350od_a_b @ A3 @ ( finser5399165414501630960od_a_b @ B @ B4 ) ) ) ).
% fsubset_finsertI2
thf(fact_258_fsubset__finsertI,axiom,
! [B4: fset_P9214369701362650254od_a_b,A: product_prod_a_b] : ( ord_le7868007465744610350od_a_b @ B4 @ ( finser5399165414501630960od_a_b @ A @ B4 ) ) ).
% fsubset_finsertI
thf(fact_259_finsert__mono,axiom,
! [C3: fset_P9214369701362650254od_a_b,D: fset_P9214369701362650254od_a_b,A: product_prod_a_b] :
( ( ord_le7868007465744610350od_a_b @ C3 @ D )
=> ( ord_le7868007465744610350od_a_b @ ( finser5399165414501630960od_a_b @ A @ C3 ) @ ( finser5399165414501630960od_a_b @ A @ D ) ) ) ).
% finsert_mono
thf(fact_260_fimage__mono,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ord_le7868007465744610350od_a_b @ ( fimage3179447126440047741od_a_b @ F @ A3 ) @ ( fimage3179447126440047741od_a_b @ F @ B4 ) ) ) ).
% fimage_mono
thf(fact_261_subset__fimage__iff,axiom,
! [B4: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,A3: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ B4 @ ( fimage3179447126440047741od_a_b @ F @ A3 ) )
= ( ? [AA: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ AA @ A3 )
& ( B4
= ( fimage3179447126440047741od_a_b @ F @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_262_fst__def,axiom,
( product_fst_a_b
= ( produc6028431345588019473_a_b_a
@ ^ [X12: a,X23: b] : X12 ) ) ).
% fst_def
thf(fact_263_sum__img__le,axiom,
! [Xs: fset_P5670320511379867111at_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( fset_P3501481629439712240at_nat @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat
@ ( groups977919841031483927at_nat
@ ( produc6842872674320459806at_nat
@ ^ [X2: nat,Y4: nat] : ( G @ X2 ) )
@ ( fset_P3501481629439712240at_nat
@ ( fimage1537860586630726333at_nat
@ ( produc2626176000494625587at_nat
@ ^ [T2: nat] : ( product_Pair_nat_nat @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups977919841031483927at_nat
@ ( produc6842872674320459806at_nat
@ ^ [X2: nat,Y4: nat] : ( G @ X2 ) )
@ ( fset_P3501481629439712240at_nat @ Xs ) ) ) ) ).
% sum_img_le
thf(fact_264_sum__img__le,axiom,
! [Xs: fset_P5367158941141162143nt_int,G: int > nat,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( fset_P322984321831570088nt_int @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat
@ ( groups3536494975961350351nt_nat
@ ( produc8213879946458358998nt_nat
@ ^ [X2: int,Y4: int] : ( G @ X2 ) )
@ ( fset_P322984321831570088nt_int
@ ( fimage8245688998986472637nt_int
@ ( produc4245557441103728435nt_int
@ ^ [T2: int] : ( product_Pair_int_int @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups3536494975961350351nt_nat
@ ( produc8213879946458358998nt_nat
@ ^ [X2: int,Y4: int] : ( G @ X2 ) )
@ ( fset_P322984321831570088nt_int @ Xs ) ) ) ) ).
% sum_img_le
thf(fact_265_sum__img__le,axiom,
! [Xs: fset_P9214369701362650254od_a_b,G: a > nat,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat
@ ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( G @ X2 ) )
@ ( fset_P2369346149119917079od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( G @ X2 ) )
@ ( fset_P2369346149119917079od_a_b @ Xs ) ) ) ) ).
% sum_img_le
thf(fact_266_fsubset__finsert,axiom,
! [X: product_prod_a_b,A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ~ ( fmembe7738088520663304047od_a_b @ X @ A3 )
=> ( ( ord_le7868007465744610350od_a_b @ A3 @ ( finser5399165414501630960od_a_b @ X @ B4 ) )
= ( ord_le7868007465744610350od_a_b @ A3 @ B4 ) ) ) ).
% fsubset_finsert
thf(fact_267_fsubset__finsert,axiom,
! [X: nat,A3: fset_nat,B4: fset_nat] :
( ~ ( fmember_nat @ X @ A3 )
=> ( ( ord_less_eq_fset_nat @ A3 @ ( finsert_nat @ X @ B4 ) )
= ( ord_less_eq_fset_nat @ A3 @ B4 ) ) ) ).
% fsubset_finsert
thf(fact_268_fcard__seteq,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( fcard_8555586198630727417od_a_b @ B4 ) @ ( fcard_8555586198630727417od_a_b @ A3 ) )
=> ( A3 = B4 ) ) ) ).
% fcard_seteq
thf(fact_269_fcard__mono,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ord_less_eq_nat @ ( fcard_8555586198630727417od_a_b @ A3 ) @ ( fcard_8555586198630727417od_a_b @ B4 ) ) ) ).
% fcard_mono
thf(fact_270_fimage__fsubsetI,axiom,
! [A3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > product_prod_a_b,B4: fset_P9214369701362650254od_a_b] :
( ! [X3: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X3 @ A3 )
=> ( fmembe7738088520663304047od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le7868007465744610350od_a_b @ ( fimage3179447126440047741od_a_b @ F @ A3 ) @ B4 ) ) ).
% fimage_fsubsetI
thf(fact_271_fimage__fsubsetI,axiom,
! [A3: fset_P9214369701362650254od_a_b,F: product_prod_a_b > nat,B4: fset_nat] :
( ! [X3: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X3 @ A3 )
=> ( fmember_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_fset_nat @ ( fimage6539872720770606011_b_nat @ F @ A3 ) @ B4 ) ) ).
% fimage_fsubsetI
thf(fact_272_fimage__fsubsetI,axiom,
! [A3: fset_nat,F: nat > product_prod_a_b,B4: fset_P9214369701362650254od_a_b] :
( ! [X3: nat] :
( ( fmember_nat @ X3 @ A3 )
=> ( fmembe7738088520663304047od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le7868007465744610350od_a_b @ ( fimage1486297136060110361od_a_b @ F @ A3 ) @ B4 ) ) ).
% fimage_fsubsetI
thf(fact_273_fimage__fsubsetI,axiom,
! [A3: fset_nat,F: nat > nat,B4: fset_nat] :
( ! [X3: nat] :
( ( fmember_nat @ X3 @ A3 )
=> ( fmember_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_fset_nat @ ( fimage_nat_nat @ F @ A3 ) @ B4 ) ) ).
% fimage_fsubsetI
thf(fact_274_Compr__fimage__eq,axiom,
! [F: nat > nat,A3: fset_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ ( fimage_nat_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_275_Compr__fimage__eq,axiom,
! [F: product_prod_a_b > a,A3: fset_P9214369701362650254od_a_b,P2: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( fmember_a @ X2 @ ( fimage1853113884124084179_a_b_a @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2802296252294471259_a_b_a @ F
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_276_Compr__fimage__eq,axiom,
! [F: nat > product_prod_nat_nat,A3: fset_nat,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( fmembe1449479052724974408at_nat @ X2 @ ( fimage5210792427819060978at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5846123807819985514at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_277_Compr__fimage__eq,axiom,
! [F: nat > product_prod_int_int,A3: fset_nat,P2: product_prod_int_int > $o] :
( ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( fmembe7494353781971608064nt_int @ X2 @ ( fimage2032295120210918826nt_int @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2667626500211843362nt_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_278_Compr__fimage__eq,axiom,
! [F: nat > product_prod_a_b,A3: fset_nat,P2: product_prod_a_b > $o] :
( ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X2 @ ( fimage1486297136060110361od_a_b @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_372941892535967121od_a_b @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_279_Compr__fimage__eq,axiom,
! [F: product_prod_nat_nat > nat,A3: fset_P5670320511379867111at_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ ( fimage1850745034776345876at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2486076414777270412at_nat @ F
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( fmembe1449479052724974408at_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_280_Compr__fimage__eq,axiom,
! [F: product_prod_int_int > nat,A3: fset_P5367158941141162143nt_int,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ ( fimage4409320169706212300nt_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5044651549707136836nt_nat @ F
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( fmembe7494353781971608064nt_int @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_281_Compr__fimage__eq,axiom,
! [F: product_prod_a_b > nat,A3: fset_P9214369701362650254od_a_b,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( fmember_nat @ X2 @ ( fimage6539872720770606011_b_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5426517477246462771_b_nat @ F
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( fmembe7738088520663304047od_a_b @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_282_Compr__fimage__eq,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A3: fset_P5670320511379867111at_nat,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( fmembe1449479052724974408at_nat @ X2 @ ( fimage1537860586630726333at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5168914502847457605at_nat @ F
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( fmembe1449479052724974408at_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_283_Compr__fimage__eq,axiom,
! [F: product_prod_int_int > product_prod_nat_nat,A3: fset_P5367158941141162143nt_int,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( fmembe1449479052724974408at_nat @ X2 @ ( fimage2200814269739838981at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5831868185956570253at_nat @ F
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( fmembe7494353781971608064nt_int @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_fimage_eq
thf(fact_284_order__antisym__conv,axiom,
! [Y5: nat,X: nat] :
( ( ord_less_eq_nat @ Y5 @ X )
=> ( ( ord_less_eq_nat @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_285_order__antisym__conv,axiom,
! [Y5: int,X: int] :
( ( ord_less_eq_int @ Y5 @ X )
=> ( ( ord_less_eq_int @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_286_order__antisym__conv,axiom,
! [Y5: real,X: real] :
( ( ord_less_eq_real @ Y5 @ X )
=> ( ( ord_less_eq_real @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_287_linorder__le__cases,axiom,
! [X: nat,Y5: nat] :
( ~ ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ).
% linorder_le_cases
thf(fact_288_linorder__le__cases,axiom,
! [X: int,Y5: int] :
( ~ ( ord_less_eq_int @ X @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X ) ) ).
% linorder_le_cases
thf(fact_289_linorder__le__cases,axiom,
! [X: real,Y5: real] :
( ~ ( ord_less_eq_real @ X @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X ) ) ).
% linorder_le_cases
thf(fact_290_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_294_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_295_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_296_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_297_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_298_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_301_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_302_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_303_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_304_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_305_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_306_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_307_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_308_linorder__linear,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
| ( ord_less_eq_nat @ Y5 @ X ) ) ).
% linorder_linear
thf(fact_309_linorder__linear,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
| ( ord_less_eq_int @ Y5 @ X ) ) ).
% linorder_linear
thf(fact_310_linorder__linear,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
| ( ord_less_eq_real @ Y5 @ X ) ) ).
% linorder_linear
thf(fact_311_order__eq__refl,axiom,
! [X: nat,Y5: nat] :
( ( X = Y5 )
=> ( ord_less_eq_nat @ X @ Y5 ) ) ).
% order_eq_refl
thf(fact_312_order__eq__refl,axiom,
! [X: int,Y5: int] :
( ( X = Y5 )
=> ( ord_less_eq_int @ X @ Y5 ) ) ).
% order_eq_refl
thf(fact_313_order__eq__refl,axiom,
! [X: real,Y5: real] :
( ( X = Y5 )
=> ( ord_less_eq_real @ X @ Y5 ) ) ).
% order_eq_refl
thf(fact_314_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_315_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_316_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_317_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_318_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_319_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_320_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_321_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_322_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_323_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_324_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_325_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_326_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_327_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_328_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_329_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_330_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_331_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_332_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_333_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_334_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
& ( ord_less_eq_real @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_335_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_336_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_337_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_338_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_339_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_340_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_341_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_342_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_343_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_344_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_345_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_346_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A6: real,B6: real] :
( ( ord_less_eq_real @ B6 @ A6 )
& ( ord_less_eq_real @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_347_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_348_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_349_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_350_order__trans,axiom,
! [X: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_351_order__trans,axiom,
! [X: int,Y5: int,Z: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ( ord_less_eq_int @ Y5 @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_352_order__trans,axiom,
! [X: real,Y5: real,Z: real] :
( ( ord_less_eq_real @ X @ Y5 )
=> ( ( ord_less_eq_real @ Y5 @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_353_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_354_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_355_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_356_order__antisym,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ X )
=> ( X = Y5 ) ) ) ).
% order_antisym
thf(fact_357_order__antisym,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ( ord_less_eq_int @ Y5 @ X )
=> ( X = Y5 ) ) ) ).
% order_antisym
thf(fact_358_order__antisym,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
=> ( ( ord_less_eq_real @ Y5 @ X )
=> ( X = Y5 ) ) ) ).
% order_antisym
thf(fact_359_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_360_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_361_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_362_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_363_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_364_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_365_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_366_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_367_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_368_le__cases3,axiom,
! [X: nat,Y5: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y5 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_369_le__cases3,axiom,
! [X: int,Y5: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y5 )
=> ~ ( ord_less_eq_int @ Y5 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y5 @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y5 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y5 )
=> ~ ( ord_less_eq_int @ Y5 @ X ) )
=> ( ( ( ord_less_eq_int @ Y5 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_370_le__cases3,axiom,
! [X: real,Y5: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y5 )
=> ~ ( ord_less_eq_real @ Y5 @ Z ) )
=> ( ( ( ord_less_eq_real @ Y5 @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y5 ) )
=> ( ( ( ord_less_eq_real @ Z @ Y5 )
=> ~ ( ord_less_eq_real @ Y5 @ X ) )
=> ( ( ( ord_less_eq_real @ Y5 @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_371_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_372_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_373_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_374_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_375_image__ident,axiom,
! [Y: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_376_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_377_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_378_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_379_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_380_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_381_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_382_size__fset__simps,axiom,
( size_fset_nat
= ( ^ [Xa2: nat > nat,X2: fset_nat] :
( groups3542108847815614940at_nat
@ ^ [Y4: nat] : ( suc @ ( Xa2 @ Y4 ) )
@ ( fset_nat2 @ X2 ) ) ) ) ).
% size_fset_simps
thf(fact_383_size__fset__simps,axiom,
( size_f6359142247592877058od_a_b
= ( ^ [Xa2: product_prod_a_b > nat,X2: fset_P9214369701362650254od_a_b] :
( groups7004838470036706110_b_nat
@ ^ [Y4: product_prod_a_b] : ( suc @ ( Xa2 @ Y4 ) )
@ ( fset_P2369346149119917079od_a_b @ X2 ) ) ) ) ).
% size_fset_simps
thf(fact_384_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_385_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_386_image__eqI,axiom,
! [B: a,F: product_prod_a_b > a,X: product_prod_a_b,A3: set_Product_prod_a_b] :
( ( B
= ( F @ X ) )
=> ( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_387_image__eqI,axiom,
! [B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b,X: product_prod_a_b,A3: set_Product_prod_a_b] :
( ( B
= ( F @ X ) )
=> ( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member1426531481828664017od_a_b @ B @ ( image_3300603549555413765od_a_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_388_image__eqI,axiom,
! [B: nat,F: product_prod_a_b > nat,X: product_prod_a_b,A3: set_Product_prod_a_b] :
( ( B
= ( F @ X ) )
=> ( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member_nat @ B @ ( image_5426517477246462771_b_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_389_image__eqI,axiom,
! [B: product_prod_a_b,F: nat > product_prod_a_b,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member1426531481828664017od_a_b @ B @ ( image_372941892535967121od_a_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_390_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_391_sum__mono,axiom,
! [K: set_nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K ) @ ( groups3542108847815614940at_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_392_sum__mono,axiom,
! [K: set_Product_prod_a_b,F: product_prod_a_b > int,G: product_prod_a_b > int] :
( ! [I2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ I2 @ K )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups7002347999527655834_b_int @ F @ K ) @ ( groups7002347999527655834_b_int @ G @ K ) ) ) ).
% sum_mono
thf(fact_393_sum__mono,axiom,
! [K: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K ) @ ( groups3539618377306564664at_int @ G @ K ) ) ) ).
% sum_mono
thf(fact_394_sum__mono,axiom,
! [K: set_Product_prod_a_b,F: product_prod_a_b > real,G: product_prod_a_b > real] :
( ! [I2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ I2 @ K )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups1618960153307602714b_real @ F @ K ) @ ( groups1618960153307602714b_real @ G @ K ) ) ) ).
% sum_mono
thf(fact_395_sum__mono,axiom,
! [K: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K )
=> ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K ) @ ( groups6591440286371151544t_real @ G @ K ) ) ) ).
% sum_mono
thf(fact_396_sum__mono,axiom,
! [K: set_Product_prod_a_b,F: product_prod_a_b > nat,G: product_prod_a_b > nat] :
( ! [I2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ I2 @ K )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups7004838470036706110_b_nat @ F @ K ) @ ( groups7004838470036706110_b_nat @ G @ K ) ) ) ).
% sum_mono
thf(fact_397_subsetI,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( member1426531481828664017od_a_b @ X3 @ B4 ) )
=> ( ord_le817736998455962536od_a_b @ A3 @ B4 ) ) ).
% subsetI
thf(fact_398_subsetI,axiom,
! [A3: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% subsetI
thf(fact_399_in__mono,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,X: product_prod_a_b] :
( ( ord_le817736998455962536od_a_b @ A3 @ B4 )
=> ( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member1426531481828664017od_a_b @ X @ B4 ) ) ) ).
% in_mono
thf(fact_400_in__mono,axiom,
! [A3: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_401_subsetD,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,C: product_prod_a_b] :
( ( ord_le817736998455962536od_a_b @ A3 @ B4 )
=> ( ( member1426531481828664017od_a_b @ C @ A3 )
=> ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).
% subsetD
thf(fact_402_subsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_403_subset__eq,axiom,
( ord_le817736998455962536od_a_b
= ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
! [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A5 )
=> ( member1426531481828664017od_a_b @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_404_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_405_subset__iff,axiom,
( ord_le817736998455962536od_a_b
= ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
! [T2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ T2 @ A5 )
=> ( member1426531481828664017od_a_b @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_406_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_407_Collect__mono,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_408_Collect__mono,axiom,
! [P2: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X3: product_prod_int_int] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P2 ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_mono
thf(fact_409_Collect__subset,axiom,
! [A3: set_Product_prod_a_b,P2: product_prod_a_b > $o] :
( ord_le817736998455962536od_a_b
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_410_Collect__subset,axiom,
! [A3: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_411_Collect__subset,axiom,
! [A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_412_Collect__subset,axiom,
! [A3: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_413_less__eq__set__def,axiom,
( ord_le817736998455962536od_a_b
= ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
( ord_le8027066870050541877_a_b_o
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ A5 )
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_414_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_415_Collect__mono__iff,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) )
= ( ! [X2: product_prod_nat_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_416_Collect__mono__iff,axiom,
! [P2: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P2 ) @ ( collec213857154873943460nt_int @ Q ) )
= ( ! [X2: product_prod_int_int] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_417_pred__subset__eq,axiom,
! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
( ( ord_le8027066870050541877_a_b_o
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ R )
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ S ) )
= ( ord_le817736998455962536od_a_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_418_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_419_imageI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,F: product_prod_a_b > a] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_2802296252294471259_a_b_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_420_imageI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ X ) @ ( image_3300603549555413765od_a_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_421_imageI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,F: product_prod_a_b > nat] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_5426517477246462771_b_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_422_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > product_prod_a_b] :
( ( member_nat @ X @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ X ) @ ( image_372941892535967121od_a_b @ F @ A3 ) ) ) ).
% imageI
thf(fact_423_imageI,axiom,
! [X: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_424_image__iff,axiom,
! [Z: a,F: product_prod_a_b > a,A3: set_Product_prod_a_b] :
( ( member_a @ Z @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
= ( ? [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_425_image__iff,axiom,
! [Z: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_426_bex__imageD,axiom,
! [F: product_prod_a_b > a,A3: set_Product_prod_a_b,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
& ( P2 @ X4 ) )
=> ? [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_427_bex__imageD,axiom,
! [F: nat > nat,A3: set_nat,P2: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A3 ) )
& ( P2 @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_428_image__cong,axiom,
! [M: set_Product_prod_a_b,N: set_Product_prod_a_b,F: product_prod_a_b > a,G: product_prod_a_b > a] :
( ( M = N )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_2802296252294471259_a_b_a @ F @ M )
= ( image_2802296252294471259_a_b_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_429_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_430_ball__imageD,axiom,
! [F: product_prod_a_b > a,A3: set_Product_prod_a_b,P2: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
=> ( P2 @ X3 ) )
=> ! [X4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X4 @ A3 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_431_ball__imageD,axiom,
! [F: nat > nat,A3: set_nat,P2: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A3 ) )
=> ( P2 @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_432_rev__image__eqI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,B: a,F: product_prod_a_b > a] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_433_rev__image__eqI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member1426531481828664017od_a_b @ B @ ( image_3300603549555413765od_a_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_434_rev__image__eqI,axiom,
! [X: product_prod_a_b,A3: set_Product_prod_a_b,B: nat,F: product_prod_a_b > nat] :
( ( member1426531481828664017od_a_b @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_5426517477246462771_b_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_435_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: product_prod_a_b,F: nat > product_prod_a_b] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member1426531481828664017od_a_b @ B @ ( image_372941892535967121od_a_b @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_436_rev__image__eqI,axiom,
! [X: nat,A3: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_437_subset__image__iff,axiom,
! [B4: set_a,F: product_prod_a_b > a,A3: set_Product_prod_a_b] :
( ( ord_less_eq_set_a @ B4 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
= ( ? [AA: set_Product_prod_a_b] :
( ( ord_le817736998455962536od_a_b @ AA @ A3 )
& ( B4
= ( image_2802296252294471259_a_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_438_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B4
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_439_image__subset__iff,axiom,
! [F: product_prod_a_b > a,A3: set_Product_prod_a_b,B4: set_a] :
( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A3 ) @ B4 )
= ( ! [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_440_image__subset__iff,axiom,
! [F: nat > nat,A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_441_subset__imageE,axiom,
! [B4: set_a,F: product_prod_a_b > a,A3: set_Product_prod_a_b] :
( ( ord_less_eq_set_a @ B4 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
=> ~ ! [C4: set_Product_prod_a_b] :
( ( ord_le817736998455962536od_a_b @ C4 @ A3 )
=> ( B4
!= ( image_2802296252294471259_a_b_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_442_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
=> ( B4
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_443_image__subsetI,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > a,B4: set_a] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_444_image__subsetI,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > product_prod_a_b,B4: set_Product_prod_a_b] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le817736998455962536od_a_b @ ( image_3300603549555413765od_a_b @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_445_image__subsetI,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > nat,B4: set_nat] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_5426517477246462771_b_nat @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_446_image__subsetI,axiom,
! [A3: set_nat,F: nat > product_prod_a_b,B4: set_Product_prod_a_b] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le817736998455962536od_a_b @ ( image_372941892535967121od_a_b @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_447_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_448_image__mono,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,F: product_prod_a_b > a] :
( ( ord_le817736998455962536od_a_b @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A3 ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_449_image__mono,axiom,
! [A3: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_450_Suc__inject,axiom,
! [X: nat,Y5: nat] :
( ( ( suc @ X )
= ( suc @ Y5 ) )
=> ( X = Y5 ) ) ).
% Suc_inject
thf(fact_451_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_452_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_453_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_454_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_455_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_456_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_457_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_458_sum_Ocong,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,G: product_prod_a_b > nat,H: product_prod_a_b > nat] :
( ( A3 = B4 )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7004838470036706110_b_nat @ G @ A3 )
= ( groups7004838470036706110_b_nat @ H @ B4 ) ) ) ) ).
% sum.cong
thf(fact_459_sum_Oeq__general,axiom,
! [B4: set_Product_prod_a_b,A3: set_nat,H: nat > product_prod_a_b,Gamma: product_prod_a_b > nat,Phi: nat > nat] :
( ! [Y3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Y3 @ B4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: nat] :
( ( ( member_nat @ Ya2 @ A3 )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( member1426531481828664017od_a_b @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A3 )
= ( groups7004838470036706110_b_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_460_sum_Oeq__general,axiom,
! [B4: set_nat,A3: set_Product_prod_a_b,H: product_prod_a_b > nat,Gamma: nat > nat,Phi: product_prod_a_b > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ? [X4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X4 @ A3 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: product_prod_a_b] :
( ( ( member1426531481828664017od_a_b @ Ya2 @ A3 )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7004838470036706110_b_nat @ Phi @ A3 )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_461_sum_Oeq__general,axiom,
! [B4: set_Product_prod_a_b,A3: set_Product_prod_a_b,H: product_prod_a_b > product_prod_a_b,Gamma: product_prod_a_b > nat,Phi: product_prod_a_b > nat] :
( ! [Y3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Y3 @ B4 )
=> ? [X4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X4 @ A3 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: product_prod_a_b] :
( ( ( member1426531481828664017od_a_b @ Ya2 @ A3 )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ( member1426531481828664017od_a_b @ ( H @ X3 ) @ B4 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7004838470036706110_b_nat @ Phi @ A3 )
= ( groups7004838470036706110_b_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general
thf(fact_462_sum_Oeq__general__inverses,axiom,
! [B4: set_Product_prod_a_b,K2: product_prod_a_b > nat,A3: set_nat,H: nat > product_prod_a_b,Gamma: product_prod_a_b > nat,Phi: nat > nat] :
( ! [Y3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Y3 @ B4 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A3 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( member1426531481828664017od_a_b @ ( H @ X3 ) @ B4 )
& ( ( K2 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A3 )
= ( groups7004838470036706110_b_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_463_sum_Oeq__general__inverses,axiom,
! [B4: set_nat,K2: nat > product_prod_a_b,A3: set_Product_prod_a_b,H: product_prod_a_b > nat,Gamma: nat > nat,Phi: product_prod_a_b > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( ( member1426531481828664017od_a_b @ ( K2 @ Y3 ) @ A3 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ( member_nat @ ( H @ X3 ) @ B4 )
& ( ( K2 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7004838470036706110_b_nat @ Phi @ A3 )
= ( groups3542108847815614940at_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_464_sum_Oeq__general__inverses,axiom,
! [B4: set_Product_prod_a_b,K2: product_prod_a_b > product_prod_a_b,A3: set_Product_prod_a_b,H: product_prod_a_b > product_prod_a_b,Gamma: product_prod_a_b > nat,Phi: product_prod_a_b > nat] :
( ! [Y3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ Y3 @ B4 )
=> ( ( member1426531481828664017od_a_b @ ( K2 @ Y3 ) @ A3 )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ( member1426531481828664017od_a_b @ ( H @ X3 ) @ B4 )
& ( ( K2 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7004838470036706110_b_nat @ Phi @ A3 )
= ( groups7004838470036706110_b_nat @ Gamma @ B4 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_465_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: product_prod_a_b > nat,J2: nat > product_prod_a_b,T4: set_Product_prod_a_b,H: product_prod_a_b > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member1426531481828664017od_a_b @ ( J2 @ A4 ) @ T4 ) )
=> ( ! [B3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ B3 @ T4 )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ B3 @ T4 )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups7004838470036706110_b_nat @ H @ T4 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_466_sum_Oreindex__bij__witness,axiom,
! [S: set_Product_prod_a_b,I3: nat > product_prod_a_b,J2: product_prod_a_b > nat,T4: set_nat,H: nat > nat,G: product_prod_a_b > nat] :
( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T4 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( member1426531481828664017od_a_b @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7004838470036706110_b_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T4 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_467_sum_Oreindex__bij__witness,axiom,
! [S: set_Product_prod_a_b,I3: product_prod_a_b > product_prod_a_b,J2: product_prod_a_b > product_prod_a_b,T4: set_Product_prod_a_b,H: product_prod_a_b > nat,G: product_prod_a_b > nat] :
( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( member1426531481828664017od_a_b @ ( J2 @ A4 ) @ T4 ) )
=> ( ! [B3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ B3 @ T4 )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ B3 @ T4 )
=> ( member1426531481828664017od_a_b @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7004838470036706110_b_nat @ G @ S )
= ( groups7004838470036706110_b_nat @ H @ T4 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_468_imageE,axiom,
! [B: a,F: product_prod_a_b > a,A3: set_Product_prod_a_b] :
( ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
=> ~ ! [X3: product_prod_a_b] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531481828664017od_a_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_469_imageE,axiom,
! [B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b,A3: set_Product_prod_a_b] :
( ( member1426531481828664017od_a_b @ B @ ( image_3300603549555413765od_a_b @ F @ A3 ) )
=> ~ ! [X3: product_prod_a_b] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531481828664017od_a_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_470_imageE,axiom,
! [B: product_prod_a_b,F: nat > product_prod_a_b,A3: set_nat] :
( ( member1426531481828664017od_a_b @ B @ ( image_372941892535967121od_a_b @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_471_imageE,axiom,
! [B: nat,F: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( member_nat @ B @ ( image_5426517477246462771_b_nat @ F @ A3 ) )
=> ~ ! [X3: product_prod_a_b] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1426531481828664017od_a_b @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_472_imageE,axiom,
! [B: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_473_image__image,axiom,
! [F: a > a,G: product_prod_a_b > a,A3: set_Product_prod_a_b] :
( ( image_a_a @ F @ ( image_2802296252294471259_a_b_a @ G @ A3 ) )
= ( image_2802296252294471259_a_b_a
@ ^ [X2: product_prod_a_b] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_474_image__image,axiom,
! [F: product_prod_a_b > a,G: product_prod_a_b > product_prod_a_b,A3: set_Product_prod_a_b] :
( ( image_2802296252294471259_a_b_a @ F @ ( image_3300603549555413765od_a_b @ G @ A3 ) )
= ( image_2802296252294471259_a_b_a
@ ^ [X2: product_prod_a_b] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_475_image__image,axiom,
! [F: nat > nat,G: nat > nat,A3: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A3 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_476_Compr__image__eq,axiom,
! [F: nat > nat,A3: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_477_Compr__image__eq,axiom,
! [F: product_prod_a_b > a,A3: set_Product_prod_a_b,P2: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2802296252294471259_a_b_a @ F
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_478_Compr__image__eq,axiom,
! [F: nat > product_prod_a_b,A3: set_nat,P2: product_prod_a_b > $o] :
( ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ ( image_372941892535967121od_a_b @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_372941892535967121od_a_b @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_479_Compr__image__eq,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_5426517477246462771_b_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5426517477246462771_b_nat @ F
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_480_Compr__image__eq,axiom,
! [F: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_2486076414777270412at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2486076414777270412at_nat @ F
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_481_Compr__image__eq,axiom,
! [F: product_prod_int_int > nat,A3: set_Pr958786334691620121nt_int,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_5044651549707136836nt_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5044651549707136836nt_nat @ F
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_482_Compr__image__eq,axiom,
! [F: nat > product_prod_nat_nat,A3: set_nat,P2: product_prod_nat_nat > $o] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( image_5846123807819985514at_nat @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_5846123807819985514at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_483_Compr__image__eq,axiom,
! [F: nat > product_prod_int_int,A3: set_nat,P2: product_prod_int_int > $o] :
( ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ ( image_2667626500211843362nt_int @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_2667626500211843362nt_int @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_484_Compr__image__eq,axiom,
! [F: product_prod_a_b > product_prod_a_b,A3: set_Product_prod_a_b,P2: product_prod_a_b > $o] :
( ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ ( image_3300603549555413765od_a_b @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_3300603549555413765od_a_b @ F
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_485_Compr__image__eq,axiom,
! [F: product_prod_nat_nat > product_prod_a_b,A3: set_Pr1261947904930325089at_nat,P2: product_prod_a_b > $o] :
( ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ ( image_4894260360327267052od_a_b @ F @ A3 ) )
& ( P2 @ X2 ) ) )
= ( image_4894260360327267052od_a_b @ F
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A3 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_486_sum_Oswap,axiom,
! [G: product_prod_a_b > product_prod_a_b > nat,B4: set_Product_prod_a_b,A3: set_Product_prod_a_b] :
( ( groups7004838470036706110_b_nat
@ ^ [I: product_prod_a_b] : ( groups7004838470036706110_b_nat @ ( G @ I ) @ B4 )
@ A3 )
= ( groups7004838470036706110_b_nat
@ ^ [J: product_prod_a_b] :
( groups7004838470036706110_b_nat
@ ^ [I: product_prod_a_b] : ( G @ I @ J )
@ A3 )
@ B4 ) ) ).
% sum.swap
thf(fact_487_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_488_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_489_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_490_Suc__le__D,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_491_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_492_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_493_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_494_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P2 @ M5 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_495_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P2 @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_496_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_497_image__Collect__subsetI,axiom,
! [P2: product_prod_a_b > $o,F: product_prod_a_b > a,B4: set_a] :
( ! [X3: product_prod_a_b] :
( ( P2 @ X3 )
=> ( member_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ ( collec3336397801687681299od_a_b @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_498_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > nat,B4: set_nat] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_499_image__Collect__subsetI,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_a_b,B4: set_Product_prod_a_b] :
( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
=> ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le817736998455962536od_a_b @ ( image_4894260360327267052od_a_b @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_500_image__Collect__subsetI,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat,B4: set_nat] :
( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_501_image__Collect__subsetI,axiom,
! [P2: product_prod_int_int > $o,F: product_prod_int_int > product_prod_a_b,B4: set_Product_prod_a_b] :
( ! [X3: product_prod_int_int] :
( ( P2 @ X3 )
=> ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
=> ( ord_le817736998455962536od_a_b @ ( image_2370028551316859444od_a_b @ F @ ( collec213857154873943460nt_int @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_502_image__Collect__subsetI,axiom,
! [P2: product_prod_int_int > $o,F: product_prod_int_int > nat,B4: set_nat] :
( ! [X3: product_prod_int_int] :
( ( P2 @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_5044651549707136836nt_nat @ F @ ( collec213857154873943460nt_int @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_503_eq__fst__iff,axiom,
! [A: nat,P: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P ) )
= ( ? [B6: nat] :
( P
= ( product_Pair_nat_nat @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_504_eq__fst__iff,axiom,
! [A: int,P: product_prod_int_int] :
( ( A
= ( product_fst_int_int @ P ) )
= ( ? [B6: int] :
( P
= ( product_Pair_int_int @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_505_eq__fst__iff,axiom,
! [A: a,P: product_prod_a_b] :
( ( A
= ( product_fst_a_b @ P ) )
= ( ? [B6: b] :
( P
= ( product_Pair_a_b @ A @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_506_fstI,axiom,
! [X: product_prod_nat_nat,Y5: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y5 @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y5 ) ) ).
% fstI
thf(fact_507_fstI,axiom,
! [X: product_prod_int_int,Y5: int,Z: int] :
( ( X
= ( product_Pair_int_int @ Y5 @ Z ) )
=> ( ( product_fst_int_int @ X )
= Y5 ) ) ).
% fstI
thf(fact_508_fstI,axiom,
! [X: product_prod_a_b,Y5: a,Z: b] :
( ( X
= ( product_Pair_a_b @ Y5 @ Z ) )
=> ( ( product_fst_a_b @ X )
= Y5 ) ) ).
% fstI
thf(fact_509_all__subset__image,axiom,
! [F: product_prod_a_b > a,A3: set_Product_prod_a_b,P2: set_a > $o] :
( ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ ( image_2802296252294471259_a_b_a @ F @ A3 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_Product_prod_a_b] :
( ( ord_le817736998455962536od_a_b @ B7 @ A3 )
=> ( P2 @ ( image_2802296252294471259_a_b_a @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_510_all__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P2: set_nat > $o] :
( ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F @ A3 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A3 )
=> ( P2 @ ( image_nat_nat @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_511_sum__img__lt,axiom,
! [Xs: fset_P5670320511379867111at_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( fset_P3501481629439712240at_nat @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( fset_P3501481629439712240at_nat @ Xs ) ) )
& ( ord_less_nat @ ( G @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( fset_P3501481629439712240at_nat @ Xs ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ord_less_nat
@ ( groups977919841031483927at_nat
@ ( produc6842872674320459806at_nat
@ ^ [X2: nat,Y4: nat] : ( G @ X2 ) )
@ ( fset_P3501481629439712240at_nat
@ ( fimage1537860586630726333at_nat
@ ( produc2626176000494625587at_nat
@ ^ [T2: nat] : ( product_Pair_nat_nat @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups977919841031483927at_nat
@ ( produc6842872674320459806at_nat
@ ^ [X2: nat,Y4: nat] : ( G @ X2 ) )
@ ( fset_P3501481629439712240at_nat @ Xs ) ) ) ) ) ) ).
% sum_img_lt
thf(fact_512_sum__img__lt,axiom,
! [Xs: fset_P5367158941141162143nt_int,G: int > nat,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( fset_P322984321831570088nt_int @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ? [X4: int] :
( ( member_int @ X4 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( fset_P322984321831570088nt_int @ Xs ) ) )
& ( ord_less_nat @ ( G @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ ( fset_P322984321831570088nt_int @ Xs ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ord_less_nat
@ ( groups3536494975961350351nt_nat
@ ( produc8213879946458358998nt_nat
@ ^ [X2: int,Y4: int] : ( G @ X2 ) )
@ ( fset_P322984321831570088nt_int
@ ( fimage8245688998986472637nt_int
@ ( produc4245557441103728435nt_int
@ ^ [T2: int] : ( product_Pair_int_int @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups3536494975961350351nt_nat
@ ( produc8213879946458358998nt_nat
@ ^ [X2: int,Y4: int] : ( G @ X2 ) )
@ ( fset_P322984321831570088nt_int @ Xs ) ) ) ) ) ) ).
% sum_img_lt
thf(fact_513_sum__img__lt,axiom,
! [Xs: fset_P9214369701362650254od_a_b,G: a > nat,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ Xs ) ) )
=> ( ord_less_eq_nat @ ( G @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ Xs ) ) )
& ( ord_less_nat @ ( G @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( fset_P2369346149119917079od_a_b @ Xs ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ord_less_nat
@ ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( G @ X2 ) )
@ ( fset_P2369346149119917079od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( F @ T2 ) ) )
@ Xs ) ) )
@ ( groups7004838470036706110_b_nat
@ ( produc5088076833887504125_b_nat
@ ^ [X2: a,Y4: b] : ( G @ X2 ) )
@ ( fset_P2369346149119917079od_a_b @ Xs ) ) ) ) ) ) ).
% sum_img_lt
thf(fact_514_conj__subset__def,axiom,
! [A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ A3
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) )
& ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_515_conj__subset__def,axiom,
! [A3: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ A3
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( P2 @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_le2843351958646193337nt_int @ A3 @ ( collec213857154873943460nt_int @ P2 ) )
& ( ord_le2843351958646193337nt_int @ A3 @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_516_prop__restrict,axiom,
! [X: product_prod_a_b,Z4: set_Product_prod_a_b,X5: set_Product_prod_a_b,P2: product_prod_a_b > $o] :
( ( member1426531481828664017od_a_b @ X @ Z4 )
=> ( ( ord_le817736998455962536od_a_b @ Z4
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_517_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_518_prop__restrict,axiom,
! [X: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X @ Z4 )
=> ( ( ord_le3146513528884898305at_nat @ Z4
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_519_prop__restrict,axiom,
! [X: product_prod_int_int,Z4: set_Pr958786334691620121nt_int,X5: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ X @ Z4 )
=> ( ( ord_le2843351958646193337nt_int @ Z4
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ X5 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_520_Collect__restrict,axiom,
! [X5: set_Product_prod_a_b,P2: product_prod_a_b > $o] :
( ord_le817736998455962536od_a_b
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_521_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_522_Collect__restrict,axiom,
! [X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_523_Collect__restrict,axiom,
! [X5: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ X5 )
& ( P2 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_524_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_525_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_526_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_527_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_528_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_529_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_530_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_531_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_532_sum_Oneutral__const,axiom,
! [A3: set_Product_prod_a_b] :
( ( groups7004838470036706110_b_nat
@ ^ [Uu: product_prod_a_b] : zero_zero_nat
@ A3 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_533_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_534_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_535_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_536_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_537_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_538_All__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_539_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M6: nat] :
( N2
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_540_Ex__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_541_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_542_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_543_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_544_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_545_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_546_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_547_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_548_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_549_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_550_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_551_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_552_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_553_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_554_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_555_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_556_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_557_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_558_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P2 @ M5 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_559_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P2 @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P2 @ M5 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_560_infinite__descent0,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P2 @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P2 @ M5 ) ) ) )
=> ( P2 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_561_linorder__neqE__nat,axiom,
! [X: nat,Y5: nat] :
( ( X != Y5 )
=> ( ~ ( ord_less_nat @ X @ Y5 )
=> ( ord_less_nat @ Y5 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_562_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_563_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_564_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_565_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_566_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_567_dense,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y5 ) ) ) ).
% dense
thf(fact_568_less__imp__neq,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( X != Y5 ) ) ).
% less_imp_neq
thf(fact_569_less__imp__neq,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ( X != Y5 ) ) ).
% less_imp_neq
thf(fact_570_less__imp__neq,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ( X != Y5 ) ) ).
% less_imp_neq
thf(fact_571_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_572_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_573_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_574_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_575_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_576_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_577_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_578_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_579_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_580_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X3 )
=> ( P2 @ Y7 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_581_antisym__conv3,axiom,
! [Y5: nat,X: nat] :
( ~ ( ord_less_nat @ Y5 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv3
thf(fact_582_antisym__conv3,axiom,
! [Y5: int,X: int] :
( ~ ( ord_less_int @ Y5 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv3
thf(fact_583_antisym__conv3,axiom,
! [Y5: real,X: real] :
( ~ ( ord_less_real @ Y5 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv3
thf(fact_584_linorder__cases,axiom,
! [X: nat,Y5: nat] :
( ~ ( ord_less_nat @ X @ Y5 )
=> ( ( X != Y5 )
=> ( ord_less_nat @ Y5 @ X ) ) ) ).
% linorder_cases
thf(fact_585_linorder__cases,axiom,
! [X: int,Y5: int] :
( ~ ( ord_less_int @ X @ Y5 )
=> ( ( X != Y5 )
=> ( ord_less_int @ Y5 @ X ) ) ) ).
% linorder_cases
thf(fact_586_linorder__cases,axiom,
! [X: real,Y5: real] :
( ~ ( ord_less_real @ X @ Y5 )
=> ( ( X != Y5 )
=> ( ord_less_real @ Y5 @ X ) ) ) ).
% linorder_cases
thf(fact_587_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_588_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_589_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_590_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_591_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_592_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_593_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N5: nat] :
( ( P4 @ N5 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N5 )
=> ~ ( P4 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_594_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_595_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P2 @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_596_linorder__less__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P2 @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_597_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_598_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_599_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_600_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y5: nat] :
( ( ~ ( ord_less_nat @ X @ Y5 ) )
= ( ( ord_less_nat @ Y5 @ X )
| ( X = Y5 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_601_not__less__iff__gr__or__eq,axiom,
! [X: int,Y5: int] :
( ( ~ ( ord_less_int @ X @ Y5 ) )
= ( ( ord_less_int @ Y5 @ X )
| ( X = Y5 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_602_not__less__iff__gr__or__eq,axiom,
! [X: real,Y5: real] :
( ( ~ ( ord_less_real @ X @ Y5 ) )
= ( ( ord_less_real @ Y5 @ X )
| ( X = Y5 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_603_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_604_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_605_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_606_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_607_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_608_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_609_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_610_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_611_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_612_linorder__neqE,axiom,
! [X: nat,Y5: nat] :
( ( X != Y5 )
=> ( ~ ( ord_less_nat @ X @ Y5 )
=> ( ord_less_nat @ Y5 @ X ) ) ) ).
% linorder_neqE
thf(fact_613_linorder__neqE,axiom,
! [X: int,Y5: int] :
( ( X != Y5 )
=> ( ~ ( ord_less_int @ X @ Y5 )
=> ( ord_less_int @ Y5 @ X ) ) ) ).
% linorder_neqE
thf(fact_614_linorder__neqE,axiom,
! [X: real,Y5: real] :
( ( X != Y5 )
=> ( ~ ( ord_less_real @ X @ Y5 )
=> ( ord_less_real @ Y5 @ X ) ) ) ).
% linorder_neqE
thf(fact_615_order__less__asym,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X ) ) ).
% order_less_asym
thf(fact_616_order__less__asym,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ~ ( ord_less_int @ Y5 @ X ) ) ).
% order_less_asym
thf(fact_617_order__less__asym,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ~ ( ord_less_real @ Y5 @ X ) ) ).
% order_less_asym
thf(fact_618_linorder__neq__iff,axiom,
! [X: nat,Y5: nat] :
( ( X != Y5 )
= ( ( ord_less_nat @ X @ Y5 )
| ( ord_less_nat @ Y5 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_619_linorder__neq__iff,axiom,
! [X: int,Y5: int] :
( ( X != Y5 )
= ( ( ord_less_int @ X @ Y5 )
| ( ord_less_int @ Y5 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_620_linorder__neq__iff,axiom,
! [X: real,Y5: real] :
( ( X != Y5 )
= ( ( ord_less_real @ X @ Y5 )
| ( ord_less_real @ Y5 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_621_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_622_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_623_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_624_order__less__trans,axiom,
! [X: nat,Y5: nat,Z: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( ( ord_less_nat @ Y5 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_625_order__less__trans,axiom,
! [X: int,Y5: int,Z: int] :
( ( ord_less_int @ X @ Y5 )
=> ( ( ord_less_int @ Y5 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_626_order__less__trans,axiom,
! [X: real,Y5: real,Z: real] :
( ( ord_less_real @ X @ Y5 )
=> ( ( ord_less_real @ Y5 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_627_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_628_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_629_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_630_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_631_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_632_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_633_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_634_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_635_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_636_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_637_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_638_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_639_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_640_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_641_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_642_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_643_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_644_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_645_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_646_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_647_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_648_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_649_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_650_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_651_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_652_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_653_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_654_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_655_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_656_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_657_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_658_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_659_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_660_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_661_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_662_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_663_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_664_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_665_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_666_order__less__not__sym,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X ) ) ).
% order_less_not_sym
thf(fact_667_order__less__not__sym,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ~ ( ord_less_int @ Y5 @ X ) ) ).
% order_less_not_sym
thf(fact_668_order__less__not__sym,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ~ ( ord_less_real @ Y5 @ X ) ) ).
% order_less_not_sym
thf(fact_669_order__less__imp__triv,axiom,
! [X: nat,Y5: nat,P2: $o] :
( ( ord_less_nat @ X @ Y5 )
=> ( ( ord_less_nat @ Y5 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_670_order__less__imp__triv,axiom,
! [X: int,Y5: int,P2: $o] :
( ( ord_less_int @ X @ Y5 )
=> ( ( ord_less_int @ Y5 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_671_order__less__imp__triv,axiom,
! [X: real,Y5: real,P2: $o] :
( ( ord_less_real @ X @ Y5 )
=> ( ( ord_less_real @ Y5 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_672_linorder__less__linear,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
| ( X = Y5 )
| ( ord_less_nat @ Y5 @ X ) ) ).
% linorder_less_linear
thf(fact_673_linorder__less__linear,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
| ( X = Y5 )
| ( ord_less_int @ Y5 @ X ) ) ).
% linorder_less_linear
thf(fact_674_linorder__less__linear,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
| ( X = Y5 )
| ( ord_less_real @ Y5 @ X ) ) ).
% linorder_less_linear
thf(fact_675_order__less__imp__not__eq,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( X != Y5 ) ) ).
% order_less_imp_not_eq
thf(fact_676_order__less__imp__not__eq,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ( X != Y5 ) ) ).
% order_less_imp_not_eq
thf(fact_677_order__less__imp__not__eq,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ( X != Y5 ) ) ).
% order_less_imp_not_eq
thf(fact_678_order__less__imp__not__eq2,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( Y5 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_679_order__less__imp__not__eq2,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ( Y5 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_680_order__less__imp__not__eq2,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ( Y5 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_681_order__less__imp__not__less,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ~ ( ord_less_nat @ Y5 @ X ) ) ).
% order_less_imp_not_less
thf(fact_682_order__less__imp__not__less,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ~ ( ord_less_int @ Y5 @ X ) ) ).
% order_less_imp_not_less
thf(fact_683_order__less__imp__not__less,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ~ ( ord_less_real @ Y5 @ X ) ) ).
% order_less_imp_not_less
thf(fact_684_ex__least__nat__less,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_685_sum__SucD,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b,N2: nat] :
( ( ( groups7004838470036706110_b_nat @ F @ A3 )
= ( suc @ N2 ) )
=> ? [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_686_order__le__imp__less__or__eq,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ord_less_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_687_order__le__imp__less__or__eq,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ( ord_less_int @ X @ Y5 )
| ( X = Y5 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_688_order__le__imp__less__or__eq,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
=> ( ( ord_less_real @ X @ Y5 )
| ( X = Y5 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_689_linorder__le__less__linear,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
| ( ord_less_nat @ Y5 @ X ) ) ).
% linorder_le_less_linear
thf(fact_690_linorder__le__less__linear,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
| ( ord_less_int @ Y5 @ X ) ) ).
% linorder_le_less_linear
thf(fact_691_linorder__le__less__linear,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
| ( ord_less_real @ Y5 @ X ) ) ).
% linorder_le_less_linear
thf(fact_692_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_693_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_694_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_695_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_696_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_697_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_698_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_699_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_700_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_701_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_702_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_703_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_704_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_706_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_707_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_708_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_709_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_710_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_711_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_712_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_713_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_714_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_715_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_716_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_717_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_718_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_719_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_720_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_721_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_722_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_723_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_724_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_725_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_726_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_727_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_728_order__less__le__trans,axiom,
! [X: nat,Y5: nat,Z: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_729_order__less__le__trans,axiom,
! [X: int,Y5: int,Z: int] :
( ( ord_less_int @ X @ Y5 )
=> ( ( ord_less_eq_int @ Y5 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_730_order__less__le__trans,axiom,
! [X: real,Y5: real,Z: real] :
( ( ord_less_real @ X @ Y5 )
=> ( ( ord_less_eq_real @ Y5 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_731_order__le__less__trans,axiom,
! [X: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ord_less_nat @ Y5 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_732_order__le__less__trans,axiom,
! [X: int,Y5: int,Z: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ( ord_less_int @ Y5 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_733_order__le__less__trans,axiom,
! [X: real,Y5: real,Z: real] :
( ( ord_less_eq_real @ X @ Y5 )
=> ( ( ord_less_real @ Y5 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_734_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_735_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_736_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_737_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_738_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_739_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_740_order__less__imp__le,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ X @ Y5 ) ) ).
% order_less_imp_le
thf(fact_741_order__less__imp__le,axiom,
! [X: int,Y5: int] :
( ( ord_less_int @ X @ Y5 )
=> ( ord_less_eq_int @ X @ Y5 ) ) ).
% order_less_imp_le
thf(fact_742_order__less__imp__le,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ X @ Y5 )
=> ( ord_less_eq_real @ X @ Y5 ) ) ).
% order_less_imp_le
thf(fact_743_linorder__not__less,axiom,
! [X: nat,Y5: nat] :
( ( ~ ( ord_less_nat @ X @ Y5 ) )
= ( ord_less_eq_nat @ Y5 @ X ) ) ).
% linorder_not_less
thf(fact_744_linorder__not__less,axiom,
! [X: int,Y5: int] :
( ( ~ ( ord_less_int @ X @ Y5 ) )
= ( ord_less_eq_int @ Y5 @ X ) ) ).
% linorder_not_less
thf(fact_745_linorder__not__less,axiom,
! [X: real,Y5: real] :
( ( ~ ( ord_less_real @ X @ Y5 ) )
= ( ord_less_eq_real @ Y5 @ X ) ) ).
% linorder_not_less
thf(fact_746_linorder__not__le,axiom,
! [X: nat,Y5: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y5 ) )
= ( ord_less_nat @ Y5 @ X ) ) ).
% linorder_not_le
thf(fact_747_linorder__not__le,axiom,
! [X: int,Y5: int] :
( ( ~ ( ord_less_eq_int @ X @ Y5 ) )
= ( ord_less_int @ Y5 @ X ) ) ).
% linorder_not_le
thf(fact_748_linorder__not__le,axiom,
! [X: real,Y5: real] :
( ( ~ ( ord_less_eq_real @ X @ Y5 ) )
= ( ord_less_real @ Y5 @ X ) ) ).
% linorder_not_le
thf(fact_749_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_750_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_751_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_752_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_753_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_int @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_754_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_755_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_756_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_757_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_758_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_759_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_760_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_761_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_762_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ~ ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_763_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B6: real,A6: real] :
( ( ord_less_eq_real @ B6 @ A6 )
& ~ ( ord_less_eq_real @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_764_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_765_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_766_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_767_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_768_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_769_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_770_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_771_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_772_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B6: real,A6: real] :
( ( ord_less_eq_real @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_773_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_nat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_774_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_int @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_775_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B6: real,A6: real] :
( ( ord_less_real @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_776_dense__le__bounded,axiom,
! [X: real,Y5: real,Z: real] :
( ( ord_less_real @ X @ Y5 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y5 )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y5 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_777_dense__ge__bounded,axiom,
! [Z: real,X: real,Y5: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y5 @ W ) ) )
=> ( ord_less_eq_real @ Y5 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_778_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_779_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ~ ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_780_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
& ~ ( ord_less_eq_real @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_781_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_782_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_783_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_784_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_785_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_786_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_787_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_788_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_789_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A6: real,B6: real] :
( ( ord_less_eq_real @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_790_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_791_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_792_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B6: real] :
( ( ord_less_real @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_793_not__le__imp__less,axiom,
! [Y5: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y5 @ X )
=> ( ord_less_nat @ X @ Y5 ) ) ).
% not_le_imp_less
thf(fact_794_not__le__imp__less,axiom,
! [Y5: int,X: int] :
( ~ ( ord_less_eq_int @ Y5 @ X )
=> ( ord_less_int @ X @ Y5 ) ) ).
% not_le_imp_less
thf(fact_795_not__le__imp__less,axiom,
! [Y5: real,X: real] :
( ~ ( ord_less_eq_real @ Y5 @ X )
=> ( ord_less_real @ X @ Y5 ) ) ).
% not_le_imp_less
thf(fact_796_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_797_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y4: int] :
( ( ord_less_eq_int @ X2 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_798_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_799_dense__le,axiom,
! [Y5: real,Z: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y5 @ Z ) ) ).
% dense_le
thf(fact_800_dense__ge,axiom,
! [Z: real,Y5: real] :
( ! [X3: real] :
( ( ord_less_real @ Z @ X3 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( ord_less_eq_real @ Y5 @ Z ) ) ).
% dense_ge
thf(fact_801_antisym__conv2,axiom,
! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ~ ( ord_less_nat @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv2
thf(fact_802_antisym__conv2,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ( ~ ( ord_less_int @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv2
thf(fact_803_antisym__conv2,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ X @ Y5 )
=> ( ( ~ ( ord_less_real @ X @ Y5 ) )
= ( X = Y5 ) ) ) ).
% antisym_conv2
thf(fact_804_antisym__conv1,axiom,
! [X: nat,Y5: nat] :
( ~ ( ord_less_nat @ X @ Y5 )
=> ( ( ord_less_eq_nat @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% antisym_conv1
thf(fact_805_antisym__conv1,axiom,
! [X: int,Y5: int] :
( ~ ( ord_less_int @ X @ Y5 )
=> ( ( ord_less_eq_int @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% antisym_conv1
thf(fact_806_antisym__conv1,axiom,
! [X: real,Y5: real] :
( ~ ( ord_less_real @ X @ Y5 )
=> ( ( ord_less_eq_real @ X @ Y5 )
= ( X = Y5 ) ) ) ).
% antisym_conv1
thf(fact_807_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_808_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_809_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_810_leI,axiom,
! [X: nat,Y5: nat] :
( ~ ( ord_less_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ).
% leI
thf(fact_811_leI,axiom,
! [X: int,Y5: int] :
( ~ ( ord_less_int @ X @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X ) ) ).
% leI
thf(fact_812_leI,axiom,
! [X: real,Y5: real] :
( ~ ( ord_less_real @ X @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X ) ) ).
% leI
thf(fact_813_leD,axiom,
! [Y5: nat,X: nat] :
( ( ord_less_eq_nat @ Y5 @ X )
=> ~ ( ord_less_nat @ X @ Y5 ) ) ).
% leD
thf(fact_814_leD,axiom,
! [Y5: int,X: int] :
( ( ord_less_eq_int @ Y5 @ X )
=> ~ ( ord_less_int @ X @ Y5 ) ) ).
% leD
thf(fact_815_leD,axiom,
! [Y5: real,X: real] :
( ( ord_less_eq_real @ Y5 @ X )
=> ~ ( ord_less_real @ X @ Y5 ) ) ).
% leD
thf(fact_816_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_817_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P2: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_818_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P2 @ I2 @ J3 )
=> ( ( P2 @ J3 @ K3 )
=> ( P2 @ I2 @ K3 ) ) ) ) )
=> ( P2 @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_819_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_820_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_821_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_822_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_823_All__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_824_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_825_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_826_Ex__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_827_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_828_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_829_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_830_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_831_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_832_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_833_nat__descend__induct,axiom,
! [N2: nat,P2: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P2 @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K3 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_834_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_835_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_836_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_837_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N5: nat] :
( ( ord_less_nat @ M6 @ N5 )
| ( M6 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_838_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_839_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
& ( M6 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_840_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A3: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A3 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_841_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: product_prod_a_b > int,A3: set_Product_prod_a_b] :
( ( ( groups7002347999527655834_b_int @ G @ A3 )
!= zero_zero_int )
=> ~ ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_842_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A3: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A3 )
!= zero_zero_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_843_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: product_prod_a_b > real,A3: set_Product_prod_a_b] :
( ( ( groups1618960153307602714b_real @ G @ A3 )
!= zero_zero_real )
=> ~ ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_844_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A3: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A3 )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_845_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( ( groups7004838470036706110_b_nat @ G @ A3 )
!= zero_zero_nat )
=> ~ ! [A4: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ A4 @ A3 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_846_sum_Oneutral,axiom,
! [A3: set_Product_prod_a_b,G: product_prod_a_b > nat] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups7004838470036706110_b_nat @ G @ A3 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_847_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_848_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_849_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_850_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_851_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N4: nat] :
( ( P2 @ ( suc @ N4 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_852_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_853_nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) )
=> ( P2 @ N2 ) ) ) ).
% nat_induct
thf(fact_854_old_Onat_Oexhaust,axiom,
! [Y5: nat] :
( ( Y5 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y5
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_855_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_856_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_857_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_858_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_859_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_860_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_861_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_862_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_863_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_864_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_865_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_866_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_867_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_868_inc__induct,axiom,
! [I3: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I3 @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P2 @ ( suc @ N4 ) )
=> ( P2 @ N4 ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% inc_induct
thf(fact_869_dec__induct,axiom,
! [I3: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P2 @ I3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I3 @ N4 )
=> ( ( ord_less_nat @ N4 @ J2 )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_870_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_871_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_872_sum__nonneg,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_873_sum__nonneg,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > int] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups7002347999527655834_b_int @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_874_sum__nonneg,axiom,
! [A3: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_875_sum__nonneg,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > real] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups1618960153307602714b_real @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_876_sum__nonneg,axiom,
! [A3: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_877_sum__nonneg,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > nat] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups7004838470036706110_b_nat @ F @ A3 ) ) ) ).
% sum_nonneg
thf(fact_878_sum__nonpos,axiom,
! [A3: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_879_sum__nonpos,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > int] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups7002347999527655834_b_int @ F @ A3 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_880_sum__nonpos,axiom,
! [A3: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_881_sum__nonpos,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > real] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups1618960153307602714b_real @ F @ A3 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_882_sum__nonpos,axiom,
! [A3: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A3 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_883_sum__nonpos,axiom,
! [A3: set_Product_prod_a_b,F: product_prod_a_b > nat] :
( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ A3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups7004838470036706110_b_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_884_ssubst__Pair__rhs,axiom,
! [R3: a,S2: b,R: set_Product_prod_a_b,S3: b] :
( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_885_ssubst__Pair__rhs,axiom,
! [R3: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_886_ssubst__Pair__rhs,axiom,
! [R3: int,S2: int,R: set_Pr958786334691620121nt_int,S3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_887_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_888_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_889_size__fset__overloaded__simps,axiom,
( size_size_fset_nat
= ( ^ [X2: fset_nat] :
( groups3542108847815614940at_nat
@ ^ [Y4: nat] : ( suc @ zero_zero_nat )
@ ( fset_nat2 @ X2 ) ) ) ) ).
% size_fset_overloaded_simps
thf(fact_890_size__fset__overloaded__simps,axiom,
( size_s2479735812177618682od_a_b
= ( ^ [X2: fset_P9214369701362650254od_a_b] :
( groups7004838470036706110_b_nat
@ ^ [Y4: product_prod_a_b] : ( suc @ zero_zero_nat )
@ ( fset_P2369346149119917079od_a_b @ X2 ) ) ) ) ).
% size_fset_overloaded_simps
thf(fact_891_zero__notin__Suc__image,axiom,
! [A3: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_892_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_893_size__neq__size__imp__neq,axiom,
! [X: char,Y5: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y5 ) )
=> ( X != Y5 ) ) ).
% size_neq_size_imp_neq
thf(fact_894_pfsubsetD,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b,C: product_prod_a_b] :
( ( ord_le1958214202730575162od_a_b @ A3 @ B4 )
=> ( ( fmembe7738088520663304047od_a_b @ C @ A3 )
=> ( fmembe7738088520663304047od_a_b @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_895_pfsubsetD,axiom,
! [A3: fset_nat,B4: fset_nat,C: nat] :
( ( ord_less_fset_nat @ A3 @ B4 )
=> ( ( fmember_nat @ C @ A3 )
=> ( fmember_nat @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_896_pfsubset__fcard__mono,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( ord_le1958214202730575162od_a_b @ A3 @ B4 )
=> ( ord_less_nat @ ( fcard_8555586198630727417od_a_b @ A3 ) @ ( fcard_8555586198630727417od_a_b @ B4 ) ) ) ).
% pfsubset_fcard_mono
thf(fact_897_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_898_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_899_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_900_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_901_fcard__pfsubset,axiom,
! [A3: fset_P9214369701362650254od_a_b,B4: fset_P9214369701362650254od_a_b] :
( ( ord_le7868007465744610350od_a_b @ A3 @ B4 )
=> ( ( ord_less_nat @ ( fcard_8555586198630727417od_a_b @ A3 ) @ ( fcard_8555586198630727417od_a_b @ B4 ) )
=> ( ord_le1958214202730575162od_a_b @ A3 @ B4 ) ) ) ).
% fcard_pfsubset
thf(fact_902_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_903_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_904_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_905_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_906_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_907_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P2 @ X_12 )
=> ? [N4: nat] :
( ~ ( P2 @ N4 )
& ( P2 @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_908_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_909_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_910_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_911_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_912_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_913_psubsetD,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,C: product_prod_a_b] :
( ( ord_le6891031760732254900od_a_b @ A3 @ B4 )
=> ( ( member1426531481828664017od_a_b @ C @ A3 )
=> ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_914_psubsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_915_less__set__def,axiom,
( ord_le6891031760732254900od_a_b
= ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
( ord_le896731050099966249_a_b_o
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ A5 )
@ ^ [X2: product_prod_a_b] : ( member1426531481828664017od_a_b @ X2 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_916_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B7: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_917_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_918_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_919_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_920_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_921_minf_I8_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_922_minf_I8_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_eq_int @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_923_minf_I8_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_924_minf_I7_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_925_minf_I7_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_int @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_926_minf_I7_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ~ ( ord_less_real @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_927_minf_I5_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_928_minf_I5_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_int @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_929_minf_I5_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_real @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_930_minf_I4_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_931_minf_I4_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_932_minf_I4_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_933_minf_I3_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_934_minf_I3_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_935_minf_I3_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_936_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_937_minf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_938_minf_I2_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_939_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_940_minf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_941_minf_I1_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_942_pinf_I7_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_943_pinf_I7_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_int @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_944_pinf_I7_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_real @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_945_pinf_I5_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_946_pinf_I5_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_int @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_947_pinf_I5_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_real @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_948_pinf_I4_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_949_pinf_I4_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_950_pinf_I4_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_951_pinf_I3_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_952_pinf_I3_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_953_pinf_I3_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_954_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_955_pinf_I2_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_956_pinf_I2_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_957_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_958_pinf_I1_J,axiom,
! [P2: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_959_pinf_I1_J,axiom,
! [P2: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_960_pinf_I6_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_961_pinf_I6_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_962_pinf_I6_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_963_pinf_I8_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_964_pinf_I8_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_eq_int @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_965_pinf_I8_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_966_minf_I6_J,axiom,
! [T3: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_967_minf_I6_J,axiom,
! [T3: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_eq_int @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_968_minf_I6_J,axiom,
! [T3: real] :
? [Z2: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z2 )
=> ( ord_less_eq_real @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_969_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_970_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C6: nat] :
( ( ord_less_eq_nat @ A @ C6 )
& ( ord_less_eq_nat @ C6 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C6 ) )
=> ( P2 @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C6 ) ) ) ) ) ) ).
% complete_interval
thf(fact_971_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C6: int] :
( ( ord_less_eq_int @ A @ C6 )
& ( ord_less_eq_int @ C6 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C6 ) )
=> ( P2 @ X4 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C6 ) ) ) ) ) ) ).
% complete_interval
thf(fact_972_complete__interval,axiom,
! [A: real,B: real,P2: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C6: real] :
( ( ord_less_eq_real @ A @ C6 )
& ( ord_less_eq_real @ C6 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C6 ) )
=> ( P2 @ X4 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C6 ) ) ) ) ) ) ).
% complete_interval
thf(fact_973_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_974_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_975_verit__comp__simplify1_I3_J,axiom,
! [B2: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_976_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_977_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_978_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_979_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_980_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_981_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_982_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_983_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_984_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_985_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_986_mlex__eq,axiom,
( mlex_prod_nat
= ( ^ [F2: nat > nat,R4: set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
| ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_987_mlex__eq,axiom,
( mlex_prod_int
= ( ^ [F2: int > nat,R4: set_Pr958786334691620121nt_int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X2: int,Y4: int] :
( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
| ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_988_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X2 )
@ ^ [X2: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_989_subset__Collect__iff,axiom,
! [B4: set_Product_prod_a_b,A3: set_Product_prod_a_b,P2: product_prod_a_b > $o] :
( ( ord_le817736998455962536od_a_b @ B4 @ A3 )
=> ( ( ord_le817736998455962536od_a_b @ B4
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_990_subset__Collect__iff,axiom,
! [B4: set_nat,A3: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_991_subset__Collect__iff,axiom,
! [B4: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B4 @ A3 )
=> ( ( ord_le3146513528884898305at_nat @ B4
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_992_subset__Collect__iff,axiom,
! [B4: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B4 @ A3 )
=> ( ( ord_le2843351958646193337nt_int @ B4
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_993_subset__CollectI,axiom,
! [B4: set_Product_prod_a_b,A3: set_Product_prod_a_b,Q: product_prod_a_b > $o,P2: product_prod_a_b > $o] :
( ( ord_le817736998455962536od_a_b @ B4 @ A3 )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le817736998455962536od_a_b
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec3336397801687681299od_a_b
@ ^ [X2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_994_subset__CollectI,axiom,
! [B4: set_nat,A3: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_995_subset__CollectI,axiom,
! [B4: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B4 @ A3 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_996_subset__CollectI,axiom,
! [B4: set_Pr958786334691620121nt_int,A3: set_Pr958786334691620121nt_int,Q: product_prod_int_int > $o,P2: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B4 @ A3 )
=> ( ! [X3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collec213857154873943460nt_int
@ ^ [X2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_997_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_998_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_999_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_1000_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_1001_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_1002_mlex__leq,axiom,
! [F: nat > nat,X: nat,Y5: nat,R: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( mlex_prod_nat @ F @ R ) ) ) ) ).
% mlex_leq
thf(fact_1003_mlex__leq,axiom,
! [F: int > nat,X: int,Y5: int,R: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ ( mlex_prod_int @ F @ R ) ) ) ) ).
% mlex_leq
thf(fact_1004_mlex__iff,axiom,
! [X: nat,Y5: nat,F: nat > nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( mlex_prod_nat @ F @ R ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) )
| ( ( ( F @ X )
= ( F @ Y5 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_1005_mlex__iff,axiom,
! [X: int,Y5: int,F: int > nat,R: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ ( mlex_prod_int @ F @ R ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) )
| ( ( ( F @ X )
= ( F @ Y5 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_1006_mlex__less,axiom,
! [F: nat > nat,X: nat,Y5: nat,R: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( mlex_prod_nat @ F @ R ) ) ) ).
% mlex_less
thf(fact_1007_mlex__less,axiom,
! [F: int > nat,X: int,Y5: int,R: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ ( mlex_prod_int @ F @ R ) ) ) ).
% mlex_less
thf(fact_1008_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A3: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_1009_Sup_OSUP__cong,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,C3: product_prod_a_b > a,D: product_prod_a_b > a,Sup: set_a > a] :
( ( A3 = B4 )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_2802296252294471259_a_b_a @ C3 @ A3 ) )
= ( Sup @ ( image_2802296252294471259_a_b_a @ D @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1010_Sup_OSUP__cong,axiom,
! [A3: set_nat,B4: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A3 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C3 @ A3 ) )
= ( Sup @ ( image_nat_nat @ D @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1011_Inf_OINF__cong,axiom,
! [A3: set_Product_prod_a_b,B4: set_Product_prod_a_b,C3: product_prod_a_b > a,D: product_prod_a_b > a,Inf: set_a > a] :
( ( A3 = B4 )
=> ( ! [X3: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_2802296252294471259_a_b_a @ C3 @ A3 ) )
= ( Inf @ ( image_2802296252294471259_a_b_a @ D @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1012_Inf_OINF__cong,axiom,
! [A3: set_nat,B4: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A3 = B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C3 @ A3 ) )
= ( Inf @ ( image_nat_nat @ D @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1013_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A3: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1014_in__measure,axiom,
! [X: nat,Y5: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) ) ) ).
% in_measure
thf(fact_1015_in__measure,axiom,
! [X: int,Y5: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y5 ) ) ) ).
% in_measure
thf(fact_1016_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M6: nat,N5: nat] :
( N5
= ( suc @ M6 ) ) ) ) ) ).
% pred_nat_def
thf(fact_1017_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1018_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1019_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1020_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1021_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1022_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1023_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1024_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1025_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1026_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1027_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1028_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1029_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1030_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_1031_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1032_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1033_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1034_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1035_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1036_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1037_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1038_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1039_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1040_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1041_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1042_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1043_real__arch__simple,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% real_arch_simple
thf(fact_1044_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_1045_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_less_as_int
thf(fact_1046_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1047_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_1048_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_1049_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_1050_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_1051_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_1052_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_1053_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1054_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1055_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_1056_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1057_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1058_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_1059_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_1060_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_1061_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_1062_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_1063_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_1064_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_1065_of__nat__sum,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( semiri1314217659103216013at_int @ ( groups7004838470036706110_b_nat @ F @ A3 ) )
= ( groups7002347999527655834_b_int
@ ^ [X2: product_prod_a_b] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A3 ) ) ).
% of_nat_sum
thf(fact_1066_of__nat__sum,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( semiri5074537144036343181t_real @ ( groups7004838470036706110_b_nat @ F @ A3 ) )
= ( groups1618960153307602714b_real
@ ^ [X2: product_prod_a_b] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A3 ) ) ).
% of_nat_sum
thf(fact_1067_of__nat__sum,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( semiri1316708129612266289at_nat @ ( groups7004838470036706110_b_nat @ F @ A3 ) )
= ( groups7004838470036706110_b_nat
@ ^ [X2: product_prod_a_b] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
@ A3 ) ) ).
% of_nat_sum
thf(fact_1068_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1069_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1070_int__sum,axiom,
! [F: product_prod_a_b > nat,A3: set_Product_prod_a_b] :
( ( semiri1314217659103216013at_int @ ( groups7004838470036706110_b_nat @ F @ A3 ) )
= ( groups7002347999527655834_b_int
@ ^ [X2: product_prod_a_b] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A3 ) ) ).
% int_sum
thf(fact_1071_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1072_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1073_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1074_conj__le__cong,axiom,
! [X: int,X7: int,P2: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1075_imp__le__cong,axiom,
! [X: int,X7: int,P2: $o,P5: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1076_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1077_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_1078_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z7: int] :
( ( ord_less_eq_int @ D4 @ Z6 )
& ( ord_less_int @ Z6 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_1079_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z7: int] :
( ( ord_less_eq_int @ D4 @ Z7 )
& ( ord_less_int @ Z6 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_1080_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1081_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1082_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_1083_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_1084_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_1085_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_1086_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_1087_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_1088_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1089_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1090_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1091_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1092_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_1093_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_1094_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_1095_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1096_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1097_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1098_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_1099_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_1100_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_1101_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1102_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_1103_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1104_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_1105_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1106_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_1107_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1108_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_1109_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1110_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_1111_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_1112_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_1113_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1114_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_1115_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1116_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1117_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1118_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1119_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1120_negative__zless,axiom,
! [N2: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1121_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1122_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1123_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1124_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_1125_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_1126_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_1127_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_1128_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1129_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_1130_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_1131_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_1132_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_1133_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1134_nat__mono,axiom,
! [X: int,Y5: int] :
( ( ord_less_eq_int @ X @ Y5 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y5 ) ) ) ).
% nat_mono
thf(fact_1135_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1136_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1137_eq__nat__nat__iff,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z8 ) )
= ( Z = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1138_int__cases,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% int_cases
thf(fact_1139_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N4: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N4 ) )
=> ( ! [N4: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1140_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_1141_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1142_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1143_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1144_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1145_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1146_int__cases4,axiom,
! [M2: int] :
( ! [N4: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1147_int__zle__neg,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1148_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N4: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1149_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1150_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1151_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1152_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1153_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1154_split__nat,axiom,
! [P2: nat > $o,I3: int] :
( ( P2 @ ( nat2 @ I3 ) )
= ( ! [N5: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N5 ) )
=> ( P2 @ N5 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1155_le__nat__iff,axiom,
! [K2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_1156_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1157_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_1158_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N4: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% negD
thf(fact_1159_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1160_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_1161_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1162_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_1163_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1164_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1165_nat__induct__non__zero,axiom,
! [N2: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P2 @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P2 @ N4 )
=> ( P2 @ ( suc @ N4 ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1166_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_1167_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1168_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1169_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_1170_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1171_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1172_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1173_int__gr__induct,axiom,
! [K2: int,I3: int,P2: int > $o] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1174_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_1175_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1176_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z7: int] :
? [N5: nat] :
( Z7
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1177_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_1178_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1179_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1180_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_1181_Suc__as__int,axiom,
( suc
= ( ^ [A6: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A6 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1182_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1183_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1184_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1185_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1186_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1187_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1188_real__0__le__add__iff,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y5 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y5 ) ) ).
% real_0_le_add_iff
thf(fact_1189_real__add__le__0__iff,axiom,
! [X: real,Y5: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y5 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y5 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1190_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N5: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N5 ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% int_less_real_le
thf(fact_1191_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N5: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N5 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1192_less__add__eq__less,axiom,
! [K2: nat,L2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1193_trans__less__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1194_trans__less__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1195_add__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1196_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1197_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_1198_add__less__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K2 @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1199_add__lessD1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_1200_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1201_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1202_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1203_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1204_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1205_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1206_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1207_le__Suc__ex,axiom,
! [K2: nat,L2: nat] :
( ( ord_less_eq_nat @ K2 @ L2 )
=> ? [N4: nat] :
( L2
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1208_add__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1209_add__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1210_trans__le__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1211_trans__le__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1212_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N5: nat] :
? [K4: nat] :
( N5
= ( plus_plus_nat @ M6 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1213_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1214_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1215_nat__arith_Osuc1,axiom,
! [A3: nat,K2: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1216_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N5: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N5 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% nat_less_real_le
thf(fact_1217_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N5: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N5 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1218_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y4: real] :
( ( ord_less_real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1219_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1220_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1221_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1222_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1223_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N5: nat] :
? [K4: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M6 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1224_less__add__Suc2,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M2 @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1225_less__add__Suc1,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1226_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q4: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1227_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1228_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1229_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1230_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1231_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B6: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1232_nat__add__distrib,axiom,
! [Z: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z8 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1233_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1234_real__add__less__0__iff,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y5 ) @ zero_zero_real )
= ( ord_less_real @ Y5 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1235_real__0__less__add__iff,axiom,
! [X: real,Y5: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y5 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y5 ) ) ).
% real_0_less_add_iff
thf(fact_1236_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P2 @ A4 @ B3 )
= ( P2 @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ A4 @ B3 )
=> ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1237_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_1238_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1239_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Xa2: int] :
( produc8739625826339149834_nat_o
@ ^ [Y4: nat,Z7: nat] :
( produc6081775807080527818_nat_o
@ ^ [U: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U @ Z7 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa2 ) ) ) ) ).
% less_int.rep_eq
thf(fact_1240_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Xa2: int] :
( produc8739625826339149834_nat_o
@ ^ [Y4: nat,Z7: nat] :
( produc6081775807080527818_nat_o
@ ^ [U: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U @ Z7 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa2 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_1241_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
@ ^ [X2: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ U @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_1242_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
@ ^ [X2: nat,Y4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ U @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_1243_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_1244_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_1245_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N5 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_1246_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
@ ^ [X2: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U ) @ ( plus_plus_nat @ Y4 @ V2 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_1247_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_1248_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1249_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K3: nat,M4: nat] :
( X
!= ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_1250_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_1251_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_1252_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y4 @ U ) ) ) ) ) ) ).
% minus_int_def
thf(fact_1253_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
@ ^ [X2: nat,Y4: nat] :
( produc2626176000494625587at_nat
@ ^ [U: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y4 @ U ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_1254_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1255_minus__int__code_I2_J,axiom,
! [L2: int] :
( ( minus_minus_int @ zero_zero_int @ L2 )
= ( uminus_uminus_int @ L2 ) ) ).
% minus_int_code(2)
thf(fact_1256_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_1257_int__less__induct,axiom,
! [I3: int,K2: int,P2: int > $o] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P2 @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_1258_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1259_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1260_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1261_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1262_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y5: nat] :
( ( if_nat @ $false @ X @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y5: nat] :
( ( if_nat @ $true @ X @ Y5 )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
! [X: fset_nat,Y5: fset_nat] :
( ( if_fset_nat @ $false @ X @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
! [X: fset_nat,Y5: fset_nat] :
( ( if_fset_nat @ $true @ X @ Y5 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_T,axiom,
! [X: product_prod_a_b,Y5: product_prod_a_b] :
( ( if_Product_prod_a_b @ $false @ X @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_T,axiom,
! [X: product_prod_a_b,Y5: product_prod_a_b] :
( ( if_Product_prod_a_b @ $true @ X @ Y5 )
= X ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
! [X: fset_P9214369701362650254od_a_b,Y5: fset_P9214369701362650254od_a_b] :
( ( if_fse8748871985657954388od_a_b @ $false @ X @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
! [X: fset_P9214369701362650254od_a_b,Y5: fset_P9214369701362650254od_a_b] :
( ( if_fse8748871985657954388od_a_b @ $true @ X @ Y5 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ ( f @ t ) @ e )
@ ( fset_P2369346149119917079od_a_b
@ ( fimage3179447126440047741od_a_b
@ ( produc8992199381948149691od_a_b
@ ^ [T2: a] : ( product_Pair_a_b @ ( f @ T2 ) ) )
@ xsa ) ) ) ).
%------------------------------------------------------------------------------