TPTP Problem File: SLH0583^1.p
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%------------------------------------------------------------------------------
% 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 : Hales_Jewett/0002_Hales_Jewett/prob_00876_035363__5735524_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1527 ( 566 unt; 254 typ; 0 def)
% Number of atoms : 3893 (1294 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 11162 ( 340 ~; 59 |; 267 &;8870 @)
% ( 0 <=>;1626 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 2606 (2606 >; 0 *; 0 +; 0 <<)
% Number of symbols : 224 ( 221 usr; 26 con; 0-6 aty)
% Number of variables : 3891 ( 460 ^;3328 !; 103 ?;3891 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:39:59.449
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8175506400003264433at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na6857298508006588994at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na3764207730537033026at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8778986904112484418at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8843485148432118594at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_na7938001796681673538at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_se3022870823424313865at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_na2445831480116662482_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_nat_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat3: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat4: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat5: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_nat_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_nat_o2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J_J,type,
set_nat_nat_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $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__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (221)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
comple2115216063353097951_nat_o: set_na2445831480116662482_nat_o > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
comple8231226574009213710_nat_o: set_nat_nat_nat_o2 > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
comple3396693796109600270_nat_o: set_nat_nat_nat_o > ( nat > nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
comple8312177224774716605_nat_o: set_nat_nat_o > ( nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
comple2450677804321093138at_nat: set_nat_nat > nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple2605510978757769510at_nat: set_se3022870823424313865at_nat > set_nat_nat_nat_nat3 ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
comple1667856448326461495at_nat: set_set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple8167887107183641911at_nat: set_set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Nat__Onat,type,
disjoi6798895846410478970at_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fun_upd_nat_nat_nat: ( nat > nat > nat ) > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
the_in5300466440149791684at_nat: set_nat_nat > ( ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6564615839001774232at_nat: set_nat_nat_nat_nat3 > ( ( ( nat > nat ) > nat > nat ) > set_nat_nat ) > set_na8175506400003264433at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_na4548495224246695081at_nat: set_nat_nat_nat_nat3 > ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_na7938001796681673538at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6840239867990089257at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_na8843485148432118594at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat_nat5 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na7122919648973241129at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat ) > set_na8778986904112484418at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat2: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat_nat4 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8678869062391380393at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat ) > set_na3764207730537033026at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat3: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
piE_nat_nat_int: set_nat_nat > ( ( nat > nat ) > set_int ) > set_nat_nat_int ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na2748089427378204713at_nat: set_nat > ( nat > set_nat_nat_nat_nat3 ) > set_na6857298508006588994at_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat4: set_nat > ( nat > set_nat_nat_nat2 ) > set_nat_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_nat_nat_nat_nat5: set_nat > ( nat > set_nat_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
piE_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri4446420529079022766at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restrict_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restrict_nat_nat_nat2: ( nat > nat > nat ) > set_nat > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
restrict_nat_set_nat: ( nat > set_nat ) > set_nat > nat > set_nat ).
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_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_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_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
hales_4783935871306402712at_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > ( nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Int__Oint,type,
hales_4259056829518216709ce_int: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > int > ( ( nat > nat ) > int ) > $o ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Nat__Onat,type,
hales_4261547300027266985ce_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > nat > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_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__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1568108970253895006_nat_o: ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo6348804412059337741_nat_o: ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1514271634159724301_nat_o: ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo3618716324728726597at_nat: set_na8175506400003264433at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
bot_bo3386126977483763158at_nat: set_na7938001796681673538at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo4291610329234208214at_nat: set_na8843485148432118594at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo4508028030728203495at_nat: set_nat_nat_nat_nat5 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo4227112084914574038at_nat: set_na8778986904112484418at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
bot_bo3013702615682746855at_nat: set_nat_nat_nat_nat4 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo3919185967433191911at_nat: set_nat_nat_nat_nat3 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo945813143650711160at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7445843802507891576at_nat: set_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le4961065272816086430_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le8812218136411540557_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le3977685358511927117_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le4629963735342356977at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_less_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_less_nat_nat_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_nat_nat_nat2: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
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__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6177938698872215975at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le371403230139555384at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6871433888996735800at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5430825838364970130_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le996020443555834177_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le747776305331315197at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le2017632242545079438at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3127000006974329230at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $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__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le9041126503034175505at_nat: set_na8175506400003264433at_nat > set_na8175506400003264433at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3190276326201062306at_nat: set_na8843485148432118594at_nat > set_na8843485148432118594at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3125778081881428130at_nat: set_na8778986904112484418at_nat > set_na8778986904112484418at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5934964663421696068at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3567154360959927026at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_4065302347126311296at_nat: ( ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ) > set_na2445831480116662482_nat_o > set_se3022870823424313865at_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_5425260358592644672at_nat: ( ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ) > set_nat_nat_nat_o2 > set_set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_3610001086604609088at_nat: ( ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ) > set_nat_nat_nat_o > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_8194121248528334964at_nat: ( ( ( nat > nat ) > nat > nat ) > nat ) > set_nat_nat_nat_nat3 > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1946857609996606506et_nat: ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_nat_nat_nat_nat3 > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7977807581451749376at_nat: ( ( ( nat > nat ) > $o ) > set_nat_nat ) > set_nat_nat_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7565631143590340539et_nat: ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat2 > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_6782468043973903547et_nat: ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_470123710477037866at_nat: ( ( nat > nat ) > set_nat_nat_nat ) > set_nat_nat > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6905811865970898491at_nat: ( ( nat > nat ) > set_nat_nat ) > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6393715451659844596at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_nat > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_5809701139083627781at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6919068903512877573at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J,type,
image_nat_nat_o: ( nat > nat > $o ) > set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
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__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_3332361743537024938at_nat: ( nat > set_nat_nat_nat_nat3 ) > set_nat > set_se3022870823424313865at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_8854229838293529787at_nat: ( nat > set_nat_nat_nat2 ) > set_nat > set_set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_6130888460295934395at_nat: ( nat > set_nat_nat_nat ) > set_nat > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
image_4040409651686222360_nat_o: ( set_nat_nat_nat_nat3 > ( ( nat > nat ) > nat > nat ) > $o ) > set_se3022870823424313865at_nat > set_na2445831480116662482_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_Eo,type,
image_7580978635682194622_nat_o: ( set_nat_nat_nat_nat3 > $o ) > set_se3022870823424313865at_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
image_6357918107393578614_nat_o: ( set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > $o ) > set_set_nat_nat_nat2 > set_nat_nat_nat_o2 ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_Eo,type,
image_8774134582277556973_nat_o: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
image_2840114971476761718_nat_o: ( set_nat_nat_nat > ( nat > nat > nat ) > $o ) > set_set_nat_nat_nat > set_nat_nat_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_Eo,type,
image_5198217506544545261_nat_o: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
image_1242417779249009364_nat_o: ( set_nat_nat > ( nat > nat ) > $o ) > set_set_nat_nat > set_nat_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
image_set_nat_nat_o: ( set_nat_nat > $o ) > set_set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3832368097948589297at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o2: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7054278410236665602at_nat: ( set_nat > set_nat_nat ) > set_set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3591701359631937174at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or5033131092550408871at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or6142498856979658663at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or9140604705432621368at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or250740698829186286at_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or7562748684798938298at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or2699333443382148811at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3808701207811398603at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member2991261302380110260at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat_nat5 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
member5318315686745620148at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat_nat4 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member1679187572556404771at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_na3764207730537033026at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
member_nat_nat_int: ( ( nat > nat ) > int ) > set_nat_nat_int > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8743709692935548195at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_na6857298508006588994at_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_B____,type,
b: nat > set_nat ).
thf(sy_v_L_H____,type,
l: nat > nat > nat ).
thf(sy_v_L____,type,
l2: nat > nat > nat ).
thf(sy_v_N_H____,type,
n: nat ).
thf(sy_v_N____,type,
n2: nat ).
thf(sy_v_S1____,type,
s1: ( nat > nat ) > nat > nat ).
thf(sy_v__092_060chi_062____,type,
chi: ( nat > nat ) > nat ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_s____,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1268)
thf(fact_0_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_1_j__prop,axiom,
( ( member_nat @ j @ ( b @ zero_zero_nat ) )
& ( ord_less_nat @ j @ n ) ) ).
% j_prop
thf(fact_2_that,axiom,
ord_less_nat @ s @ t ).
% that
thf(fact_3__092_060open_062_092_060And_062s_Aj_O_A_092_060lbrakk_062is__line_AL_AN_H_At_059_As_A_060_At_059_Aj_A_060_AN_H_092_060rbrakk_062_A_092_060Longrightarrow_062_AL_As_Aj_A_092_060in_062_A_123_O_O_060t_125_092_060close_062,axiom,
! [S: nat,J: nat] :
( ( hales_is_line @ l2 @ n @ t )
=> ( ( ord_less_nat @ S @ t )
=> ( ( ord_less_nat @ J @ n )
=> ( member_nat @ ( l2 @ S @ J ) @ ( set_ord_lessThan_nat @ t ) ) ) ) ) ).
% \<open>\<And>s j. \<lbrakk>is_line L N' t; s < t; j < N'\<rbrakk> \<Longrightarrow> L s j \<in> {..<t}\<close>
thf(fact_4_assms_I2_J,axiom,
! [R: nat] : ( hales_hj @ R @ t ) ).
% assms(2)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062j_O_Aj_A_092_060in_062_AB_A0_A_092_060and_062_Aj_A_060_AN_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [J2: nat] :
~ ( ( member_nat @ J2 @ ( b @ zero_zero_nat ) )
& ( ord_less_nat @ J2 @ n ) ) ).
% \<open>\<And>thesis. (\<And>j. j \<in> B 0 \<and> j < N' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_calculation,axiom,
! [S: nat] :
( ( ord_less_nat @ S @ t )
=> ( ( l @ S @ j )
= ( l2 @ S @ j ) ) ) ).
% calculation
thf(fact_7_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_8_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_9_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_10_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_11_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_12_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_13_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_14_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_15_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_16_mask__0,axiom,
( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
= zero_zero_int ) ).
% mask_0
thf(fact_17_True,axiom,
t = one_one_nat ).
% True
thf(fact_18_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_19_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_20_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2000444600071755411sk_int @ N )
= zero_zero_int )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_21_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_22_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_23_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_24_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_25_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_26_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_27_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_28_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_29_line__points__in__cube__unfolded,axiom,
! [L: nat > nat > nat,N: nat,T: nat,S: nat,J: nat] :
( ( hales_is_line @ L @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L @ S @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_30_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_31_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_32_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_33_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_34_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_35_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_36_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_37_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_38_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_39_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_40_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_41_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_42_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A @ ( collect_nat_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_lessThan__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_53_lessThan__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I @ K ) ) ).
% lessThan_iff
thf(fact_54_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_55_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or7562748684798938298at_nat @ K ) )
= ( ord_le4629963735342356977at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_56_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_57_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_58_line__points__in__cube,axiom,
! [L: nat > nat > nat,N: nat,T: nat,S: nat] :
( ( hales_is_line @ L @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( member_nat_nat @ ( L @ S ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_59_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_60_asm_I1_J,axiom,
ord_less_eq_nat @ n2 @ n ).
% asm(1)
thf(fact_61_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_62_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_63_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_64_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_65_L__def,axiom,
( ( hales_is_line @ l2 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ l2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X3 )
= C2 ) ) ) ) ).
% L_def
thf(fact_66_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_67_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_68_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_69_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X4 @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_70_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_71_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_72_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_73_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_74_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_75_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_76_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_77_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_78_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_79_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_80_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_81_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_82_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_83_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_84_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_85_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_86_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_87_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_88_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_89_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_90_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_91_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_92_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_93_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062L_O_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060exists_062c_060r_O_A_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_Arestrict_A_092_060chi_062_A_Icube_AN_H_At_J_Ay_A_061_Ac_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L2: nat > nat > nat] :
~ ( ( hales_is_line @ L2 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X3 )
= C2 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>L. is_line L N' t \<and> (\<exists>c<r. \<forall>y\<in>L ` {..<t}. restrict \<chi> (cube N' t) y = c) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_95_N__def,axiom,
( ( ord_less_nat @ zero_zero_nat @ n2 )
& ! [N3: nat] :
( ( ord_less_eq_nat @ n2 @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L2: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L2 @ N3 @ t )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X3 )
= C2 ) ) ) ) ) ) ).
% N_def
thf(fact_96_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_97_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_98_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_99_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_100_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_101_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_102_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_103_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_104_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_105_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_106_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_107_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_108_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_109_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_110_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_111_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_112_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_113_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_114_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_115_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_116_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_117_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_118_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_119_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_120_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_121_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_122_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_123_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_124_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_125_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_126_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2002935070580805687sk_nat @ N ) ) ).
% of_nat_mask_eq
thf(fact_127_of__nat__mask__eq,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( bit_se2000444600071755411sk_int @ N ) ) ).
% of_nat_mask_eq
thf(fact_128_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_129_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_130_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_131_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_132_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_133_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_134_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_135_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_136_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_137_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_138_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_139_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_140_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_141_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_142_N_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L2: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L2 @ n @ t )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X3 )
= C2 ) ) ) ) ) ).
% N'_props
thf(fact_143__092_060open_062restrict_A_092_060chi_062_A_Icube_AN_H_At_J_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_092_060close_062,axiom,
( member_nat_nat_nat @ ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) )
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>restrict \<chi> (cube N' t) \<in> cube N' t \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_144__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062N_O_A0_A_060_AN_A_092_060and_062_A_I_092_060forall_062N_H_092_060ge_062N_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_A_092_060longrightarrow_062_A_I_092_060exists_062L_Ac_O_Ac_A_060_Ar_A_092_060and_062_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_A_092_060chi_062_Ay_A_061_Ac_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N5: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
& ! [N3: nat] :
( ( ord_less_eq_nat @ N5 @ N3 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L2: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L2 @ N3 @ t )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X3 )
= C2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>N. 0 < N \<and> (\<forall>N'\<ge>N. \<forall>\<chi>. \<chi> \<in> cube N' t \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>L c. c < r \<and> is_line L N' t \<and> (\<forall>y\<in>L ` {..<t}. \<chi> y = c))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_145_image__restrict__eq,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ ( restrict_nat_set_nat @ F @ A2 ) @ A2 )
= ( image_nat_set_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_146_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_147_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_148_image__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_149_image__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_150_asm_I2_J,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% asm(2)
thf(fact_151_image__ident,axiom,
! [Y4: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_152_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_153_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_154_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_155_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_156_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_157_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_158_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_159_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_160_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_161_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_162_add__right__cancel,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C3 @ A ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_163_add__right__cancel,axiom,
! [B: int,A: int,C3: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C3 @ A ) )
= ( B = C3 ) ) ).
% add_right_cancel
thf(fact_164_add__left__cancel,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_165_add__left__cancel,axiom,
! [A: int,B: int,C3: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C3 ) )
= ( B = C3 ) ) ).
% add_left_cancel
thf(fact_166_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_167_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_168_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_169_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_170_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_171_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_172_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_173_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_174_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_175_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_176_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_177_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_178_subsetI,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ X4 @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_179_subsetI,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ X4 @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_180_subsetI,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A2 )
=> ( member952132173341509300at_nat @ X4 @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_181_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ X4 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_182_S1__def,axiom,
( s1
= ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( l @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% S1_def
thf(fact_183_add__le__cancel__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_184_add__le__cancel__right,axiom,
! [A: int,C3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_185_add__le__cancel__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_186_add__le__cancel__left,axiom,
! [C3: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_187_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_188_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_189_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_190_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_191_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_192_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_193_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_194_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_195_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_196_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_197_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_198_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_199_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_200_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_201_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_202_add__less__cancel__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_203_add__less__cancel__left,axiom,
! [C3: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_204_add__less__cancel__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_205_add__less__cancel__right,axiom,
! [A: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_206_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_207_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_208_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_209_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_210_psubsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_211_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_212_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_213_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_214_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_215_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_216_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_217_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_218_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_219_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_220_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_221_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_222_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_223_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_224_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_225_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_226_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_227_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_228_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_229_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_230_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_231_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_232_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_233_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_234_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_235_image__add__0,axiom,
! [S2: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_236_image__add__0,axiom,
! [S2: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_237_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_238_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_239_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_240_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_241_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_242_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A3 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_243_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A3 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_244_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ord_le5430825838364970130_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A3 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_245_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_246_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_247_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_248_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_249_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_250_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_251_set__eq__subset,axiom,
( ( ^ [Y6: set_nat_nat,Z: set_nat_nat] : ( Y6 = Z ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_252_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_253_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_254_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_255_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: nat > set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_256_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B2 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_257_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B2 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_258_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le9041126503034175505at_nat @ ( piE_na6564615839001774232at_nat @ A2 @ B2 ) @ ( piE_na6564615839001774232at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_259_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_260_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_261_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_262_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_263_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A3 )
=> ( member_nat_nat_nat2 @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_264_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A3 )
=> ( member_nat_nat_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_265_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
! [T2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T2 @ A3 )
=> ( member952132173341509300at_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_266_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A3 )
=> ( member_nat_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_267_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_268_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_269_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_270_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A3 )
=> ( member_nat_nat_nat2 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_271_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A3 )
=> ( member_nat_nat_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_272_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A3 )
=> ( member952132173341509300at_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_273_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_274_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_275_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C3 @ A2 )
=> ( member_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_276_subsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C3: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C3 @ A2 )
=> ( member_nat_nat_nat2 @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_277_subsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C3: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C3 @ A2 )
=> ( member_nat_nat_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_278_subsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C3: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C3 @ A2 )
=> ( member952132173341509300at_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_279_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C3 @ A2 )
=> ( member_nat_nat @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_280_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_281_in__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,X: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_282_in__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,X: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_283_in__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,X: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_284_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_285_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C3 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C3 ) ) ) ).
% is_num_normalize(1)
thf(fact_286_PiE__ext,axiom,
! [X: ( nat > nat ) > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat,Y: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ( member_nat_nat_nat @ Y @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_287_PiE__ext,axiom,
! [X: nat > nat > nat,K: set_nat,S: nat > set_nat_nat,Y: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ( member_nat_nat_nat2 @ Y @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_288_PiE__ext,axiom,
! [X: nat > nat,K: set_nat,S: nat > set_nat,Y: nat > nat] :
( ( member_nat_nat @ X @ ( piE_nat_nat @ K @ S ) )
=> ( ( member_nat_nat @ Y @ ( piE_nat_nat @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_289_PiE__ext,axiom,
! [X: ( nat > nat ) > nat > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat_nat,Y: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ( member952132173341509300at_nat @ Y @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X @ I3 )
= ( Y @ I3 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_290_PiE__mem,axiom,
! [F: nat > nat,S2: set_nat,T3: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_291_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat,X: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_292_PiE__mem,axiom,
! [F: nat > nat > nat,S2: set_nat,T3: nat > set_nat_nat,X: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_293_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S2: set_nat,T3: nat > set_nat_nat_nat,X: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_294_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S2: set_nat,T3: nat > set_nat_nat_nat2,X: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_295_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S2: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat,X: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S2 @ T3 ) )
=> ( ( member_nat_nat_nat2 @ X @ S2 )
=> ( member_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_296_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S2: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat,X: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat_nat @ X @ S2 )
=> ( member_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_297_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat_nat,X: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_298_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat > nat,S2: set_nat,T3: nat > set_nat_nat_nat_nat3,X: nat] :
( ( member8743709692935548195at_nat @ F @ ( piE_na2748089427378204713at_nat @ S2 @ T3 ) )
=> ( ( member_nat @ X @ S2 )
=> ( member952132173341509300at_nat @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_299_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S2: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat,X: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S2 @ T3 ) )
=> ( ( member_nat_nat @ X @ S2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( T3 @ X ) ) ) ) ).
% PiE_mem
thf(fact_300_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat,B2: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat_nat @ I5 @ A2 )
= ( piE_nat_nat_nat @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_301_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat_nat2 @ I5 @ A2 )
= ( piE_nat_nat_nat2 @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_302_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat,B2: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat @ I5 @ A2 )
= ( piE_nat_nat @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_303_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B2 @ I3 ) ) )
=> ( ( piE_nat_nat_nat_nat3 @ I5 @ A2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ B2 ) ) ) ).
% PiE_cong
thf(fact_304_add__right__imp__eq,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C3 @ A ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_305_add__right__imp__eq,axiom,
! [B: int,A: int,C3: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C3 @ A ) )
=> ( B = C3 ) ) ).
% add_right_imp_eq
thf(fact_306_add__left__imp__eq,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_307_add__left__imp__eq,axiom,
! [A: int,B: int,C3: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C3 ) )
=> ( B = C3 ) ) ).
% add_left_imp_eq
thf(fact_308_add_Oleft__commute,axiom,
! [B: nat,A: nat,C3: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C3 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_309_add_Oleft__commute,axiom,
! [B: int,A: int,C3: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C3 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add.left_commute
thf(fact_310_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_311_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_312_add_Oright__cancel,axiom,
! [B: int,A: int,C3: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C3 @ A ) )
= ( B = C3 ) ) ).
% add.right_cancel
thf(fact_313_add_Oleft__cancel,axiom,
! [A: int,B: int,C3: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C3 ) )
= ( B = C3 ) ) ).
% add.left_cancel
thf(fact_314_add_Oassoc,axiom,
! [A: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C3 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_315_add_Oassoc,axiom,
! [A: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C3 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add.assoc
thf(fact_316_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_317_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_318_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_319_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_320_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( I = J )
& ( K = L3 ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_321_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( I = J )
& ( K = L3 ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_322_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C3 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_323_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C3: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C3 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_324_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_325_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_326_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_327_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X4: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X4
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I2: nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_328_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X4: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X4
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I2: nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_329_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X4: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X4
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I2: nat > nat > nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_330_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B2: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X4: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X4
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I2: ( nat > nat ) > nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_331_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_332_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ B2 )
=> ? [X4: nat > ( nat > nat ) > nat > nat] :
( ( member8743709692935548195at_nat @ X4
@ ( piE_na2748089427378204713at_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_333_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X4: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X4
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ( ( X4 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_334_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_335_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_336_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_337_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_338_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_339_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_340_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_341_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_342_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_343_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B2: set_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_344_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_345_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_346_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_347_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_348_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_349_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_350_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_351_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_352_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_353_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_354_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_355_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_356_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I5: set_nat_nat,X5: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I5 ) @ ( piE_nat_nat_nat @ I5 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_357_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I5: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_358_restrict__PiE__iff,axiom,
! [F: nat > nat,I5: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I5 ) @ ( piE_nat_nat @ I5 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_359_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I5: set_nat,X5: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I5 ) @ ( piE_nat_nat_nat2 @ I5 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_360_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_361_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_362_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_363_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_364_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_365_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_366_psubsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_367_PiE__uniqueness,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ? [X4: nat > set_nat] :
( ( member_nat_set_nat @ X4
@ ( piE_nat_set_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > set_nat] :
( ( ( member_nat_set_nat @ Y3
@ ( piE_nat_set_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X4 ) ) ) ) ).
% PiE_uniqueness
thf(fact_368_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 )
=> ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y3
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X4 ) ) ) ) ).
% PiE_uniqueness
thf(fact_369_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > nat] :
( ( ( member_nat_nat @ Y3
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X4 ) ) ) ) ).
% PiE_uniqueness
thf(fact_370_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
=> ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X4 ) ) ) ) ).
% PiE_uniqueness
thf(fact_371_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ? [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X4 ) ) ) ) ).
% PiE_uniqueness
thf(fact_372_cube__subset,axiom,
! [N: nat,T: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N @ T ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_373_add__le__imp__le__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_374_add__le__imp__le__right,axiom,
! [A: int,C3: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_375_add__le__imp__le__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_376_add__le__imp__le__left,axiom,
! [C3: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_377_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A4 @ C ) ) ) ) ).
% le_iff_add
thf(fact_378_add__right__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add_right_mono
thf(fact_379_add__right__mono,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add_right_mono
thf(fact_380_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_381_add__left__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% add_left_mono
thf(fact_382_add__left__mono,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% add_left_mono
thf(fact_383_add__mono,axiom,
! [A: nat,B: nat,C3: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C3 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_384_add__mono,axiom,
! [A: int,B: int,C3: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C3 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_385_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_386_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L3 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_387_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_388_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L3 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_389_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_390_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L3 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_391_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_392_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_393_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_394_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_395_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_396_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_397_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L3 ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_398_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_399_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L3 ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_400_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_401_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L3 ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_402_add__strict__mono,axiom,
! [A: nat,B: nat,C3: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C3 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_403_add__strict__mono,axiom,
! [A: int,B: int,C3: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C3 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_404_add__strict__left__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_405_add__strict__left__mono,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_406_add__strict__right__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_407_add__strict__right__mono,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_408_add__less__imp__less__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_409_add__less__imp__less__left,axiom,
! [C3: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C3 @ A ) @ ( plus_plus_int @ C3 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_410_add__less__imp__less__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_411_add__less__imp__less__right,axiom,
! [A: int,C3: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ C3 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_412_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_413_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_414_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_415_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ) ).
% add_less_mono
thf(fact_416_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_417_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_418_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_419_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_420_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_421_less__add__eq__less,axiom,
! [K: nat,L3: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L3 )
=> ( ( ( plus_plus_nat @ M @ L3 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_422_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_423_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_424_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_425_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_426_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ) ).
% add_le_mono
thf(fact_427_le__Suc__ex,axiom,
! [K: nat,L3: nat] :
( ( ord_less_eq_nat @ K @ L3 )
=> ? [N2: nat] :
( L3
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_428_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_429_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_430_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_431_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_432_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_433_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_434_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_435_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_436_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_437_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_438_split__cube_I1_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_439_split__cube_I2_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y5: nat] : ( X @ ( plus_plus_nat @ Y5 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_440_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_441_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_442_add__decreasing,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ B ) ) ) ).
% add_decreasing
thf(fact_443_add__decreasing,axiom,
! [A: int,C3: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ B ) ) ) ).
% add_decreasing
thf(fact_444_add__increasing,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_increasing
thf(fact_445_add__increasing,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C3 ) ) ) ) ).
% add_increasing
thf(fact_446_add__decreasing2,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_447_add__decreasing2,axiom,
! [C3: int,A: int,B: int] :
( ( ord_less_eq_int @ C3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C3 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_448_add__increasing2,axiom,
! [C3: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C3 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_increasing2
thf(fact_449_add__increasing2,axiom,
! [C3: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C3 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C3 ) ) ) ) ).
% add_increasing2
thf(fact_450_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_451_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_452_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_453_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_454_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_455_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_456_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_457_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_458_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_459_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L3 ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_460_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L3: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L3 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_461_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L3: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L3 ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L3 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_462_add__le__less__mono,axiom,
! [A: nat,B: nat,C3: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C3 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_463_add__le__less__mono,axiom,
! [A: int,B: int,C3: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C3 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_464_add__less__le__mono,axiom,
! [A: nat,B: nat,C3: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C3 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_465_add__less__le__mono,axiom,
! [A: int,B: int,C3: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C3 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C3 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_466_pos__add__strict,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% pos_add_strict
thf(fact_467_pos__add__strict,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C3 ) ) ) ) ).
% pos_add_strict
thf(fact_468_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_469_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_470_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_471_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_472_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_473_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_474_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_475_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_476_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_477_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_478_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_479_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_480_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_481_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_482_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_483_add__strict__increasing,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_strict_increasing
thf(fact_484_add__strict__increasing,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C3 ) ) ) ) ).
% add_strict_increasing
thf(fact_485_add__strict__increasing2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_strict_increasing2
thf(fact_486_add__strict__increasing2,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C3 ) ) ) ) ).
% add_strict_increasing2
thf(fact_487_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_488_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_489_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_490_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_491_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_492_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_493_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_494_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_495_imageI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_496_imageI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_497_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X ) @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_498_image__iff,axiom,
! [Z2: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_499_image__iff,axiom,
! [Z2: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_500_image__iff,axiom,
! [Z2: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_501_bex__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_502_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_503_bex__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X3 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_504_image__cong,axiom,
! [M3: set_nat,N6: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M3 = N6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M3 )
= ( image_nat_nat_nat2 @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_505_image__cong,axiom,
! [M3: set_nat,N6: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M3 = N6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M3 )
= ( image_nat_set_nat @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_506_image__cong,axiom,
! [M3: set_nat_nat,N6: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M3 = N6 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ N6 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M3 )
= ( image_3205354838064109189at_nat @ G @ N6 ) ) ) ) ).
% image_cong
thf(fact_507_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_508_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_509_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_510_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_511_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_512_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_513_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_514_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_515_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_516_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_517_rev__image__eqI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_518_rev__image__eqI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_519_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_520_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C3: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C3 @ A2 )
=> ( member_nat @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_521_psubsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C3: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C3 @ A2 )
=> ( member_nat_nat @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_522_psubsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C3: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C3 @ A2 )
=> ( member_nat_nat_nat2 @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_523_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C3: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C3 @ A2 )
=> ( member_nat_nat_nat @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_524_psubsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C3: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C3 @ A2 )
=> ( member952132173341509300at_nat @ C3 @ B2 ) ) ) ).
% psubsetD
thf(fact_525_cube__restrict,axiom,
! [J: nat,N: nat,Y: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_526_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_527_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_528_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat > nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_529_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_530_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X4: nat > nat > nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat_nat_nat2 @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_531_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X4: ( nat > nat ) > nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat_nat_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_532_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X4: nat > nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_533_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_534_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_535_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3] :
( ( member_nat @ B @ ( image_8194121248528334964at_nat @ F @ A2 ) )
=> ~ ! [X4: ( nat > nat ) > nat > nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member952132173341509300at_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_536_image__image,axiom,
! [F: ( nat > nat ) > set_nat,G: nat > nat > nat,A2: set_nat] :
( ( image_7432509271690132940et_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_537_image__image,axiom,
! [F: set_nat > nat > nat,G: nat > set_nat,A2: set_nat] :
( ( image_8569768528772619084at_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_538_image__image,axiom,
! [F: set_nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_539_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_540_image__image,axiom,
! [F: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_541_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_542_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_3205354838064109189at_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_543_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_544_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_545_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_546_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_547_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_548_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_913610194320715013at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_549_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_550_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_551_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( image_6919068903512877573at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_552_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_553_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_8194121248528334964at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_8194121248528334964at_nat @ F
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_554_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_555_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_556_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A3 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_557_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A3 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_558_less__set__def,axiom,
( ord_le6177938698872215975at_nat
= ( ^ [A3: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ord_le4961065272816086430_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A3 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_559_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_560_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_561_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_562_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_563_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_564_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_565_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_566_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_567_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_568_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_569_hj__def,axiom,
( hales_hj
= ( ^ [R2: nat,T2: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ T2 )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [L4: nat > nat > nat,C: nat] :
( ( ord_less_nat @ C @ R2 )
& ( hales_is_line @ L4 @ N8 @ T2 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi2 @ X2 )
= C ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_570__092_060open_062is__subspace_A_I_092_060lambda_062y_092_060in_062cube_A1_At_O_AL_A_Iy_A0_J_J_A1_AN_H_At_092_060close_062,axiom,
( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( l2 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t ) )
@ one_one_nat
@ n
@ t ) ).
% \<open>is_subspace (\<lambda>y\<in>cube 1 t. L (y 0)) 1 N' t\<close>
thf(fact_571_A1,axiom,
( member_nat_nat_nat2 @ l
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat] : ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% A1
thf(fact_572_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_573_dim0__layered__subspace__ex,axiom,
! [Chi3: ( nat > nat ) > nat,N: nat,T: nat,R3: nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ zero_zero_nat @ N @ T @ R3 @ Chi3 ) ) ).
% dim0_layered_subspace_ex
thf(fact_574_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T2: nat,K3: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S4: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S4 @ K3 @ N8 @ T2 @ R2 @ Chi2 ) ) ) ) ) ) ).
% lhj_def
thf(fact_575_join__cubes,axiom,
! [F: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_576_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
@ N4
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_577_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] :
( semiri8420488043553186161ux_int
@ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
@ N4
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_578_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_579_subspace__elems__embed,axiom,
! [S2: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S2 @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S2 @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_580_cube__def,axiom,
( hales_cube
= ( ^ [N4: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N4 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_581_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_582_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S3 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_583_is__line__def,axiom,
( hales_is_line
= ( ^ [L4: nat > nat > nat,N4: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L4
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I2: nat] : ( hales_cube @ N4 @ T2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N4 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T2 )
=> ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ T2 )
=> ( ( L4 @ X2 @ J3 )
= ( L4 @ Y5 @ J3 ) ) ) )
| ! [S5: nat] :
( ( ord_less_nat @ S5 @ T2 )
=> ( ( L4 @ S5 @ J3 )
= S5 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N4 )
& ! [S5: nat] :
( ( ord_less_nat @ S5 @ T2 )
=> ( ( L4 @ S5 @ J3 )
= S5 ) ) ) ) ) ) ).
% is_line_def
thf(fact_584_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_585_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_586_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( L @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_587_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( L @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ one_one_nat ) )
@ one_one_nat
@ N
@ one_one_nat ) ) ) ).
% line_is_dim1_subspace_t_1
thf(fact_588_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( L @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_589_classes__subset__cube,axiom,
! [N: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_590_pred__subset__eq,axiom,
! [R4: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S2 ) )
= ( ord_less_eq_set_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_591_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat,S2: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ R4 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ S2 ) )
= ( ord_le3211623285424100676at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_592_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat2,S2: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ R4 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ S2 ) )
= ( ord_le5934964663421696068at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_593_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat_nat3,S2: set_nat_nat_nat_nat3] :
( ( ord_le5430825838364970130_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ R4 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ S2 ) )
= ( ord_le5260717879541182899at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_594_pred__subset__eq,axiom,
! [R4: set_nat_nat,S2: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R4 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S2 ) )
= ( ord_le9059583361652607317at_nat @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_595_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_596_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_597_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_598_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_599_order__refl,axiom,
! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).
% order_refl
thf(fact_600_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_601_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_602_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_603_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_604_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_605_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_606_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_607_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_608_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_609_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_610_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_611_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_612_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_613_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_614_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_615_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_616_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_617_plus__int__code_I2_J,axiom,
! [L3: int] :
( ( plus_plus_int @ zero_zero_int @ L3 )
= L3 ) ).
% plus_int_code(2)
thf(fact_618_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_619_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_620_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_621_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_622_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_623_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_624_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_625_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_626_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z: nat] : ( Y6 = Z ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_627_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z: set_nat_nat] : ( Y6 = Z ) )
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ( ord_le9059583361652607317at_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_628_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z: int] : ( Y6 = Z ) )
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_629_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_630_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ord_le9059583361652607317at_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_631_ord__eq__le__trans,axiom,
! [A: int,B: int,C3: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_632_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_633_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_le9059583361652607317at_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_634_ord__le__eq__trans,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_int @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_635_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_636_order__antisym,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_637_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_638_order_Otrans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_639_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ord_le9059583361652607317at_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_640_order_Otrans,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_eq_int @ A @ C3 ) ) ) ).
% order.trans
thf(fact_641_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_642_order__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_643_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_644_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_645_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_646_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z: nat] : ( Y6 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_647_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z: set_nat_nat] : ( Y6 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_648_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z: int] : ( Y6 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_649_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_650_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_651_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_652_dual__order_Otrans,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_653_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C3 @ B )
=> ( ord_le9059583361652607317at_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_654_dual__order_Otrans,axiom,
! [B: int,A: int,C3: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_eq_int @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_655_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_656_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_657_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_658_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z: nat] : ( Y6 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_659_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z: set_nat_nat] : ( Y6 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_660_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z: int] : ( Y6 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_661_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_662_order__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_663_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_664_order__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C3: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_665_order__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_666_order__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C3: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_667_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_668_order__subst1,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_669_order__subst1,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_670_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_671_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_672_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_673_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C3: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_674_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_675_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C3: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_676_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_677_order__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_678_order__subst2,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_679_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_680_order__eq__refl,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( X = Y )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_681_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_682_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_683_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_684_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_685_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C3: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_686_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_687_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C3: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_688_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_689_ord__eq__le__subst,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C3: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_690_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_691_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C3: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_692_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_693_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_694_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_695_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_696_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C3: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_697_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_698_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C3: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_699_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_700_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_701_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_702_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_703_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_704_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_705_order__antisym__conv,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_706_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_707_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_708_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_709_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_710_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_711_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_712_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_713_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_714_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_715_ord__eq__less__trans,axiom,
! [A: int,B: int,C3: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_716_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_717_ord__less__eq__trans,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_int @ A @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_718_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X4 )
=> ( P @ Y3 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_719_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_720_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_721_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_722_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_723_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_724_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_725_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_726_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_727_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_728_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_729_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_730_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans
thf(fact_731_order_Ostrict__trans,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A @ C3 ) ) ) ).
% order.strict_trans
thf(fact_732_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_733_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_734_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_735_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C3: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_736_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_737_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_738_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_739_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_740_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_741_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_742_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_743_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_744_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_745_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_746_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_747_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_748_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_749_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_750_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_751_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_752_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_753_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_754_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_755_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_756_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_757_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_758_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_759_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_760_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_761_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_762_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_763_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_764_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_765_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_766_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_767_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_768_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_769_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_770_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_771_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_772_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_773_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_774_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_775_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_776_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_777_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_778_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_779_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_780_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_781_leD,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ~ ( ord_less_set_nat_nat @ X @ Y ) ) ).
% leD
thf(fact_782_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_783_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_784_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_785_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_786_nless__le,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_787_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_788_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_789_antisym__conv1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_790_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_791_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_792_antisym__conv2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_793_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_794_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_795_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ~ ( ord_le9059583361652607317at_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_796_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_797_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_798_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_799_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_800_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_801_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_802_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_803_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_804_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_805_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_806_order_Ostrict__trans1,axiom,
! [A: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C3 )
=> ( ord_less_set_nat_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_807_order_Ostrict__trans1,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C3 )
=> ( ord_less_int @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_808_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_809_order_Ostrict__trans2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ord_less_set_nat_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_810_order_Ostrict__trans2,axiom,
! [A: int,B: int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ord_less_int @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_811_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_812_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ~ ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_813_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_814_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_815_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( ord_less_set_nat_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_816_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_817_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_818_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_819_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_820_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_821_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C3 @ B )
=> ( ord_less_set_nat_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_822_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C3: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_823_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_824_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C3 @ B )
=> ( ord_less_set_nat_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_825_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C3: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C3 @ B )
=> ( ord_less_int @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_826_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_827_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ~ ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_828_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_829_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_830_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_831_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_832_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_833_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_834_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_835_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_836_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_837_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_int @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_838_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_839_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_840_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y5: int] :
( ( ord_less_eq_int @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_841_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_842_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_843_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_844_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_845_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_846_order__less__imp__le,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_847_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_848_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_849_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_850_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_851_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_852_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_853_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_854_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_855_order__le__less__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_less_set_nat_nat @ Y @ Z2 )
=> ( ord_less_set_nat_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_856_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_857_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_858_order__less__le__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z2 )
=> ( ord_less_set_nat_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_859_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_860_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_861_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_862_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C3: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_863_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C3: int] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_864_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_865_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_866_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_867_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_868_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_869_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C3: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_870_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_871_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > int,C3: int] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_872_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_873_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_874_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_875_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_876_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C3: nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_877_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C3: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_878_order__less__le__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_879_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_880_order__less__le__subst1,axiom,
! [A: int,F: set_nat_nat > int,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_881_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C3: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_882_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: int > set_nat_nat,B: int,C3: int] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_883_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C3: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_eq_int @ X4 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_884_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_885_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C3: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_886_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_887_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > set_nat_nat,C3: set_nat_nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_888_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C3: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_889_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C3 )
=> ( ! [X4: int,Y2: int] :
( ( ord_less_int @ X4 @ Y2 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_890_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_891_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_892_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_893_order__le__imp__less__or__eq,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_less_set_nat_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_894_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_895_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_896_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_897_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_898_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_899_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_900_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_901_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_902_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_903_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_904_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_905_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_906_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_907_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_908_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_909_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z: nat] : ( Y6 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_910_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_911_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_912_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_913_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_914_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_915_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_916_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_917_layered__subspace__def,axiom,
( hales_4783935871306402712at_nat
= ( ^ [S4: ( nat > nat ) > nat > nat,K3: nat,N4: nat,T2: nat,R2: nat > nat,Chi2: ( nat > nat ) > nat > nat] :
( ( hales_is_subspace @ S4 @ K3 @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C: nat > nat] :
( ( ord_less_nat_nat @ C @ R2 )
& ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ ( hales_classes @ K3 @ T2 @ X2 ) )
=> ( ( Chi2 @ ( S4 @ Y5 ) )
= C ) ) ) )
& ( member952132173341509300at_nat @ Chi2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_or1140352010380016476at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_918_layered__subspace__def,axiom,
( hales_4259056829518216709ce_int
= ( ^ [S4: ( nat > nat ) > nat > nat,K3: nat,N4: nat,T2: nat,R2: int,Chi2: ( nat > nat ) > int] :
( ( hales_is_subspace @ S4 @ K3 @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C: int] :
( ( ord_less_int @ C @ R2 )
& ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ ( hales_classes @ K3 @ T2 @ X2 ) )
=> ( ( Chi2 @ ( S4 @ Y5 ) )
= C ) ) ) )
& ( member_nat_nat_int @ Chi2
@ ( piE_nat_nat_int @ ( hales_cube @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_int @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_919_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S4: ( nat > nat ) > nat > nat,K3: nat,N4: nat,T2: nat,R2: nat,Chi2: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S4 @ K3 @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C: nat] :
( ( ord_less_nat @ C @ R2 )
& ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ ( hales_classes @ K3 @ T2 @ X2 ) )
=> ( ( Chi2 @ ( S4 @ Y5 ) )
= C ) ) ) )
& ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_920_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_921_atMost__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_922_atMost__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or6142498856979658663at_nat @ K ) )
= ( ord_le3127000006974329230at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_923_atMost__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or5033131092550408871at_nat @ K ) )
= ( ord_le2017632242545079438at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_924_atMost__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or3591701359631937174at_nat @ K ) )
= ( ord_le747776305331315197at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_925_atMost__iff,axiom,
! [I: set_nat_nat,K: set_nat_nat] :
( ( member_set_nat_nat @ I @ ( set_or250740698829186286at_nat @ K ) )
= ( ord_le9059583361652607317at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_926_atMost__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I @ K ) ) ).
% atMost_iff
thf(fact_927_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_928_atMost__subset__iff,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X ) @ ( set_or250740698829186286at_nat @ Y ) )
= ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_929_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_930_atMost__subset__iff,axiom,
! [X: nat > nat,Y: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X ) @ ( set_or9140604705432621368at_nat @ Y ) )
= ( ord_less_eq_nat_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_931_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_932_image__add__atMost,axiom,
! [C3: int,A: int] :
( ( image_int_int @ ( plus_plus_int @ C3 ) @ ( set_ord_atMost_int @ A ) )
= ( set_ord_atMost_int @ ( plus_plus_int @ C3 @ A ) ) ) ).
% image_add_atMost
thf(fact_933_atMost__def,axiom,
( set_or250740698829186286at_nat
= ( ^ [U: set_nat_nat] :
( collect_set_nat_nat
@ ^ [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_934_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_935_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_936_layered__eq__classes,axiom,
! [S2: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( hales_4261547300027266985ce_nat @ S2 @ K @ N @ T @ R3 @ Chi3 )
=> ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_classes @ K @ T @ X3 ) )
=> ! [Xb: nat > nat] :
( ( member_nat_nat @ Xb @ ( hales_classes @ K @ T @ X3 ) )
=> ( ( Chi3 @ ( S2 @ Xa ) )
= ( Chi3 @ ( S2 @ Xb ) ) ) ) ) ) ) ).
% layered_eq_classes
thf(fact_937_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_938_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_939_Bf__defs,axiom,
( ( disjoi6798895846410478970at_nat @ b @ ( set_ord_atMost_nat @ one_one_nat ) )
& ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ b @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ n ) )
& ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ b @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
& ( member_nat_nat @ f
@ ( piE_nat_nat @ ( b @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ t ) ) )
& ( member952132173341509300at_nat
@ ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( l2 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t ) )
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ t )
@ ^ [I2: nat > nat] : ( hales_cube @ n @ t ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ t ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ one_one_nat ) )
=> ( ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( l2 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t )
@ X3
@ Xa )
= ( f @ Xa ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ J4 ) )
=> ( ( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( l2 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t )
@ X3
@ Xa )
= ( X3 @ J4 ) ) ) ) ) ) ) ).
% Bf_defs
thf(fact_940_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B2: set_set_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_941_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_942_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_943_dim1__subspace__is__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S2 @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T ) )
@ N
@ T ) ) ) ).
% dim1_subspace_is_line
thf(fact_944_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R3: int,Chi3: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ? [C1: int,C22: int] :
( ( ord_less_int @ C1 @ R3 )
& ( ord_less_int @ C22 @ R3 )
& ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
= C1 ) )
& ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_945_dim1__layered__subspace__as__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R3 )
& ( ord_less_nat @ C22 @ R3 )
& ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
= C1 ) )
& ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_946_empty__iff,axiom,
! [C3: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ C3 @ bot_bo7445843802507891576at_nat ) ).
% empty_iff
thf(fact_947_empty__iff,axiom,
! [C3: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ C3 @ bot_bo945813143650711160at_nat ) ).
% empty_iff
thf(fact_948_empty__iff,axiom,
! [C3: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C3 @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_949_empty__iff,axiom,
! [C3: nat] :
~ ( member_nat @ C3 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_950_empty__iff,axiom,
! [C3: nat > nat] :
~ ( member_nat_nat @ C3 @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_951_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ! [X2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X2 @ A2 ) )
= ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% all_not_in_conv
thf(fact_952_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ! [X2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X2 @ A2 ) )
= ( A2 = bot_bo945813143650711160at_nat ) ) ).
% all_not_in_conv
thf(fact_953_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ! [X2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X2 @ A2 ) )
= ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_954_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_955_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_956_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_957_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_958_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_959_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_960_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_961_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_962_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_963_image__is__empty,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_964_image__is__empty,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ( image_3205354838064109189at_nat @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_965_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_966_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_967_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_968_empty__is__image,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_969_empty__is__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_970_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_971_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_972_image__empty,axiom,
! [F: nat > nat > nat] :
( ( image_nat_nat_nat2 @ F @ bot_bot_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_973_image__empty,axiom,
! [F: ( nat > nat ) > nat] :
( ( image_nat_nat_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_974_image__empty,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ( image_3205354838064109189at_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_975_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_976_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_977_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_978_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_979_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_980_PiE__empty__range,axiom,
! [I: nat > nat > nat,I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat2 @ I5 @ F2 )
= bot_bo3013702615682746855at_nat ) ) ) ).
% PiE_empty_range
thf(fact_981_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat,I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat @ I5 @ F2 )
= bot_bo4508028030728203495at_nat ) ) ) ).
% PiE_empty_range
thf(fact_982_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat > nat,I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat )
=> ( ( piE_na4548495224246695081at_nat @ I5 @ F2 )
= bot_bo3386126977483763158at_nat ) ) ) ).
% PiE_empty_range
thf(fact_983_PiE__empty__range,axiom,
! [I: nat > nat,I5: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat @ I5 @ F2 )
= bot_bo945813143650711160at_nat ) ) ) ).
% PiE_empty_range
thf(fact_984_PiE__empty__range,axiom,
! [I: nat,I5: set_nat,F2: nat > set_nat] :
( ( member_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat @ I5 @ F2 )
= bot_bot_set_nat_nat ) ) ) ).
% PiE_empty_range
thf(fact_985_PiE__empty__range,axiom,
! [I: nat > nat > nat,I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na7122919648973241129at_nat @ I5 @ F2 )
= bot_bo4227112084914574038at_nat ) ) ) ).
% PiE_empty_range
thf(fact_986_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat,I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na6840239867990089257at_nat @ I5 @ F2 )
= bot_bo4291610329234208214at_nat ) ) ) ).
% PiE_empty_range
thf(fact_987_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat > nat,I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na6564615839001774232at_nat @ I5 @ F2 )
= bot_bo3618716324728726597at_nat ) ) ) ).
% PiE_empty_range
thf(fact_988_PiE__empty__range,axiom,
! [I: nat,I5: set_nat,F2: nat > set_nat_nat] :
( ( member_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat2 @ I5 @ F2 )
= bot_bo7445843802507891576at_nat ) ) ) ).
% PiE_empty_range
thf(fact_989_PiE__empty__range,axiom,
! [I: nat > nat,I5: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I @ I5 )
=> ( ( ( F2 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat_nat3 @ I5 @ F2 )
= bot_bo3919185967433191911at_nat ) ) ) ).
% PiE_empty_range
thf(fact_990_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_991_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat > nat] :
( bot_bot_set_nat_nat
!= ( set_or9140604705432621368at_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_992_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_993_PiE__eq__empty__iff,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ F2 )
= bot_bo945813143650711160at_nat )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_994_PiE__eq__empty__iff,axiom,
! [I5: set_nat,F2: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ F2 )
= bot_bo7445843802507891576at_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_995_PiE__eq__empty__iff,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ F2 )
= bot_bo3919185967433191911at_nat )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_996_PiE__eq__empty__iff,axiom,
! [I5: set_nat,F2: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ F2 )
= bot_bot_set_nat_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_997_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_998_not__psubset__empty,axiom,
! [A2: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_999_subset__emptyI,axiom,
! [A2: set_nat_nat_nat] :
( ! [X4: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X4 @ A2 )
=> ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat ) ) ).
% subset_emptyI
thf(fact_1000_subset__emptyI,axiom,
! [A2: set_nat_nat_nat2] :
( ! [X4: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X4 @ A2 )
=> ( ord_le5934964663421696068at_nat @ A2 @ bot_bo945813143650711160at_nat ) ) ).
% subset_emptyI
thf(fact_1001_subset__emptyI,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ! [X4: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X4 @ A2 )
=> ( ord_le5260717879541182899at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ) ).
% subset_emptyI
thf(fact_1002_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1003_subset__emptyI,axiom,
! [A2: set_nat_nat] :
( ! [X4: nat > nat] :
~ ( member_nat_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% subset_emptyI
thf(fact_1004_emptyE,axiom,
! [A: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ A @ bot_bo7445843802507891576at_nat ) ).
% emptyE
thf(fact_1005_emptyE,axiom,
! [A: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ A @ bot_bo945813143650711160at_nat ) ).
% emptyE
thf(fact_1006_emptyE,axiom,
! [A: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_1007_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1008_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_1009_equals0D,axiom,
! [A2: set_nat_nat_nat,A: nat > nat > nat] :
( ( A2 = bot_bo7445843802507891576at_nat )
=> ~ ( member_nat_nat_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_1010_equals0D,axiom,
! [A2: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( A2 = bot_bo945813143650711160at_nat )
=> ~ ( member_nat_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1011_equals0D,axiom,
! [A2: set_nat_nat_nat_nat3,A: ( nat > nat ) > nat > nat] :
( ( A2 = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1012_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1013_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1014_equals0I,axiom,
! [A2: set_nat_nat_nat] :
( ! [Y2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ Y2 @ A2 )
=> ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% equals0I
thf(fact_1015_equals0I,axiom,
! [A2: set_nat_nat_nat2] :
( ! [Y2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ Y2 @ A2 )
=> ( A2 = bot_bo945813143650711160at_nat ) ) ).
% equals0I
thf(fact_1016_equals0I,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ! [Y2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y2 @ A2 )
=> ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_1017_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1018_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y2: nat > nat] :
~ ( member_nat_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_1019_ex__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ? [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= ( A2 != bot_bo7445843802507891576at_nat ) ) ).
% ex_in_conv
thf(fact_1020_ex__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ? [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= ( A2 != bot_bo945813143650711160at_nat ) ) ).
% ex_in_conv
thf(fact_1021_ex__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ? [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A2 ) )
= ( A2 != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_1022_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1023_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_1024_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_1025_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_1026_empty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X2: nat > nat] : $false ) ) ).
% empty_def
thf(fact_1027_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1028_bot_Oextremum__strict,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% bot.extremum_strict
thf(fact_1029_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1030_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1031_bot_Onot__eq__extremum,axiom,
! [A: set_nat_nat] :
( ( A != bot_bot_set_nat_nat )
= ( ord_less_set_nat_nat @ bot_bot_set_nat_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1032_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1033_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1034_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1035_bot_Oextremum,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% bot.extremum
thf(fact_1036_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1037_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1038_bot_Oextremum__unique,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_1039_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1040_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1041_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1042_verit__sko__ex_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P @ ( fChoice_nat_nat @ P ) )
= A2 )
=> ( ( ? [X7: nat > nat] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_1043_verit__sko__forall,axiom,
( ( ^ [P2: ( nat > nat ) > $o] :
! [X6: nat > nat] : ( P2 @ X6 ) )
= ( ^ [P3: ( nat > nat ) > $o] :
( P3
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_1044_verit__sko__forall_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P
@ ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
= A2 )
=> ( ( ! [X7: nat > nat] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_1045_verit__sko__forall_H_H,axiom,
! [B2: nat > nat,A2: nat > nat,P: ( nat > nat ) > $o] :
( ( B2 = A2 )
=> ( ( ( fChoice_nat_nat @ P )
= A2 )
= ( ( fChoice_nat_nat @ P )
= B2 ) ) ) ).
% verit_sko_forall''
thf(fact_1046_verit__sko__ex__indirect,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat @ P ) )
=> ( ( ? [X7: nat > nat] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_1047_verit__sko__ex__indirect2,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o,P5: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat @ P ) )
=> ( ! [X4: nat > nat] :
( ( P @ X4 )
= ( P5 @ X4 ) )
=> ( ( ? [X7: nat > nat] : ( P5 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_1048_verit__sko__forall__indirect,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
=> ( ( ! [X7: nat > nat] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_1049_verit__sko__forall__indirect2,axiom,
! [X: nat > nat,P: ( nat > nat ) > $o,P5: ( nat > nat ) > $o] :
( ( X
= ( fChoice_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) )
=> ( ! [X4: nat > nat] :
( ( P @ X4 )
= ( P5 @ X4 ) )
=> ( ( ! [X7: nat > nat] : ( P5 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_1050_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat,F3: ( nat > nat > nat ) > set_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I5 @ F2 )
= ( piE_nat_nat_nat_nat2 @ I5 @ F3 ) )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1051_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat,F3: ( ( nat > nat ) > nat ) > set_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I5 @ F2 )
= ( piE_nat_nat_nat_nat @ I5 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1052_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I5 @ F2 )
= ( piE_na4548495224246695081at_nat @ I5 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1053_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I5 @ F2 )
= ( piE_nat_nat_nat @ I5 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1054_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I5 @ F2 )
= ( piE_nat_nat @ I5 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1055_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na7122919648973241129at_nat @ I5 @ F2 )
= ( piE_na7122919648973241129at_nat @ I5 @ F3 ) )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1056_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6840239867990089257at_nat @ I5 @ F2 )
= ( piE_na6840239867990089257at_nat @ I5 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1057_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6564615839001774232at_nat @ I5 @ F2 )
= ( piE_na6564615839001774232at_nat @ I5 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1058_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I5 @ F2 )
= ( piE_nat_nat_nat2 @ I5 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1059_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I5 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1060_PiE__eq__iff,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ F2 )
= ( piE_nat_nat_nat @ I5 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) )
& ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F3 @ X2 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_1061_PiE__eq__iff,axiom,
! [I5: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ F2 )
= ( piE_nat_nat @ I5 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) )
& ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F3 @ X2 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_1062_PiE__eq__iff,axiom,
! [I5: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ F2 )
= ( piE_nat_nat_nat2 @ I5 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) )
& ? [X2: nat] :
( ( member_nat @ X2 @ I5 )
& ( ( F3 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_1063_PiE__eq__iff,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) )
& ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
& ( ( F3 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_1064_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1065_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat,F3: ( nat > nat > nat ) > set_nat,I: nat > nat > nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I5 @ F2 )
= ( piE_nat_nat_nat_nat2 @ I5 @ F3 ) )
=> ( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1066_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat,F3: ( ( nat > nat ) > nat ) > set_nat,I: ( nat > nat ) > nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I5 @ F2 )
= ( piE_nat_nat_nat_nat @ I5 @ F3 ) )
=> ( ( member_nat_nat_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1067_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat,I: ( nat > nat ) > nat > nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I5 @ F2 )
= ( piE_na4548495224246695081at_nat @ I5 @ F3 ) )
=> ( ( member952132173341509300at_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1068_PiE__eq__subset,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat,I: nat > nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I5 @ F2 )
= ( piE_nat_nat_nat @ I5 @ F3 ) )
=> ( ( member_nat_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1069_PiE__eq__subset,axiom,
! [I5: set_nat,F2: nat > set_nat,F3: nat > set_nat,I: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I5 @ F2 )
= ( piE_nat_nat @ I5 @ F3 ) )
=> ( ( member_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1070_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat,I: nat > nat > nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na7122919648973241129at_nat @ I5 @ F2 )
= ( piE_na7122919648973241129at_nat @ I5 @ F3 ) )
=> ( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1071_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat ) > set_nat_nat,I: ( nat > nat ) > nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6840239867990089257at_nat @ I5 @ F2 )
= ( piE_na6840239867990089257at_nat @ I5 @ F3 ) )
=> ( ( member_nat_nat_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1072_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,I: ( nat > nat ) > nat > nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6564615839001774232at_nat @ I5 @ F2 )
= ( piE_na6564615839001774232at_nat @ I5 @ F3 ) )
=> ( ( member952132173341509300at_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1073_PiE__eq__subset,axiom,
! [I5: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat,I: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I5 @ F2 )
= ( piE_nat_nat_nat2 @ I5 @ F3 ) )
=> ( ( member_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1074_PiE__eq__subset,axiom,
! [I5: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat,I: nat > nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I5 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F3 ) )
=> ( ( member_nat_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I ) @ ( F3 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1075_is__subspace__def,axiom,
( hales_is_subspace
= ( ^ [S4: ( nat > nat ) > nat > nat,K3: nat,N4: nat,T2: nat] :
? [B3: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ K3 ) )
& ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ K3 ) ) )
= ( set_ord_lessThan_nat @ N4 ) )
& ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ K3 ) ) )
& ? [F4: nat > nat] :
( ( member_nat_nat @ F4
@ ( piE_nat_nat @ ( B3 @ K3 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
& ( member952132173341509300at_nat @ S4
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ K3 @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N4 @ T2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ K3 @ T2 ) )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B3 @ K3 ) )
=> ( ( S4 @ X2 @ Y5 )
= ( F4 @ Y5 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ K3 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B3 @ J3 ) )
=> ( ( S4 @ X2 @ Y5 )
= ( X2 @ J3 ) ) ) ) ) ) ) ) ) ) ).
% is_subspace_def
thf(fact_1076_dim1__subspace__elims_I4_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J2 ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( B2 @ zero_zero_nat )
!= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(4)
thf(fact_1077_dim1__subspace__elims_I3_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J2 ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B2 @ one_one_nat ) )
=> ( ( S2 @ X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( B2 @ zero_zero_nat ) )
=> ( ( S2 @ X3 @ Xa )
= ( X3 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(3)
thf(fact_1078_cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) )
@ zero_zero_nat )
= S ) ) ).
% cube_props(2)
thf(fact_1079_cube__props_I4_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_1080_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1081_prop__restrict,axiom,
! [X: nat > nat > nat,Z4: set_nat_nat_nat,X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ X @ Z4 )
=> ( ( ord_le3211623285424100676at_nat @ Z4
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1082_prop__restrict,axiom,
! [X: ( nat > nat ) > nat,Z4: set_nat_nat_nat2,X5: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ X @ Z4 )
=> ( ( ord_le5934964663421696068at_nat @ Z4
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1083_prop__restrict,axiom,
! [X: ( nat > nat ) > nat > nat,Z4: set_nat_nat_nat_nat3,X5: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ X @ Z4 )
=> ( ( ord_le5260717879541182899at_nat @ Z4
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1084_prop__restrict,axiom,
! [X: nat > nat,Z4: set_nat_nat,X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X @ Z4 )
=> ( ( ord_le9059583361652607317at_nat @ Z4
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1085_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1086_Collect__restrict,axiom,
! [X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1087_Collect__restrict,axiom,
! [X5: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1088_Collect__restrict,axiom,
! [X5: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1089_Collect__restrict,axiom,
! [X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1090_cube__props_I3_J,axiom,
! [S: nat,T: nat,S2: ( nat > nat ) > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat
@ ^ [S5: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat
@ ^ [S5: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_1091_cube__props_I3_J,axiom,
! [S: nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat_nat2
@ ^ [S5: nat] :
( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S5 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_1092_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R3: int,Chi3: ( nat > nat ) > int] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4259056829518216709ce_int @ S2 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ! [L5: nat] :
( ( ord_less_nat @ L5 @ T )
=> ( ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
= ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= L5 ) ) ) ) ) )
& ( ord_less_int
@ ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
@ R3 ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_1093_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S2: ( nat > nat ) > nat > nat,N: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S2 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ! [L5: nat] :
( ( ord_less_nat @ L5 @ T )
=> ( ( ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
= ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= L5 ) ) ) ) ) )
& ( ord_less_nat
@ ( Chi3
@ ( S2
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) ) )
@ R3 ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_1094_SUP__apply,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ X )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [Y5: nat] : ( F @ Y5 @ X )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_1095_SUP__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,X: nat] :
( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ X )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [Y5: nat > nat] : ( F @ Y5 @ X )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_1096_UN__constant,axiom,
! [A2: set_nat,C3: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y5: nat] : C3
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y5: nat] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_1097_UN__constant,axiom,
! [A2: set_nat_nat,C3: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y5: nat > nat] : C3
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y5: nat > nat] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_1098_UN__constant,axiom,
! [A2: set_nat,C3: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C3
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_1099_UN__constant,axiom,
! [A2: set_nat_nat,C3: set_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y5: nat > nat] : C3
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y5: nat > nat] : C3
@ A2 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_1100_SUP__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_1101_SUP__const,axiom,
! [A2: set_nat_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [I2: nat > nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_1102_cSUP__const,axiom,
! [A2: set_nat,C3: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : C3
@ A2 ) )
= C3 ) ) ).
% cSUP_const
thf(fact_1103_cSUP__const,axiom,
! [A2: set_nat_nat,C3: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [X2: nat > nat] : C3
@ A2 ) )
= C3 ) ) ).
% cSUP_const
thf(fact_1104_cSUP__const,axiom,
! [A2: set_nat,C3: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C3
@ A2 ) )
= C3 ) ) ).
% cSUP_const
thf(fact_1105_cSUP__const,axiom,
! [A2: set_nat_nat,C3: nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [X2: nat > nat] : C3
@ A2 ) )
= C3 ) ) ).
% cSUP_const
thf(fact_1106_ball__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B2 @ X2 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_1107_bex__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y5: nat] :
( ( member_nat @ Y5 @ ( B2 @ X2 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_1108_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B2 @ X2 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_1109_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y5: nat] :
( ( member_nat @ Y5 @ ( B2 @ X2 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_1110_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1111_SUP__identity__eq,axiom,
! [A2: set_nat_nat] :
( ( comple2450677804321093138at_nat
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( comple2450677804321093138at_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1112_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1113_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1114_UN__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_1115_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1116_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1117_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1118_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B2: nat > set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1119_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B2: nat > set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat @ B @ ( comple1667856448326461495at_nat @ ( image_8854229838293529787at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1120_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1121_UN__I,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B2: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1122_UN__I,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B2: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1123_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B2: nat > set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ ( B2 @ A ) )
=> ( member952132173341509300at_nat @ B @ ( comple2605510978757769510at_nat @ ( image_3332361743537024938at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1124_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: ( nat > nat ) > set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_470123710477037866at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1125_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_1126_SUP__bot__conv_I1_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_1127_SUP__bot__conv_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_1128_all__PiE__elements,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat,P: ( nat > nat ) > nat > $o] :
( ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( piE_nat_nat_nat @ I5 @ S2 ) )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ I5 )
=> ( P @ Y5 @ ( X2 @ Y5 ) ) ) ) )
= ( ( ( piE_nat_nat_nat @ I5 @ S2 )
= bot_bo945813143650711160at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( S2 @ X2 ) )
=> ( P @ X2 @ Y5 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_1129_all__PiE__elements,axiom,
! [I5: set_nat,S2: nat > set_nat_nat,P: nat > ( nat > nat ) > $o] :
( ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( piE_nat_nat_nat2 @ I5 @ S2 ) )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ I5 )
=> ( P @ Y5 @ ( X2 @ Y5 ) ) ) ) )
= ( ( ( piE_nat_nat_nat2 @ I5 @ S2 )
= bot_bo7445843802507891576at_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ ( S2 @ X2 ) )
=> ( P @ X2 @ Y5 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_1130_all__PiE__elements,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat_nat,P: ( nat > nat ) > ( nat > nat ) > $o] :
( ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ ( piE_nat_nat_nat_nat3 @ I5 @ S2 ) )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ I5 )
=> ( P @ Y5 @ ( X2 @ Y5 ) ) ) ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S2 )
= bot_bo3919185967433191911at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ ( S2 @ X2 ) )
=> ( P @ X2 @ Y5 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_1131_all__PiE__elements,axiom,
! [I5: set_nat,S2: nat > set_nat,P: nat > nat > $o] :
( ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( piE_nat_nat @ I5 @ S2 ) )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ I5 )
=> ( P @ Y5 @ ( X2 @ Y5 ) ) ) ) )
= ( ( ( piE_nat_nat @ I5 @ S2 )
= bot_bot_set_nat_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( S2 @ X2 ) )
=> ( P @ X2 @ Y5 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_1132_PiE__eq,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat,T3: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ S2 )
= ( piE_nat_nat_nat @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat @ I5 @ S2 )
= bot_bo945813143650711160at_nat )
& ( ( piE_nat_nat_nat @ I5 @ T3 )
= bot_bo945813143650711160at_nat ) )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( S2 @ X2 )
= ( T3 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_1133_PiE__eq,axiom,
! [I5: set_nat,S2: nat > set_nat_nat,T3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ S2 )
= ( piE_nat_nat_nat2 @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat2 @ I5 @ S2 )
= bot_bo7445843802507891576at_nat )
& ( ( piE_nat_nat_nat2 @ I5 @ T3 )
= bot_bo7445843802507891576at_nat ) )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( S2 @ X2 )
= ( T3 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_1134_PiE__eq,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat_nat,T3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ S2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S2 )
= bot_bo3919185967433191911at_nat )
& ( ( piE_nat_nat_nat_nat3 @ I5 @ T3 )
= bot_bo3919185967433191911at_nat ) )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ( S2 @ X2 )
= ( T3 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_1135_PiE__eq,axiom,
! [I5: set_nat,S2: nat > set_nat,T3: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ S2 )
= ( piE_nat_nat @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat @ I5 @ S2 )
= bot_bot_set_nat_nat )
& ( ( piE_nat_nat @ I5 @ T3 )
= bot_bot_set_nat_nat ) )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ( S2 @ X2 )
= ( T3 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_1136_bot__empty__eq,axiom,
( bot_bo1514271634159724301_nat_o
= ( ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ bot_bo7445843802507891576at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1137_bot__empty__eq,axiom,
( bot_bo6348804412059337741_nat_o
= ( ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ bot_bo945813143650711160at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1138_bot__empty__eq,axiom,
( bot_bo1568108970253895006_nat_o
= ( ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ bot_bo3919185967433191911at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1139_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1140_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_1141_Sup__SUP__eq,axiom,
( comple8312177224774716605_nat_o
= ( ^ [S4: set_nat_nat_o,X2: nat > nat] : ( member_nat_nat @ X2 @ ( comple5448282615319421384at_nat @ ( image_7977807581451749376at_nat @ collect_nat_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1142_Sup__SUP__eq,axiom,
( comple3396693796109600270_nat_o
= ( ^ [S4: set_nat_nat_nat_o,X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ ( comple8167887107183641911at_nat @ ( image_3610001086604609088at_nat @ collect_nat_nat_nat2 @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1143_Sup__SUP__eq,axiom,
( comple8231226574009213710_nat_o
= ( ^ [S4: set_nat_nat_nat_o2,X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ ( comple1667856448326461495at_nat @ ( image_5425260358592644672at_nat @ collect_nat_nat_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1144_Sup__SUP__eq,axiom,
( comple2115216063353097951_nat_o
= ( ^ [S4: set_na2445831480116662482_nat_o,X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ ( comple2605510978757769510at_nat @ ( image_4065302347126311296at_nat @ collec3567154360959927026at_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1145_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1146_Sup__set__def,axiom,
( comple5448282615319421384at_nat
= ( ^ [A3: set_set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] : ( complete_Sup_Sup_o @ ( image_set_nat_nat_o @ ( member_nat_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1147_Sup__set__def,axiom,
( comple8167887107183641911at_nat
= ( ^ [A3: set_set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( complete_Sup_Sup_o @ ( image_5198217506544545261_nat_o @ ( member_nat_nat_nat2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1148_Sup__set__def,axiom,
( comple1667856448326461495at_nat
= ( ^ [A3: set_set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( complete_Sup_Sup_o @ ( image_8774134582277556973_nat_o @ ( member_nat_nat_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1149_Sup__set__def,axiom,
( comple2605510978757769510at_nat
= ( ^ [A3: set_se3022870823424313865at_nat] :
( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( complete_Sup_Sup_o @ ( image_7580978635682194622_nat_o @ ( member952132173341509300at_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1150_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1151_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1152_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_1153_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1154_SUP__Sup__eq,axiom,
! [S2: set_set_nat_nat] :
( ( comple8312177224774716605_nat_o
@ ( image_1242417779249009364_nat_o
@ ^ [I2: set_nat_nat,X2: nat > nat] : ( member_nat_nat @ X2 @ I2 )
@ S2 ) )
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( comple5448282615319421384at_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1155_SUP__Sup__eq,axiom,
! [S2: set_set_nat_nat_nat] :
( ( comple3396693796109600270_nat_o
@ ( image_2840114971476761718_nat_o
@ ^ [I2: set_nat_nat_nat,X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ I2 )
@ S2 ) )
= ( ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ ( comple8167887107183641911at_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1156_SUP__Sup__eq,axiom,
! [S2: set_set_nat_nat_nat2] :
( ( comple8231226574009213710_nat_o
@ ( image_6357918107393578614_nat_o
@ ^ [I2: set_nat_nat_nat2,X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ I2 )
@ S2 ) )
= ( ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ ( comple1667856448326461495at_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1157_SUP__Sup__eq,axiom,
! [S2: set_se3022870823424313865at_nat] :
( ( comple2115216063353097951_nat_o
@ ( image_4040409651686222360_nat_o
@ ^ [I2: set_nat_nat_nat_nat3,X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ I2 )
@ S2 ) )
= ( ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ ( comple2605510978757769510at_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1158_SUP__Sup__eq,axiom,
! [S2: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o2
@ ^ [I2: set_nat,X2: nat] : ( member_nat @ X2 @ I2 )
@ S2 ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1159_SUP__UN__eq,axiom,
! [R3: nat > set_nat,S2: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I2: nat,X2: nat] : ( member_nat @ X2 @ ( R3 @ I2 ) )
@ S2 ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R3 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_1160_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D2: nat > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat_nat2 @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1161_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D2: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1162_Inf_OINF__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D2: ( nat > nat ) > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Inf @ ( image_3205354838064109189at_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1163_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D2: nat > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat_nat2 @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1164_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D2: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1165_Sup_OSUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D2: ( nat > nat ) > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Sup @ ( image_3205354838064109189at_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1166_subset__PiE,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat,T3: ( nat > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ I5 @ S2 ) @ ( piE_nat_nat_nat @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat @ I5 @ S2 )
= bot_bo945813143650711160at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( S2 @ X2 ) @ ( T3 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_1167_subset__PiE,axiom,
! [I5: set_nat,S2: nat > set_nat_nat,T3: nat > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ I5 @ S2 ) @ ( piE_nat_nat_nat2 @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat2 @ I5 @ S2 )
= bot_bo7445843802507891576at_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( S2 @ X2 ) @ ( T3 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_1168_subset__PiE,axiom,
! [I5: set_nat_nat,S2: ( nat > nat ) > set_nat_nat,T3: ( nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ I5 @ S2 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S2 )
= bot_bo3919185967433191911at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( S2 @ X2 ) @ ( T3 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_1169_subset__PiE,axiom,
! [I5: set_nat,S2: nat > set_nat,T3: nat > set_nat] :
( ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ I5 @ S2 ) @ ( piE_nat_nat @ I5 @ T3 ) )
= ( ( ( piE_nat_nat @ I5 @ S2 )
= bot_bot_set_nat_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( S2 @ X2 ) @ ( T3 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_1170_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Inf
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_1171_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Sup
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1172_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1173_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1174_cSup__eq,axiom,
! [X5: set_int,A: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ X4 @ A ) )
=> ( ! [Y2: int] :
( ! [X3: int] :
( ( member_int @ X3 @ X5 )
=> ( ord_less_eq_int @ X3 @ Y2 ) )
=> ( ord_less_eq_int @ A @ Y2 ) )
=> ( ( complete_Sup_Sup_int @ X5 )
= A ) ) ) ).
% cSup_eq
thf(fact_1175_cSup__eq__maximum,axiom,
! [Z2: set_nat_nat,X5: set_set_nat_nat] :
( ( member_set_nat_nat @ Z2 @ X5 )
=> ( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z2 ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_1176_cSup__eq__maximum,axiom,
! [Z2: int,X5: set_int] :
( ( member_int @ Z2 @ X5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X5 )
=> ( ord_less_eq_int @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_int @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_1177_cSup__eq__maximum,axiom,
! [Z2: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z2 @ X5 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_1178_cSup__eq__maximum,axiom,
! [Z2: nat,X5: set_nat] :
( ( member_nat @ Z2 @ X5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_1179_Sup__upper2,axiom,
! [U2: set_nat_nat,A2: set_set_nat_nat,V: set_nat_nat] :
( ( member_set_nat_nat @ U2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ V @ U2 )
=> ( ord_le9059583361652607317at_nat @ V @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1180_Sup__upper2,axiom,
! [U2: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U2 @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U2 )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1181_Sup__le__iff,axiom,
! [A2: set_set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ B )
= ( ! [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1182_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1183_Sup__upper,axiom,
! [X: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1184_Sup__upper,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1185_Sup__least,axiom,
! [A2: set_set_nat_nat,Z2: set_nat_nat] :
( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1186_Sup__least,axiom,
! [A2: set_set_nat,Z2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_1187_Sup__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [A5: set_nat_nat] :
( ( member_set_nat_nat @ A5 @ A2 )
=> ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ B2 )
& ( ord_le9059583361652607317at_nat @ A5 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1188_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A5 @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1189_Sup__eqI,axiom,
! [A2: set_set_nat_nat,X: set_nat_nat] :
( ! [Y2: set_nat_nat] :
( ( member_set_nat_nat @ Y2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Y2 @ X ) )
=> ( ! [Y2: set_nat_nat] :
( ! [Z5: set_nat_nat] :
( ( member_set_nat_nat @ Z5 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Z5 @ Y2 ) )
=> ( ord_le9059583361652607317at_nat @ X @ Y2 ) )
=> ( ( comple5448282615319421384at_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1190_Sup__eqI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ! [Y2: set_nat] :
( ! [Z5: set_nat] :
( ( member_set_nat @ Z5 @ A2 )
=> ( ord_less_eq_set_nat @ Z5 @ Y2 ) )
=> ( ord_less_eq_set_nat @ X @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1191_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D2: nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1192_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D2: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1193_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D2: nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1194_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat,D2: ( nat > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1195_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > set_nat,D2: ( nat > nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1196_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D2: ( nat > nat ) > nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1197_SUP__cong,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C4: ( nat > nat > nat ) > nat,D2: ( nat > nat > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1198_SUP__cong,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C4: ( ( nat > nat ) > nat ) > nat,D2: ( ( nat > nat ) > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1199_SUP__cong,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C4: ( nat > nat > nat ) > set_nat,D2: ( nat > nat > nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1200_SUP__cong,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C4: ( ( nat > nat ) > nat ) > set_nat,D2: ( ( nat > nat ) > nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1201_Union__subsetI,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A2 )
=> ? [Y3: set_nat_nat] :
( ( member_set_nat_nat @ Y3 @ B2 )
& ( ord_le9059583361652607317at_nat @ X4 @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1202_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ? [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B2 )
& ( ord_less_eq_set_nat @ X4 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1203_Union__upper,axiom,
! [B2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1204_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1205_Union__least,axiom,
! [A2: set_set_nat_nat,C4: set_nat_nat] :
( ! [X8: set_nat_nat] :
( ( member_set_nat_nat @ X8 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X8 @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_1206_Union__least,axiom,
! [A2: set_set_nat,C4: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ X8 @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_1207_Union__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1208_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1209_SUP__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J3: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( F @ I2 @ J3 )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_1210_image__Union,axiom,
! [F: ( nat > nat ) > nat > nat,S2: set_set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( comple5448282615319421384at_nat @ S2 ) )
= ( comple5448282615319421384at_nat @ ( image_3832368097948589297at_nat @ ( image_3205354838064109189at_nat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_1211_image__Union,axiom,
! [F: nat > nat > nat,S2: set_set_nat] :
( ( image_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple5448282615319421384at_nat @ ( image_7054278410236665602at_nat @ ( image_nat_nat_nat2 @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_1212_image__Union,axiom,
! [F: nat > set_nat,S2: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_1213_image__Union,axiom,
! [F: nat > nat,S2: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_1214_UN__UN__flatten,axiom,
! [C4: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B2 @ Y5 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_1215_UN__E,axiom,
! [B: nat,B2: ( nat > nat > nat ) > set_nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_1216_UN__E,axiom,
! [B: nat,B2: ( ( nat > nat ) > nat ) > set_nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_1217_UN__E,axiom,
! [B: nat,B2: ( ( nat > nat ) > nat > nat ) > set_nat,A2: set_nat_nat_nat_nat3] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_1218_dim1__subspace__elims_I2_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J2 ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( inf_inf_set_nat @ ( B2 @ zero_zero_nat ) @ ( B2 @ one_one_nat ) )
= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(2)
thf(fact_1219_dim1__subspace__elims_I1_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S2: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J2 ) )
=> ( ( S2 @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( sup_sup_set_nat @ ( B2 @ zero_zero_nat ) @ ( B2 @ one_one_nat ) )
= ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(1)
thf(fact_1220_B__props,axiom,
( ( ( sup_sup_set_nat @ ( b @ zero_zero_nat ) @ ( b @ one_one_nat ) )
= ( set_ord_lessThan_nat @ n ) )
& ( ( inf_inf_set_nat @ ( b @ zero_zero_nat ) @ ( b @ one_one_nat ) )
= bot_bot_set_nat ) ) ).
% B_props
thf(fact_1221_is__line__elim__t__1,axiom,
! [L: nat > nat > nat,N: nat,T: nat] :
( ( hales_is_line @ L @ N @ T )
=> ( ( T = one_one_nat )
=> ~ ! [B_0: set_nat,B_1: set_nat] :
~ ( ( ( sup_sup_set_nat @ B_0 @ B_1 )
= ( set_ord_lessThan_nat @ N ) )
& ( ( inf_inf_set_nat @ B_0 @ B_1 )
= bot_bot_set_nat )
& ( B_0 != bot_bot_set_nat )
& ! [X3: nat] :
( ( member_nat @ X3 @ B_1 )
=> ! [Xa: nat] :
( ( ord_less_nat @ Xa @ T )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T )
=> ( ( L @ Xa @ X3 )
= ( L @ Y3 @ X3 ) ) ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B_0 )
=> ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ( ( L @ S6 @ X3 )
= S6 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_1222__092_060open_062_123_O_O1_125_A_061_A_1230_M_A1_125_092_060close_062,axiom,
( ( set_ord_atMost_nat @ one_one_nat )
= ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).
% \<open>{..1} = {0, 1}\<close>
thf(fact_1223_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1224_cube1__alt__def,axiom,
! [N: nat] :
( ( hales_cube @ N @ one_one_nat )
= ( insert_nat_nat
@ ( restrict_nat_nat
@ ^ [X2: nat] : zero_zero_nat
@ ( set_ord_lessThan_nat @ N ) )
@ bot_bot_set_nat_nat ) ) ).
% cube1_alt_def
thf(fact_1225_some__inv__into__2,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F4: nat > nat] : ( F4 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
thf(fact_1226_inv__into__cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F4: nat > nat] : ( F4 @ zero_zero_nat )
@ S )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_1227_inv__into__cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F4: nat > nat] : ( F4 @ zero_zero_nat )
@ S
@ zero_zero_nat )
= S ) ) ).
% inv_into_cube_props(2)
thf(fact_1228_some__inv__into,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F4: nat > nat] : ( F4 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into
thf(fact_1229_cube0__alt__def,axiom,
! [T: nat] :
( ( hales_cube @ zero_zero_nat @ T )
= ( insert_nat_nat
@ ^ [X2: nat] : undefined_nat
@ bot_bot_set_nat_nat ) ) ).
% cube0_alt_def
thf(fact_1230_L_H__def,axiom,
( l
= ( fun_upd_nat_nat_nat @ l2 @ t
@ ^ [J3: nat] : ( if_nat @ ( member_nat @ J3 @ ( b @ one_one_nat ) ) @ ( l2 @ ( minus_minus_nat @ t @ one_one_nat ) @ J3 ) @ ( if_nat @ ( member_nat @ J3 @ ( b @ zero_zero_nat ) ) @ t @ undefined_nat ) ) ) ) ).
% L'_def
thf(fact_1231_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1232_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1233_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1234_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1235_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1236_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1237_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1238_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1239_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1240_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1241_diff__le__mono2,axiom,
! [M: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M ) ) ) ).
% diff_le_mono2
thf(fact_1242_le__diff__iff_H,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C3 )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C3 @ A ) @ ( minus_minus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1243_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1244_diff__le__mono,axiom,
! [M: nat,N: nat,L3: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L3 ) @ ( minus_minus_nat @ N @ L3 ) ) ) ).
% diff_le_mono
thf(fact_1245_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1246_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1247_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1248_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1249_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1250_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1251_diff__less__mono2,axiom,
! [M: nat,N: nat,L3: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L3 )
=> ( ord_less_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1252_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1253_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1254_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1255_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1256_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1257_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1258_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1259_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1260_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1261_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1262_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1263_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1264_diff__less__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C3 ) @ ( minus_minus_nat @ B @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_1265_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1266_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1267_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
% Helper facts (4)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X7: nat > nat] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( l2 @ s @ j )
= zero_zero_nat ) ).
%------------------------------------------------------------------------------