TPTP Problem File: SLH0381^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0007_Fair_Stream/prob_00087_002863__11939168_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1488 ( 470 unt; 215 typ; 0 def)
% Number of atoms : 3340 (1258 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10393 ( 187 ~; 20 |; 173 &;8534 @)
% ( 0 <=>;1479 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 1348 (1348 >; 0 *; 0 +; 0 <<)
% Number of symbols : 185 ( 184 usr; 23 con; 0-3 aty)
% Number of variables : 3387 ( 93 ^;3167 !; 127 ?;3387 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:32.495
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
set_Pr4193341848836149977_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
set_Sum_sum_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Nat__Onat_J_J,type,
stream_stream_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
set_Sum_sum_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Nat__Onat_J_J,type,
set_stream_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J,type,
stream_stream_a: $tType ).
thf(ty_n_t__Stream__Ostream_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
stream_a_nat: $tType ).
thf(ty_n_t__Stream__Ostream_I_062_It__Nat__Onat_Mtf__a_J_J,type,
stream_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
set_stream_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Stream__Ostream_I_062_Itf__a_Mtf__a_J_J,type,
stream_a_a: $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__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
stream_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Stream__Ostream_Itf__a_J,type,
stream_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (184)
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_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
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_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Fair__Stream_Ofair_001_062_It__Nat__Onat_Mtf__a_J,type,
fair_fair_nat_a: stream_nat_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001_062_Itf__a_Mt__Nat__Onat_J,type,
fair_fair_a_nat: stream_a_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001_062_Itf__a_Mtf__a_J,type,
fair_fair_a_a: stream_a_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001t__Nat__Onat,type,
fair_fair_nat: stream_nat > $o ).
thf(sy_c_Fair__Stream_Ofair_001tf__a,type,
fair_fair_a: stream_a > $o ).
thf(sy_c_Fair__Stream_Ofair__nats,type,
fair_fair_nats: stream_nat ).
thf(sy_c_Fair__Stream_Ofair__stream_001_062_It__Nat__Onat_Mtf__a_J,type,
fair_f187036196379226704_nat_a: ( nat > nat > a ) > stream_nat_a ).
thf(sy_c_Fair__Stream_Ofair__stream_001_062_Itf__a_Mt__Nat__Onat_J,type,
fair_f6172244769510325546_a_nat: ( nat > a > nat ) > stream_a_nat ).
thf(sy_c_Fair__Stream_Ofair__stream_001_062_Itf__a_Mtf__a_J,type,
fair_fair_stream_a_a: ( nat > a > a ) > stream_a_a ).
thf(sy_c_Fair__Stream_Ofair__stream_001t__Nat__Onat,type,
fair_fair_stream_nat: ( nat > nat ) > stream_nat ).
thf(sy_c_Fair__Stream_Ofair__stream_001tf__a,type,
fair_fair_stream_a: ( nat > a ) > stream_a ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Nat__Onat_J,type,
finite_finite_a_nat: set_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mtf__a_J,type,
finite_finite_a_a: set_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
finite1674126218327898605tion_a: set_option_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
finite659689790015031866_nat_a: set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
finite6644898363146130708_a_nat: set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
finite6544458595007987280od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J,type,
finite3740268481367103950_nat_a: set_Sum_sum_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite502105017643426984_a_nat: set_Sum_sum_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
finite51705147264084924um_a_a: set_Sum_sum_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001tf__a,type,
comp_nat_nat_a: ( nat > nat ) > ( a > nat ) > a > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001tf__a,type,
comp_nat_a_a: ( nat > a ) > ( a > nat ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001tf__a,type,
comp_a_nat_a: ( a > nat ) > ( a > a ) > a > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
id_nat: nat > nat ).
thf(sy_c_Fun_Oid_001t__Set__Oset_It__Nat__Onat_J,type,
id_set_nat: set_nat > set_nat ).
thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__a_J,type,
id_set_a: set_a > set_a ).
thf(sy_c_Fun_Oid_001tf__a,type,
id_a: a > a ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_set_a_set_nat: ( set_a > set_nat ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_a_set_a: ( set_a > set_a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Stream__Ostream_It__Nat__Onat_J_001t__Stream__Ostream_It__Nat__Onat_J,type,
inj_on1381642877210728371am_nat: ( stream_nat > stream_nat ) > set_stream_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Nat__Onat,type,
hilber5277034221543178913on_nat: ( nat > nat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001tf__a,type,
hilbert_bijection_a: ( a > a ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001tf__a,type,
hilber2795491120104822624_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__Nat__Onat,type,
hilber7986931655781312002_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: 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_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > 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_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $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_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_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
top_top_nat_a_o: ( nat > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
top_top_a_nat_o: ( a > nat ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
top_top_a_a_o: ( a > a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
top_top_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
top_top_set_a_nat: set_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
top_top_set_a_a: set_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
top_to2612598781856825737_nat_a: set_Pr4193341848836149977_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
top_to3353692345378799459_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
top_to8063371432257647191od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_It__Nat__Onat_J_J,type,
top_to7548458143485696966am_nat: set_stream_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
top_top_set_stream_a: set_stream_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mtf__a_J_J,type,
top_to54524901450547413_nat_a: set_Sum_sum_nat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J,type,
top_to8848906000605539851um_a_a: set_Sum_sum_a_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Nat__Onat_J,type,
collect_a_nat: ( ( a > nat ) > $o ) > set_a_nat ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
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_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
image_nat_nat_a: ( nat > nat > a ) > set_nat > set_nat_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_Itf__a_Mt__Nat__Onat_J,type,
image_nat_a_nat: ( nat > a > nat ) > set_nat > set_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_Itf__a_Mtf__a_J,type,
image_nat_a_a: ( nat > a > a ) > set_nat > set_a_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__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__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Stream__Ostream_It__Nat__Onat_J,type,
image_nat_stream_nat: ( nat > stream_nat ) > set_nat > set_stream_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Stream__Ostream_Itf__a_J,type,
image_nat_stream_a: ( nat > stream_a ) > set_nat > set_stream_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
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_Itf__a_J,type,
image_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_stream_a_set_a: ( stream_a > set_a ) > set_stream_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Stream__Ostream_It__Nat__Onat_J,type,
image_a_stream_nat: ( a > stream_nat ) > set_a > set_stream_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Stream__Ostream_Itf__a_J,type,
image_a_stream_a: ( a > stream_a ) > set_a > set_stream_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Stream_Ositerate_001t__Nat__Onat,type,
siterate_nat: ( nat > nat ) > nat > stream_nat ).
thf(sy_c_Stream_Ositerate_001tf__a,type,
siterate_a: ( a > a ) > a > stream_a ).
thf(sy_c_Stream_Osmember_001t__Nat__Onat,type,
smember_nat: nat > stream_nat > $o ).
thf(sy_c_Stream_Osmember_001tf__a,type,
smember_a: a > stream_a > $o ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001tf__a,type,
smerge_a: stream_stream_a > stream_a ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_Itf__a_J,type,
snth_stream_a: stream_stream_a > nat > stream_a ).
thf(sy_c_Stream_Osnth_001tf__a,type,
snth_a: stream_a > nat > a ).
thf(sy_c_Stream_Ostream_Osmap_001t__Nat__Onat_001t__Nat__Onat,type,
smap_nat_nat: ( nat > nat ) > stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Nat__Onat_001tf__a,type,
smap_nat_a: ( nat > a ) > stream_nat > stream_a ).
thf(sy_c_Stream_Ostream_Osmap_001tf__a_001t__Nat__Onat,type,
smap_a_nat: ( a > nat ) > stream_a > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001tf__a_001tf__a,type,
smap_a_a: ( a > a ) > stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_Itf__a_J,type,
sset_stream_a: stream_stream_a > set_stream_a ).
thf(sy_c_Stream_Ostream_Osset_001tf__a,type,
sset_a: stream_a > set_a ).
thf(sy_c_Stream_Ostreams_001t__Nat__Onat,type,
streams_nat: set_nat > set_stream_nat ).
thf(sy_c_Stream_Ostreams_001tf__a,type,
streams_a: set_a > set_stream_a ).
thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Nat__Onat,type,
type_d6250493948777748686at_nat: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001tf__a,type,
type_d2627918313818726784_nat_a: ( nat > a ) > ( a > nat ) > set_a > $o ).
thf(sy_c_Typedef_Otype__definition_001tf__a_001t__Nat__Onat,type,
type_d7819358849495216162_a_nat: ( a > nat ) > ( nat > a ) > set_nat > $o ).
thf(sy_c_Typedef_Otype__definition_001tf__a_001tf__a,type,
type_definition_a_a: ( a > a ) > ( a > a ) > set_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_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_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Nat__Onat_J,type,
member_stream_nat: stream_nat > set_stream_nat > $o ).
thf(sy_c_member_001t__Stream__Ostream_Itf__a_J,type,
member_stream_a: stream_a > set_stream_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_f,type,
f: nat > a ).
% Relevant facts (1271)
thf(fact_0_fair__stream,axiom,
! [F: nat > nat > a] :
( ( ( image_nat_nat_a @ F @ top_top_set_nat )
= top_top_set_nat_a )
=> ( fair_fair_nat_a @ ( fair_f187036196379226704_nat_a @ F ) ) ) ).
% fair_stream
thf(fact_1_fair__stream,axiom,
! [F: nat > a > nat] :
( ( ( image_nat_a_nat @ F @ top_top_set_nat )
= top_top_set_a_nat )
=> ( fair_fair_a_nat @ ( fair_f6172244769510325546_a_nat @ F ) ) ) ).
% fair_stream
thf(fact_2_fair__stream,axiom,
! [F: nat > a > a] :
( ( ( image_nat_a_a @ F @ top_top_set_nat )
= top_top_set_a_a )
=> ( fair_fair_a_a @ ( fair_fair_stream_a_a @ F ) ) ) ).
% fair_stream
thf(fact_3_fair__stream,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( fair_fair_nat @ ( fair_fair_stream_nat @ F ) ) ) ).
% fair_stream
thf(fact_4_fair__stream,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( fair_fair_a @ ( fair_fair_stream_a @ F ) ) ) ).
% fair_stream
thf(fact_5_UNIV__I,axiom,
! [X: stream_nat] : ( member_stream_nat @ X @ top_to7548458143485696966am_nat ) ).
% UNIV_I
thf(fact_6_UNIV__I,axiom,
! [X: stream_a] : ( member_stream_a @ X @ top_top_set_stream_a ) ).
% UNIV_I
thf(fact_7_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_8_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_9_UNIV__I,axiom,
! [X: nat > a] : ( member_nat_a @ X @ top_top_set_nat_a ) ).
% UNIV_I
thf(fact_10_UNIV__I,axiom,
! [X: a > nat] : ( member_a_nat @ X @ top_top_set_a_nat ) ).
% UNIV_I
thf(fact_11_UNIV__I,axiom,
! [X: a > a] : ( member_a_a @ X @ top_top_set_a_a ) ).
% UNIV_I
thf(fact_12_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_13_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_14_iso__tuple__UNIV__I,axiom,
! [X: stream_nat] : ( member_stream_nat @ X @ top_to7548458143485696966am_nat ) ).
% iso_tuple_UNIV_I
thf(fact_15_iso__tuple__UNIV__I,axiom,
! [X: stream_a] : ( member_stream_a @ X @ top_top_set_stream_a ) ).
% iso_tuple_UNIV_I
thf(fact_16_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_17_iso__tuple__UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_18_iso__tuple__UNIV__I,axiom,
! [X: nat > a] : ( member_nat_a @ X @ top_top_set_nat_a ) ).
% iso_tuple_UNIV_I
thf(fact_19_iso__tuple__UNIV__I,axiom,
! [X: a > nat] : ( member_a_nat @ X @ top_top_set_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_20_iso__tuple__UNIV__I,axiom,
! [X: a > a] : ( member_a_a @ X @ top_top_set_a_a ) ).
% iso_tuple_UNIV_I
thf(fact_21_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_22_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_23_top__apply,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : top_top_o ) ) ).
% top_apply
thf(fact_24_top__apply,axiom,
( top_top_a_o
= ( ^ [X2: a] : top_top_o ) ) ).
% top_apply
thf(fact_25_image__eqI,axiom,
! [B: a,F: nat > a,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_26_image__eqI,axiom,
! [B: a,F: a > a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_27_image__eqI,axiom,
! [B: nat,F: a > nat,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_28_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_29_image__eqI,axiom,
! [B: stream_nat,F: a > stream_nat,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_stream_nat @ B @ ( image_a_stream_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_30_image__eqI,axiom,
! [B: stream_a,F: a > stream_a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_stream_a @ B @ ( image_a_stream_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_31_image__eqI,axiom,
! [B: set_nat,F: a > set_nat,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_set_nat @ B @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_32_image__eqI,axiom,
! [B: set_a,F: a > set_a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_33_image__eqI,axiom,
! [B: stream_nat,F: nat > stream_nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_stream_nat @ B @ ( image_nat_stream_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_34_image__eqI,axiom,
! [B: stream_a,F: nat > stream_a,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_stream_a @ B @ ( image_nat_stream_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_35_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_36_surjD,axiom,
! [F: nat > a,Y: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_37_surjD,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ? [X3: a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_38_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X3: a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_39_surjD,axiom,
! [F: set_nat > set_a,Y: set_a] :
( ( ( image_set_nat_set_a @ F @ top_top_set_set_nat )
= top_top_set_set_a )
=> ? [X3: set_nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_40_surjD,axiom,
! [F: set_a > set_nat,Y: set_nat] :
( ( ( image_set_a_set_nat @ F @ top_top_set_set_a )
= top_top_set_set_nat )
=> ? [X3: set_a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_41_surjD,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ? [X3: set_a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_42_surjD,axiom,
! [F: nat > nat > a,Y: nat > a] :
( ( ( image_nat_nat_a @ F @ top_top_set_nat )
= top_top_set_nat_a )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_43_surjD,axiom,
! [F: nat > a > nat,Y: a > nat] :
( ( ( image_nat_a_nat @ F @ top_top_set_nat )
= top_top_set_a_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_44_surjD,axiom,
! [F: nat > a > a,Y: a > a] :
( ( ( image_nat_a_a @ F @ top_top_set_nat )
= top_top_set_a_a )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_45_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_46_surjE,axiom,
! [F: nat > a,Y: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_47_surjE,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X3: a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_48_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X3: a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_49_surjE,axiom,
! [F: set_nat > set_a,Y: set_a] :
( ( ( image_set_nat_set_a @ F @ top_top_set_set_nat )
= top_top_set_set_a )
=> ~ ! [X3: set_nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_50_surjE,axiom,
! [F: set_a > set_nat,Y: set_nat] :
( ( ( image_set_a_set_nat @ F @ top_top_set_set_a )
= top_top_set_set_nat )
=> ~ ! [X3: set_a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_51_surjE,axiom,
! [F: set_a > set_a,Y: set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ~ ! [X3: set_a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_52_surjE,axiom,
! [F: nat > nat > a,Y: nat > a] :
( ( ( image_nat_nat_a @ F @ top_top_set_nat )
= top_top_set_nat_a )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_53_surjE,axiom,
! [F: nat > a > nat,Y: a > nat] :
( ( ( image_nat_a_nat @ F @ top_top_set_nat )
= top_top_set_a_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_54_surjE,axiom,
! [F: nat > a > a,Y: a > a] :
( ( ( image_nat_a_a @ F @ top_top_set_nat )
= top_top_set_a_a )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_55_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_56_surjI,axiom,
! [G: nat > a,F: a > nat] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a ) ) ).
% surjI
thf(fact_57_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_58_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_59_surjI,axiom,
! [G: set_nat > set_a,F: set_a > set_nat] :
( ! [X3: set_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_set_nat_set_a @ G @ top_top_set_set_nat )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_60_surjI,axiom,
! [G: set_a > set_nat,F: set_nat > set_a] :
( ! [X3: set_nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_set_a_set_nat @ G @ top_top_set_set_a )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_61_surjI,axiom,
! [G: set_a > set_a,F: set_a > set_a] :
( ! [X3: set_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_set_a_set_a @ G @ top_top_set_set_a )
= top_top_set_set_a ) ) ).
% surjI
thf(fact_62_surjI,axiom,
! [G: nat > nat > a,F: ( nat > a ) > nat] :
( ! [X3: nat > a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat_a @ G @ top_top_set_nat )
= top_top_set_nat_a ) ) ).
% surjI
thf(fact_63_surjI,axiom,
! [G: nat > a > nat,F: ( a > nat ) > nat] :
( ! [X3: a > nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_a_nat @ G @ top_top_set_nat )
= top_top_set_a_nat ) ) ).
% surjI
thf(fact_64_surjI,axiom,
! [G: nat > a > a,F: ( a > a ) > nat] :
( ! [X3: a > a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_a_a @ G @ top_top_set_nat )
= top_top_set_a_a ) ) ).
% surjI
thf(fact_65_rangeI,axiom,
! [F: nat > a,X: nat] : ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_66_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_67_rangeI,axiom,
! [F: a > a,X: a] : ( member_a @ ( F @ X ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_68_rangeI,axiom,
! [F: a > nat,X: a] : ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_69_rangeI,axiom,
! [F: nat > stream_nat,X: nat] : ( member_stream_nat @ ( F @ X ) @ ( image_nat_stream_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_70_rangeI,axiom,
! [F: nat > stream_a,X: nat] : ( member_stream_a @ ( F @ X ) @ ( image_nat_stream_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_71_rangeI,axiom,
! [F: nat > set_nat,X: nat] : ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_72_rangeI,axiom,
! [F: nat > set_a,X: nat] : ( member_set_a @ ( F @ X ) @ ( image_nat_set_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_73_rangeI,axiom,
! [F: a > stream_nat,X: a] : ( member_stream_nat @ ( F @ X ) @ ( image_a_stream_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_74_rangeI,axiom,
! [F: a > stream_a,X: a] : ( member_stream_a @ ( F @ X ) @ ( image_a_stream_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_75_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_76_surj__def,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [Y2: a] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_77_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_78_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y2: a] :
? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_79_surj__def,axiom,
! [F: set_nat > set_a] :
( ( ( image_set_nat_set_a @ F @ top_top_set_set_nat )
= top_top_set_set_a )
= ( ! [Y2: set_a] :
? [X2: set_nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_80_surj__def,axiom,
! [F: set_a > set_nat] :
( ( ( image_set_a_set_nat @ F @ top_top_set_set_a )
= top_top_set_set_nat )
= ( ! [Y2: set_nat] :
? [X2: set_a] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_81_surj__def,axiom,
! [F: set_a > set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
= ( ! [Y2: set_a] :
? [X2: set_a] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_82_surj__def,axiom,
! [F: nat > nat > a] :
( ( ( image_nat_nat_a @ F @ top_top_set_nat )
= top_top_set_nat_a )
= ( ! [Y2: nat > a] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_83_surj__def,axiom,
! [F: nat > a > nat] :
( ( ( image_nat_a_nat @ F @ top_top_set_nat )
= top_top_set_a_nat )
= ( ! [Y2: a > nat] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_84_surj__def,axiom,
! [F: nat > a > a] :
( ( ( image_nat_a_a @ F @ top_top_set_nat )
= top_top_set_a_a )
= ( ! [Y2: a > a] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_85_range__eqI,axiom,
! [B: a,F: nat > a,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_86_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_87_range__eqI,axiom,
! [B: a,F: a > a,X: a] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_88_range__eqI,axiom,
! [B: nat,F: a > nat,X: a] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_89_range__eqI,axiom,
! [B: stream_nat,F: nat > stream_nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_stream_nat @ B @ ( image_nat_stream_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_90_range__eqI,axiom,
! [B: stream_a,F: nat > stream_a,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_stream_a @ B @ ( image_nat_stream_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_91_range__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_92_range__eqI,axiom,
! [B: set_a,F: nat > set_a,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_nat_set_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_93_range__eqI,axiom,
! [B: stream_nat,F: a > stream_nat,X: a] :
( ( B
= ( F @ X ) )
=> ( member_stream_nat @ B @ ( image_a_stream_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_94_range__eqI,axiom,
! [B: stream_a,F: a > stream_a,X: a] :
( ( B
= ( F @ X ) )
=> ( member_stream_a @ B @ ( image_a_stream_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_95_UNIV__eq__I,axiom,
! [A: set_stream_nat] :
( ! [X3: stream_nat] : ( member_stream_nat @ X3 @ A )
=> ( top_to7548458143485696966am_nat = A ) ) ).
% UNIV_eq_I
thf(fact_96_UNIV__eq__I,axiom,
! [A: set_stream_a] :
( ! [X3: stream_a] : ( member_stream_a @ X3 @ A )
=> ( top_top_set_stream_a = A ) ) ).
% UNIV_eq_I
thf(fact_97_UNIV__eq__I,axiom,
! [A: set_set_nat] :
( ! [X3: set_nat] : ( member_set_nat @ X3 @ A )
=> ( top_top_set_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_98_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X3: set_a] : ( member_set_a @ X3 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_99_UNIV__eq__I,axiom,
! [A: set_nat_a] :
( ! [X3: nat > a] : ( member_nat_a @ X3 @ A )
=> ( top_top_set_nat_a = A ) ) ).
% UNIV_eq_I
thf(fact_100_UNIV__eq__I,axiom,
! [A: set_a_nat] :
( ! [X3: a > nat] : ( member_a_nat @ X3 @ A )
=> ( top_top_set_a_nat = A ) ) ).
% UNIV_eq_I
thf(fact_101_UNIV__eq__I,axiom,
! [A: set_a_a] :
( ! [X3: a > a] : ( member_a_a @ X3 @ A )
=> ( top_top_set_a_a = A ) ) ).
% UNIV_eq_I
thf(fact_102_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_103_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_104_top__set__def,axiom,
( top_top_set_nat_a
= ( collect_nat_a @ top_top_nat_a_o ) ) ).
% top_set_def
thf(fact_105_top__set__def,axiom,
( top_top_set_a_nat
= ( collect_a_nat @ top_top_a_nat_o ) ) ).
% top_set_def
thf(fact_106_top__set__def,axiom,
( top_top_set_a_a
= ( collect_a_a @ top_top_a_a_o ) ) ).
% top_set_def
thf(fact_107_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_108_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_109_rev__image__eqI,axiom,
! [X: a,A: set_a,B: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_110_rev__image__eqI,axiom,
! [X: a,A: set_a,B: nat,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_111_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: a,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_112_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_113_rev__image__eqI,axiom,
! [X: a,A: set_a,B: stream_nat,F: a > stream_nat] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_stream_nat @ B @ ( image_a_stream_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_114_rev__image__eqI,axiom,
! [X: a,A: set_a,B: stream_a,F: a > stream_a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_stream_a @ B @ ( image_a_stream_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_115_rev__image__eqI,axiom,
! [X: a,A: set_a,B: set_nat,F: a > set_nat] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_116_rev__image__eqI,axiom,
! [X: a,A: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_117_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: stream_nat,F: nat > stream_nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_stream_nat @ B @ ( image_nat_stream_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_118_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: stream_a,F: nat > stream_a] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_stream_a @ B @ ( image_nat_stream_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_119_ball__imageD,axiom,
! [F: set_nat > set_a,A: set_set_nat,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_set_nat_set_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_120_ball__imageD,axiom,
! [F: set_a > set_nat,A: set_set_a,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_set_a_set_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_121_ball__imageD,axiom,
! [F: set_a > set_a,A: set_set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_set_a_set_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_122_ball__imageD,axiom,
! [F: ( nat > $o ) > set_nat,A: set_nat_o,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_o_set_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat > $o] :
( ( member_nat_o @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_123_ball__imageD,axiom,
! [F: ( a > $o ) > set_a,A: set_a_o,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_o_set_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a > $o] :
( ( member_a_o @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_124_ball__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_nat_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_125_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_126_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_127_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_128_image__cong,axiom,
! [M: set_nat_o,N: set_nat_o,F: ( nat > $o ) > set_nat,G: ( nat > $o ) > set_nat] :
( ( M = N )
=> ( ! [X3: nat > $o] :
( ( member_nat_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_o_set_nat @ F @ M )
= ( image_nat_o_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_129_image__cong,axiom,
! [M: set_a_o,N: set_a_o,F: ( a > $o ) > set_a,G: ( a > $o ) > set_a] :
( ( M = N )
=> ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_o_set_a @ F @ M )
= ( image_a_o_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_130_image__cong,axiom,
! [M: set_set_nat,N: set_set_nat,F: set_nat > set_a,G: set_nat > set_a] :
( ( M = N )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_nat_set_a @ F @ M )
= ( image_set_nat_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_131_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > set_nat,G: set_a > set_nat] :
( ( M = N )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_a_set_nat @ F @ M )
= ( image_set_a_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_132_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( M = N )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_a_set_a @ F @ M )
= ( image_set_a_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_133_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_134_image__cong,axiom,
! [M: set_a,N: set_a,F: a > nat,G: a > nat] :
( ( M = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_nat @ F @ M )
= ( image_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_135_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > a,G: nat > a] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_a @ F @ M )
= ( image_nat_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_136_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_137_bex__imageD,axiom,
! [F: set_nat > set_a,A: set_set_nat,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_set_nat_set_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_138_bex__imageD,axiom,
! [F: set_a > set_nat,A: set_set_a,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_set_a_set_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_139_bex__imageD,axiom,
! [F: set_a > set_a,A: set_set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_set_a_set_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_140_bex__imageD,axiom,
! [F: ( nat > $o ) > set_nat,A: set_nat_o,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_o_set_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat > $o] :
( ( member_nat_o @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_141_bex__imageD,axiom,
! [F: ( a > $o ) > set_a,A: set_a_o,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_o_set_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a > $o] :
( ( member_a_o @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_142_bex__imageD,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_nat_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_143_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_144_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_145_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_a_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_146_image__iff,axiom,
! [Z: set_nat,F: set_a > set_nat,A: set_set_a] :
( ( member_set_nat @ Z @ ( image_set_a_set_nat @ F @ A ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_147_image__iff,axiom,
! [Z: set_nat,F: ( nat > $o ) > set_nat,A: set_nat_o] :
( ( member_set_nat @ Z @ ( image_nat_o_set_nat @ F @ A ) )
= ( ? [X2: nat > $o] :
( ( member_nat_o @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_148_image__iff,axiom,
! [Z: set_a,F: set_nat > set_a,A: set_set_nat] :
( ( member_set_a @ Z @ ( image_set_nat_set_a @ F @ A ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_149_image__iff,axiom,
! [Z: set_a,F: set_a > set_a,A: set_set_a] :
( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_150_image__iff,axiom,
! [Z: set_a,F: ( a > $o ) > set_a,A: set_a_o] :
( ( member_set_a @ Z @ ( image_a_o_set_a @ F @ A ) )
= ( ? [X2: a > $o] :
( ( member_a_o @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_151_image__iff,axiom,
! [Z: a,F: nat > a,A: set_nat] :
( ( member_a @ Z @ ( image_nat_a @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_152_image__iff,axiom,
! [Z: a,F: a > a,A: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_153_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_154_image__iff,axiom,
! [Z: nat,F: a > nat,A: set_a] :
( ( member_nat @ Z @ ( image_a_nat @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_155_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_156_imageI,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_157_imageI,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_158_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_159_imageI,axiom,
! [X: a,A: set_a,F: a > stream_nat] :
( ( member_a @ X @ A )
=> ( member_stream_nat @ ( F @ X ) @ ( image_a_stream_nat @ F @ A ) ) ) ).
% imageI
thf(fact_160_imageI,axiom,
! [X: a,A: set_a,F: a > stream_a] :
( ( member_a @ X @ A )
=> ( member_stream_a @ ( F @ X ) @ ( image_a_stream_a @ F @ A ) ) ) ).
% imageI
thf(fact_161_imageI,axiom,
! [X: a,A: set_a,F: a > set_nat] :
( ( member_a @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ ( image_a_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_162_imageI,axiom,
! [X: a,A: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_163_imageI,axiom,
! [X: nat,A: set_nat,F: nat > stream_nat] :
( ( member_nat @ X @ A )
=> ( member_stream_nat @ ( F @ X ) @ ( image_nat_stream_nat @ F @ A ) ) ) ).
% imageI
thf(fact_164_imageI,axiom,
! [X: nat,A: set_nat,F: nat > stream_a] :
( ( member_nat @ X @ A )
=> ( member_stream_a @ ( F @ X ) @ ( image_nat_stream_a @ F @ A ) ) ) ).
% imageI
thf(fact_165_UNIV__witness,axiom,
? [X3: stream_a] : ( member_stream_a @ X3 @ top_top_set_stream_a ) ).
% UNIV_witness
thf(fact_166_UNIV__witness,axiom,
? [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_167_UNIV__witness,axiom,
? [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_168_UNIV__witness,axiom,
? [X3: nat > a] : ( member_nat_a @ X3 @ top_top_set_nat_a ) ).
% UNIV_witness
thf(fact_169_UNIV__witness,axiom,
? [X3: a > nat] : ( member_a_nat @ X3 @ top_top_set_a_nat ) ).
% UNIV_witness
thf(fact_170_UNIV__witness,axiom,
? [X3: a > a] : ( member_a_a @ X3 @ top_top_set_a_a ) ).
% UNIV_witness
thf(fact_171_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_172_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_173_fair__surj,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( fair_fair_nat @ ( smap_nat_nat @ F @ fair_fair_nats ) ) ) ).
% fair_surj
thf(fact_174_fair__surj,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( fair_fair_a @ ( smap_nat_a @ F @ fair_fair_nats ) ) ) ).
% fair_surj
thf(fact_175_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_176_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_177_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > a,D: a > a,Sup: set_a > a] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_a @ C @ A ) )
= ( Sup @ ( image_a_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_178_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > nat,D: a > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_a_nat @ C @ A ) )
= ( Sup @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_179_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > a,D: nat > a,Sup: set_a > a] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_a @ C @ A ) )
= ( Sup @ ( image_nat_a @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_180_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_181_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > a,D: a > a,Inf: set_a > a] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_a @ C @ A ) )
= ( Inf @ ( image_a_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_182_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C: a > nat,D: a > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_a_nat @ C @ A ) )
= ( Inf @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_183_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > a,D: nat > a,Inf: set_a > a] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_a @ C @ A ) )
= ( Inf @ ( image_nat_a @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_184_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_185_sset__range,axiom,
( sset_nat
= ( ^ [S: stream_nat] : ( image_nat_nat @ ( snth_nat @ S ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_186_sset__range,axiom,
( sset_a
= ( ^ [S: stream_a] : ( image_nat_a @ ( snth_a @ S ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_187_range__subsetD,axiom,
! [F: nat > a,B2: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_188_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_189_range__subsetD,axiom,
! [F: a > a,B2: set_a,I: a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ top_top_set_a ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_190_range__subsetD,axiom,
! [F: a > nat,B2: set_nat,I: a] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ top_top_set_a ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_191_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_192_surj__id,axiom,
( ( image_a_a @ id_a @ top_top_set_a )
= top_top_set_a ) ).
% surj_id
thf(fact_193_comp__surj,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_194_comp__surj,axiom,
! [F: nat > nat,G: nat > a] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a )
=> ( ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_195_comp__surj,axiom,
! [F: nat > a,G: a > nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_196_comp__surj,axiom,
! [F: nat > a,G: a > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_nat_a @ ( comp_a_a_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_197_comp__surj,axiom,
! [F: a > nat,G: nat > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_a_nat @ ( comp_nat_nat_a @ G @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_198_comp__surj,axiom,
! [F: a > nat,G: nat > a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_nat_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_199_comp__surj,axiom,
! [F: a > a,G: a > nat] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat )
=> ( ( image_a_nat @ ( comp_a_nat_a @ G @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_200_comp__surj,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_201_inj__image__mem__iff,axiom,
! [F: nat > a,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ ( F @ A2 ) @ ( image_nat_a @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_202_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_203_inj__image__mem__iff,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_204_inj__image__mem__iff,axiom,
! [F: a > nat,A2: a,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_205_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_206_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_207_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_208_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_209_image__id,axiom,
( ( image_nat_nat @ id_nat )
= id_set_nat ) ).
% image_id
thf(fact_210_image__id,axiom,
( ( image_a_a @ id_a )
= id_set_a ) ).
% image_id
thf(fact_211_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_212_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_213_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_214_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_215_snth__smap,axiom,
! [F: nat > nat,S2: stream_nat,N2: nat] :
( ( snth_nat @ ( smap_nat_nat @ F @ S2 ) @ N2 )
= ( F @ ( snth_nat @ S2 @ N2 ) ) ) ).
% snth_smap
thf(fact_216_stream_Oinj__map,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on1381642877210728371am_nat @ ( smap_nat_nat @ F ) @ top_to7548458143485696966am_nat ) ) ).
% stream.inj_map
thf(fact_217_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_218_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_219_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_220_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_221_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_222_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_223_order_Otrans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_224_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_225_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_226_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_227_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_228_inj__on__add,axiom,
! [A2: nat,A: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A2 ) @ A ) ).
% inj_on_add
thf(fact_229_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_230_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_231_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_232_in__mono,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_233_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_234_subsetD,axiom,
! [A: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_235_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_236_inj__on__id,axiom,
! [A: set_nat] : ( inj_on_nat_nat @ id_nat @ A ) ).
% inj_on_id
thf(fact_237_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_238_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_239_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A5 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_240_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_241_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_242_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_243_inj__on__diff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% inj_on_diff
thf(fact_244_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_245_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_246_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_247_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_248_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_249_inj__on__imageI2,axiom,
! [F3: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_250_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_251_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_252_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_253_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_254_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_255_inj__on__image__set__diff,axiom,
! [F: nat > a,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_a @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_256_inj__on__image__set__diff,axiom,
! [F: a > a,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( image_a_a @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_257_inj__on__image__set__diff,axiom,
! [F: a > nat,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ C )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( image_a_nat @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_258_inj__on__image__set__diff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_259_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_260_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_262_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_263_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_264_smap__alt,axiom,
! [F: nat > nat,S2: stream_nat,S3: stream_nat] :
( ( ( smap_nat_nat @ F @ S2 )
= S3 )
= ( ! [N3: nat] :
( ( F @ ( snth_nat @ S2 @ N3 ) )
= ( snth_nat @ S3 @ N3 ) ) ) ) ).
% smap_alt
thf(fact_265_comp__inj__on__iff,axiom,
! [F: nat > a,A: set_nat,F3: a > nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( inj_on_a_nat @ F3 @ ( image_nat_a @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_a_nat_nat @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_266_comp__inj__on__iff,axiom,
! [F: a > nat,A: set_a,F3: nat > nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( inj_on_nat_nat @ F3 @ ( image_a_nat @ F @ A ) )
= ( inj_on_a_nat @ ( comp_nat_nat_a @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_267_comp__inj__on__iff,axiom,
! [F: nat > nat,A: set_nat,F3: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_268_inj__on__imageI,axiom,
! [G: nat > nat,F: a > nat,A: set_a] :
( ( inj_on_a_nat @ ( comp_nat_nat_a @ G @ F ) @ A )
=> ( inj_on_nat_nat @ G @ ( image_a_nat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_269_inj__on__imageI,axiom,
! [G: a > nat,F: nat > a,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ A )
=> ( inj_on_a_nat @ G @ ( image_nat_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_270_inj__on__imageI,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A )
=> ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_271_comp__inj__on,axiom,
! [F: nat > a,A: set_nat,G: a > nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( inj_on_a_nat @ G @ ( image_nat_a @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_272_comp__inj__on,axiom,
! [F: a > nat,A: set_a,G: nat > nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_a_nat @ F @ A ) )
=> ( inj_on_a_nat @ ( comp_nat_nat_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_273_comp__inj__on,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_274_inj__compose,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_275_inj__compose,axiom,
! [F: nat > nat,G: a > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_a_nat @ G @ top_top_set_a )
=> ( inj_on_a_nat @ ( comp_nat_nat_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_276_inj__compose,axiom,
! [F: a > nat,G: nat > a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( inj_on_nat_a @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_a_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_277_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_278_inj__on__image__mem__iff,axiom,
! [F: a > nat,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_nat @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_279_inj__on__image__mem__iff,axiom,
! [F: nat > a,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_nat_a @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_280_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_281_inj__on__image__eq__iff,axiom,
! [F: nat > a,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_282_inj__on__image__eq__iff,axiom,
! [F: a > a,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_283_inj__on__image__eq__iff,axiom,
! [F: a > nat,C: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ C )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_284_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_285_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_286_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_287_image__diff__subset,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) @ ( image_nat_a @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_288_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_289_image__diff__subset,axiom,
! [F: a > a,A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_290_image__diff__subset,axiom,
! [F: a > nat,A: set_a,B2: set_a] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) @ ( image_a_nat @ F @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_291_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: nat > a,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_292_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: nat > a,F: a > nat,A: set_a] :
( ( Inf @ ( image_nat_a @ G @ ( image_a_nat @ F @ A ) ) )
= ( Inf @ ( image_a_a @ ( comp_nat_a_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_293_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_294_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: a > nat,A: set_a] :
( ( Inf @ ( image_nat_nat @ G @ ( image_a_nat @ F @ A ) ) )
= ( Inf @ ( image_a_nat @ ( comp_nat_nat_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_295_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: a > a,F: nat > a,A: set_nat] :
( ( Inf @ ( image_a_a @ G @ ( image_nat_a @ F @ A ) ) )
= ( Inf @ ( image_nat_a @ ( comp_a_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_296_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: a > a,F: a > a,A: set_a] :
( ( Inf @ ( image_a_a @ G @ ( image_a_a @ F @ A ) ) )
= ( Inf @ ( image_a_a @ ( comp_a_a_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_297_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: a > nat,F: nat > a,A: set_nat] :
( ( Inf @ ( image_a_nat @ G @ ( image_nat_a @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_298_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: a > nat,F: a > a,A: set_a] :
( ( Inf @ ( image_a_nat @ G @ ( image_a_a @ F @ A ) ) )
= ( Inf @ ( image_a_nat @ ( comp_a_nat_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_299_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: nat > a,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_300_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: nat > a,F: a > nat,A: set_a] :
( ( Sup @ ( image_nat_a @ G @ ( image_a_nat @ F @ A ) ) )
= ( Sup @ ( image_a_a @ ( comp_nat_a_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_301_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_302_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: a > nat,A: set_a] :
( ( Sup @ ( image_nat_nat @ G @ ( image_a_nat @ F @ A ) ) )
= ( Sup @ ( image_a_nat @ ( comp_nat_nat_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_303_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: a > a,F: nat > a,A: set_nat] :
( ( Sup @ ( image_a_a @ G @ ( image_nat_a @ F @ A ) ) )
= ( Sup @ ( image_nat_a @ ( comp_a_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_304_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: a > a,F: a > a,A: set_a] :
( ( Sup @ ( image_a_a @ G @ ( image_a_a @ F @ A ) ) )
= ( Sup @ ( image_a_a @ ( comp_a_a_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_305_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: a > nat,F: nat > a,A: set_nat] :
( ( Sup @ ( image_a_nat @ G @ ( image_nat_a @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_306_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: a > nat,F: a > a,A: set_a] :
( ( Sup @ ( image_a_nat @ G @ ( image_a_a @ F @ A ) ) )
= ( Sup @ ( image_a_nat @ ( comp_a_nat_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_307_Inf_OINF__id__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_nat @ id_nat @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_308_Inf_OINF__id__eq,axiom,
! [Inf: set_a > a,A: set_a] :
( ( Inf @ ( image_a_a @ id_a @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_309_Sup_OSUP__id__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_nat @ id_nat @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_310_Sup_OSUP__id__eq,axiom,
! [Sup: set_a > a,A: set_a] :
( ( Sup @ ( image_a_a @ id_a @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_311_inj__image__Compl__subset,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus_uminus_set_a @ ( image_nat_a @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_312_inj__image__Compl__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_313_inj__image__Compl__subset,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A ) ) @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_314_inj__image__Compl__subset,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( uminus_uminus_set_a @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_a_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_315_stream_Omap__ident__strong,axiom,
! [T2: stream_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( sset_nat @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( smap_nat_nat @ F @ T2 )
= T2 ) ) ).
% stream.map_ident_strong
thf(fact_316_stream_Omap__ident__strong,axiom,
! [T2: stream_a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( sset_a @ T2 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( smap_a_a @ F @ T2 )
= T2 ) ) ).
% stream.map_ident_strong
thf(fact_317_snth__sset,axiom,
! [S2: stream_nat,N2: nat] : ( member_nat @ ( snth_nat @ S2 @ N2 ) @ ( sset_nat @ S2 ) ) ).
% snth_sset
thf(fact_318_snth__sset,axiom,
! [S2: stream_a,N2: nat] : ( member_a @ ( snth_a @ S2 @ N2 ) @ ( sset_a @ S2 ) ) ).
% snth_sset
thf(fact_319_image__set__diff,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( image_nat_a @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_320_image__set__diff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_321_image__set__diff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_322_image__set__diff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( image_a_nat @ F @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_323_inj__image__subset__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_324_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_325_inj__image__subset__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_326_inj__image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_327_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_328_surj__Compl__image__subset,axiom,
! [F: nat > a,A: set_nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_nat_a @ F @ A ) ) @ ( image_nat_a @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_329_surj__Compl__image__subset,axiom,
! [F: a > nat,A: set_a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_a_nat @ F @ A ) ) @ ( image_a_nat @ F @ ( uminus_uminus_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_330_surj__Compl__image__subset,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A ) ) @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_331_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_332_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_333_injI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_334_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_335_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ).
% inj_def
thf(fact_336_image__comp,axiom,
! [F: nat > a,G: nat > nat,R: set_nat] :
( ( image_nat_a @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_a @ ( comp_nat_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_337_image__comp,axiom,
! [F: nat > a,G: a > nat,R: set_a] :
( ( image_nat_a @ F @ ( image_a_nat @ G @ R ) )
= ( image_a_a @ ( comp_nat_a_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_338_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_339_image__comp,axiom,
! [F: nat > nat,G: a > nat,R: set_a] :
( ( image_nat_nat @ F @ ( image_a_nat @ G @ R ) )
= ( image_a_nat @ ( comp_nat_nat_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_340_image__comp,axiom,
! [F: a > a,G: nat > a,R: set_nat] :
( ( image_a_a @ F @ ( image_nat_a @ G @ R ) )
= ( image_nat_a @ ( comp_a_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_341_image__comp,axiom,
! [F: a > a,G: a > a,R: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G @ R ) )
= ( image_a_a @ ( comp_a_a_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_342_image__comp,axiom,
! [F: a > nat,G: nat > a,R: set_nat] :
( ( image_a_nat @ F @ ( image_nat_a @ G @ R ) )
= ( image_nat_nat @ ( comp_a_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_343_image__comp,axiom,
! [F: a > nat,G: a > a,R: set_a] :
( ( image_a_nat @ F @ ( image_a_a @ G @ R ) )
= ( image_a_nat @ ( comp_a_nat_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_344_image__eq__imp__comp,axiom,
! [F: nat > a,A: set_nat,G: nat > a,B2: set_nat,H: a > a] :
( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ G @ B2 ) )
=> ( ( image_nat_a @ ( comp_a_a_nat @ H @ F ) @ A )
= ( image_nat_a @ ( comp_a_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_345_image__eq__imp__comp,axiom,
! [F: nat > a,A: set_nat,G: nat > a,B2: set_nat,H: a > nat] :
( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_a_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_a_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_346_image__eq__imp__comp,axiom,
! [F: nat > a,A: set_nat,G: a > a,B2: set_a,H: a > a] :
( ( ( image_nat_a @ F @ A )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_nat_a @ ( comp_a_a_nat @ H @ F ) @ A )
= ( image_a_a @ ( comp_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_347_image__eq__imp__comp,axiom,
! [F: nat > a,A: set_nat,G: a > a,B2: set_a,H: a > nat] :
( ( ( image_nat_a @ F @ A )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_a_nat_nat @ H @ F ) @ A )
= ( image_a_nat @ ( comp_a_nat_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_348_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat,H: nat > a] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_a @ ( comp_nat_a_nat @ H @ F ) @ A )
= ( image_nat_a @ ( comp_nat_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_349_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_350_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: a > nat,B2: set_a,H: nat > a] :
( ( ( image_nat_nat @ F @ A )
= ( image_a_nat @ G @ B2 ) )
=> ( ( image_nat_a @ ( comp_nat_a_nat @ H @ F ) @ A )
= ( image_a_a @ ( comp_nat_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_351_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: a > nat,B2: set_a,H: nat > nat] :
( ( ( image_nat_nat @ F @ A )
= ( image_a_nat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A )
= ( image_a_nat @ ( comp_nat_nat_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_352_image__eq__imp__comp,axiom,
! [F: a > a,A: set_a,G: nat > a,B2: set_nat,H: a > a] :
( ( ( image_a_a @ F @ A )
= ( image_nat_a @ G @ B2 ) )
=> ( ( image_a_a @ ( comp_a_a_a @ H @ F ) @ A )
= ( image_nat_a @ ( comp_a_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_353_image__eq__imp__comp,axiom,
! [F: a > a,A: set_a,G: nat > a,B2: set_nat,H: a > nat] :
( ( ( image_a_a @ F @ A )
= ( image_nat_a @ G @ B2 ) )
=> ( ( image_a_nat @ ( comp_a_nat_a @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_a_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_354_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_355_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_356_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_357_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_358_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_359_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_360_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_361_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_362_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_363_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_364_image__subsetI,axiom,
! [A: set_a,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_365_image__subsetI,axiom,
! [A: set_a,F: a > nat,B2: set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_366_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B2: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_367_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_368_subset__imageE,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_369_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_370_subset__imageE,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_371_subset__imageE,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B2
!= ( image_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_372_image__subset__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_373_image__subset__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_374_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_375_image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_376_subset__image__iff,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_377_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_378_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_379_subset__image__iff,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_380_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_381_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_382_stream_Oset__map,axiom,
! [F: nat > nat,V: stream_nat] :
( ( sset_nat @ ( smap_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( sset_nat @ V ) ) ) ).
% stream.set_map
thf(fact_383_stream_Oset__map,axiom,
! [F: a > nat,V: stream_a] :
( ( sset_nat @ ( smap_a_nat @ F @ V ) )
= ( image_a_nat @ F @ ( sset_a @ V ) ) ) ).
% stream.set_map
thf(fact_384_stream_Oset__map,axiom,
! [F: nat > a,V: stream_nat] :
( ( sset_a @ ( smap_nat_a @ F @ V ) )
= ( image_nat_a @ F @ ( sset_nat @ V ) ) ) ).
% stream.set_map
thf(fact_385_stream_Oset__map,axiom,
! [F: a > a,V: stream_a] :
( ( sset_a @ ( smap_a_a @ F @ V ) )
= ( image_a_a @ F @ ( sset_a @ V ) ) ) ).
% stream.set_map
thf(fact_386_fair__stream__def,axiom,
( fair_fair_stream_a
= ( ^ [F2: nat > a] : ( smap_nat_a @ F2 @ fair_fair_nats ) ) ) ).
% fair_stream_def
thf(fact_387_fair__stream__def,axiom,
( fair_fair_stream_nat
= ( ^ [F2: nat > nat] : ( smap_nat_nat @ F2 @ fair_fair_nats ) ) ) ).
% fair_stream_def
thf(fact_388_range__ex1__eq,axiom,
! [F: nat > a,B: a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B
= ( F @ X2 ) )
& ! [Y2: nat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_389_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B
= ( F @ X2 ) )
& ! [Y2: nat] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_390_range__ex1__eq,axiom,
! [F: a > a,B: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B
= ( F @ X2 ) )
& ! [Y2: a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_391_range__ex1__eq,axiom,
! [F: a > nat,B: nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B
= ( F @ X2 ) )
& ! [Y2: a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_392_inj__image__eq__iff,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ( image_nat_a @ F @ A )
= ( image_nat_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_393_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_394_inj__image__eq__iff,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_395_inj__image__eq__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_396_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_397_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_398_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_399_add__diff__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_400_add__left__cancel,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_401_add__right__cancel,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_402_DiffI,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_403_DiffI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_404_Diff__iff,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_405_Diff__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_406_ComplI,axiom,
! [C2: a,A: set_a] :
( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_407_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_408_Compl__iff,axiom,
! [C2: a,A: set_a] :
( ( member_a @ C2 @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_409_Compl__iff,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_410_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_411_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_412_add__diff__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_413_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_414_DiffE,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_415_DiffE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_416_ComplD,axiom,
! [C2: a,A: set_a] :
( ( member_a @ C2 @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C2 @ A ) ) ).
% ComplD
thf(fact_417_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C2 @ A ) ) ).
% ComplD
thf(fact_418_DiffD1,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_419_DiffD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_420_DiffD2,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( member_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_421_DiffD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_422_all__ex__fair__nats,axiom,
! [M2: nat,X: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( ( snth_nat @ fair_fair_nats @ N4 )
= X ) ) ).
% all_ex_fair_nats
thf(fact_423_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_424_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_425_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_426_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_427_add_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_428_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_429_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_430_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_431_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_432_diff__right__commute,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_433_fair__def,axiom,
( fair_fair_nat
= ( ^ [S: stream_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ ( sset_nat @ S ) )
=> ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( ( snth_nat @ S @ N3 )
= X2 ) ) ) ) ) ).
% fair_def
thf(fact_434_fair__def,axiom,
( fair_fair_a
= ( ^ [S: stream_a] :
! [X2: a] :
( ( member_a @ X2 @ ( sset_a @ S ) )
=> ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( ( snth_a @ S @ N3 )
= X2 ) ) ) ) ) ).
% fair_def
thf(fact_435_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_436_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_437_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_438_add__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_439_add__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_440_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_441_add__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_442_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_443_add__le__imp__le__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_444_add__le__imp__le__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_445_add__implies__diff,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A2 )
=> ( C2
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_446_diff__diff__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_447_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_448_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_449_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_450_le__add__diff,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_451_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_452_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_453_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_454_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_455_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_456_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_457_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_458_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_459_fun_Omap__ident__strong,axiom,
! [T2: nat > a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ T2 @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_a_a_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_460_fun_Omap__ident__strong,axiom,
! [T2: nat > nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ T2 @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_nat @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_461_fun_Omap__ident__strong,axiom,
! [T2: a > a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_a_a @ T2 @ top_top_set_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_a_a_a @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_462_fun_Omap__ident__strong,axiom,
! [T2: a > nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_a_nat @ T2 @ top_top_set_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_a @ F @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_463_fun_Oset__map,axiom,
! [F: nat > a,V: nat > nat] :
( ( image_nat_a @ ( comp_nat_a_nat @ F @ V ) @ top_top_set_nat )
= ( image_nat_a @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_464_fun_Oset__map,axiom,
! [F: a > a,V: nat > a] :
( ( image_nat_a @ ( comp_a_a_nat @ F @ V ) @ top_top_set_nat )
= ( image_a_a @ F @ ( image_nat_a @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_465_fun_Oset__map,axiom,
! [F: nat > nat,V: nat > nat] :
( ( image_nat_nat @ ( comp_nat_nat_nat @ F @ V ) @ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_466_fun_Oset__map,axiom,
! [F: a > nat,V: nat > a] :
( ( image_nat_nat @ ( comp_a_nat_nat @ F @ V ) @ top_top_set_nat )
= ( image_a_nat @ F @ ( image_nat_a @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_467_fun_Oset__map,axiom,
! [F: nat > a,V: a > nat] :
( ( image_a_a @ ( comp_nat_a_a @ F @ V ) @ top_top_set_a )
= ( image_nat_a @ F @ ( image_a_nat @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_468_fun_Oset__map,axiom,
! [F: a > a,V: a > a] :
( ( image_a_a @ ( comp_a_a_a @ F @ V ) @ top_top_set_a )
= ( image_a_a @ F @ ( image_a_a @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_469_fun_Oset__map,axiom,
! [F: nat > nat,V: a > nat] :
( ( image_a_nat @ ( comp_nat_nat_a @ F @ V ) @ top_top_set_a )
= ( image_nat_nat @ F @ ( image_a_nat @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_470_fun_Oset__map,axiom,
! [F: a > nat,V: a > a] :
( ( image_a_nat @ ( comp_a_nat_a @ F @ V ) @ top_top_set_a )
= ( image_a_nat @ F @ ( image_a_a @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_471_subset__image__inj,axiom,
! [S4: set_a,F: nat > a,T3: set_nat] :
( ( ord_less_eq_set_a @ S4 @ ( image_nat_a @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_a @ F @ U )
& ( S4
= ( image_nat_a @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_472_subset__image__inj,axiom,
! [S4: set_a,F: a > a,T3: set_a] :
( ( ord_less_eq_set_a @ S4 @ ( image_a_a @ F @ T3 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T3 )
& ( inj_on_a_a @ F @ U )
& ( S4
= ( image_a_a @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_473_subset__image__inj,axiom,
! [S4: set_nat,F: a > nat,T3: set_a] :
( ( ord_less_eq_set_nat @ S4 @ ( image_a_nat @ F @ T3 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T3 )
& ( inj_on_a_nat @ F @ U )
& ( S4
= ( image_a_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_474_subset__image__inj,axiom,
! [S4: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_nat @ F @ U )
& ( S4
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_475_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_476_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_477_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_478_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_479_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_480_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_481_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_482_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N2: nat,M2: nat] : ( member_nat @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M2 ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_483_snth__sset__smerge,axiom,
! [Ss: stream_stream_a,N2: nat,M2: nat] : ( member_a @ ( snth_a @ ( snth_stream_a @ Ss @ N2 ) @ M2 ) @ ( sset_a @ ( smerge_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_484_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_485_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_486_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_487_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_488_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_489_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_490_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_491_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_492_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_493_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_494_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_495_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_496_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_497_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_498_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_499_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_500_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_501_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_502_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_503_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_504_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_505_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_506_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_507_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_508_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_509_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_510_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_511_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_512_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_513_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_514_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_515_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_516_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_517_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_518_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_519_DEADID_Oin__rel,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
? [Z4: nat] :
( ( member_nat @ Z4 @ top_top_set_nat )
& ( ( id_nat @ Z4 )
= A4 )
& ( ( id_nat @ Z4 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_520_DEADID_Oin__rel,axiom,
( ( ^ [Y3: a,Z2: a] : ( Y3 = Z2 ) )
= ( ^ [A4: a,B4: a] :
? [Z4: a] :
( ( member_a @ Z4 @ top_top_set_a )
& ( ( id_a @ Z4 )
= A4 )
& ( ( id_a @ Z4 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_521_all__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_522_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_523_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_524_all__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_525_inv__o__cancel,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% inv_o_cancel
thf(fact_526_the__inv__into__comp,axiom,
! [F: a > a,G: nat > a,A: set_nat,X: a] :
( ( inj_on_a_a @ F @ ( image_nat_a @ G @ A ) )
=> ( ( inj_on_nat_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_nat_a @ G @ A ) ) )
=> ( ( the_inv_into_nat_a @ A @ ( comp_a_a_nat @ F @ G ) @ X )
= ( comp_a_nat_a @ ( the_inv_into_nat_a @ A @ G ) @ ( the_inv_into_a_a @ ( image_nat_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_527_the__inv__into__comp,axiom,
! [F: a > a,G: a > a,A: set_a,X: a] :
( ( inj_on_a_a @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( the_inv_into_a_a @ A @ ( comp_a_a_a @ F @ G ) @ X )
= ( comp_a_a_a @ ( the_inv_into_a_a @ A @ G ) @ ( the_inv_into_a_a @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_528_the__inv__into__comp,axiom,
! [F: nat > a,G: a > nat,A: set_a,X: a] :
( ( inj_on_nat_a @ F @ ( image_a_nat @ G @ A ) )
=> ( ( inj_on_a_nat @ G @ A )
=> ( ( member_a @ X @ ( image_nat_a @ F @ ( image_a_nat @ G @ A ) ) )
=> ( ( the_inv_into_a_a @ A @ ( comp_nat_a_a @ F @ G ) @ X )
= ( comp_nat_a_a @ ( the_inv_into_a_nat @ A @ G ) @ ( the_inv_into_nat_a @ ( image_a_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_529_the__inv__into__comp,axiom,
! [F: nat > a,G: nat > nat,A: set_nat,X: a] :
( ( inj_on_nat_a @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_a @ X @ ( image_nat_a @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( the_inv_into_nat_a @ A @ ( comp_nat_a_nat @ F @ G ) @ X )
= ( comp_nat_nat_a @ ( the_inv_into_nat_nat @ A @ G ) @ ( the_inv_into_nat_a @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_530_the__inv__into__comp,axiom,
! [F: a > nat,G: nat > a,A: set_nat,X: nat] :
( ( inj_on_a_nat @ F @ ( image_nat_a @ G @ A ) )
=> ( ( inj_on_nat_a @ G @ A )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ ( image_nat_a @ G @ A ) ) )
=> ( ( the_inv_into_nat_nat @ A @ ( comp_a_nat_nat @ F @ G ) @ X )
= ( comp_a_nat_nat @ ( the_inv_into_nat_a @ A @ G ) @ ( the_inv_into_a_nat @ ( image_nat_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_531_the__inv__into__comp,axiom,
! [F: a > nat,G: a > a,A: set_a,X: nat] :
( ( inj_on_a_nat @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( the_inv_into_a_nat @ A @ ( comp_a_nat_a @ F @ G ) @ X )
= ( comp_a_a_nat @ ( the_inv_into_a_a @ A @ G ) @ ( the_inv_into_a_nat @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_532_the__inv__into__comp,axiom,
! [F: nat > nat,G: a > nat,A: set_a,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_a_nat @ G @ A ) )
=> ( ( inj_on_a_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_a_nat @ G @ A ) ) )
=> ( ( the_inv_into_a_nat @ A @ ( comp_nat_nat_a @ F @ G ) @ X )
= ( comp_nat_a_nat @ ( the_inv_into_a_nat @ A @ G ) @ ( the_inv_into_nat_nat @ ( image_a_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_533_the__inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( the_inv_into_nat_nat @ A @ ( comp_nat_nat_nat @ F @ G ) @ X )
= ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ A @ G ) @ ( the_inv_into_nat_nat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_534_the__inv__into__into,axiom,
! [F: a > a,A: set_a,X: a,B2: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( member_a @ ( the_inv_into_a_a @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_535_the__inv__into__into,axiom,
! [F: nat > a,A: set_nat,X: a,B2: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( member_a @ X @ ( image_nat_a @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_a @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_536_the__inv__into__into,axiom,
! [F: a > nat,A: set_a,X: nat,B2: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( member_a @ ( the_inv_into_a_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_537_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_538_inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( hilber3633877196798814958at_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% inv_into_f_f
thf(fact_539_inv__id,axiom,
( ( hilber3633877196798814958at_nat @ top_top_set_nat @ id_nat )
= id_nat ) ).
% inv_id
thf(fact_540_inv__id,axiom,
( ( hilbert_inv_into_a_a @ top_top_set_a @ id_a )
= id_a ) ).
% inv_id
thf(fact_541_the__inv__into__onto,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( image_nat_a @ ( the_inv_into_a_nat @ A @ F ) @ ( image_a_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_542_the__inv__into__onto,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( image_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_543_the__inv__into__onto,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( image_a_nat @ ( the_inv_into_nat_a @ A @ F ) @ ( image_nat_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_544_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_545_inv__into__image__cancel,axiom,
! [F: a > nat,A: set_a,S4: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_a @ S4 @ A )
=> ( ( image_nat_a @ ( hilber7986931655781312002_a_nat @ A @ F ) @ ( image_a_nat @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_546_inv__into__image__cancel,axiom,
! [F: a > a,A: set_a,S4: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( ord_less_eq_set_a @ S4 @ A )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_547_inv__into__image__cancel,axiom,
! [F: nat > a,A: set_nat,S4: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( ord_less_eq_set_nat @ S4 @ A )
=> ( ( image_a_nat @ ( hilber2795491120104822624_nat_a @ A @ F ) @ ( image_nat_a @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_548_inv__into__image__cancel,axiom,
! [F: nat > nat,A: set_nat,S4: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ S4 @ A )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ ( image_nat_nat @ F @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_549_inv__into__injective,axiom,
! [A: set_nat,F: nat > a,X: a,Y: a] :
( ( ( hilber2795491120104822624_nat_a @ A @ F @ X )
= ( hilber2795491120104822624_nat_a @ A @ F @ Y ) )
=> ( ( member_a @ X @ ( image_nat_a @ F @ A ) )
=> ( ( member_a @ Y @ ( image_nat_a @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_550_inv__into__injective,axiom,
! [A: set_a,F: a > a,X: a,Y: a] :
( ( ( hilbert_inv_into_a_a @ A @ F @ X )
= ( hilbert_inv_into_a_a @ A @ F @ Y ) )
=> ( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( ( member_a @ Y @ ( image_a_a @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_551_inv__into__injective,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( ( hilber3633877196798814958at_nat @ A @ F @ X )
= ( hilber3633877196798814958at_nat @ A @ F @ Y ) )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_552_inv__into__injective,axiom,
! [A: set_a,F: a > nat,X: nat,Y: nat] :
( ( ( hilber7986931655781312002_a_nat @ A @ F @ X )
= ( hilber7986931655781312002_a_nat @ A @ F @ Y ) )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
=> ( ( member_nat @ Y @ ( image_a_nat @ F @ A ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_553_inv__into__into,axiom,
! [X: a,F: a > a,A: set_a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( member_a @ ( hilbert_inv_into_a_a @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_554_inv__into__into,axiom,
! [X: a,F: nat > a,A: set_nat] :
( ( member_a @ X @ ( image_nat_a @ F @ A ) )
=> ( member_nat @ ( hilber2795491120104822624_nat_a @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_555_inv__into__into,axiom,
! [X: nat,F: a > nat,A: set_a] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
=> ( member_a @ ( hilber7986931655781312002_a_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_556_inv__into__into,axiom,
! [X: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_557_f__inv__into__f,axiom,
! [Y: a,F: nat > a,A: set_nat] :
( ( member_a @ Y @ ( image_nat_a @ F @ A ) )
=> ( ( F @ ( hilber2795491120104822624_nat_a @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_558_f__inv__into__f,axiom,
! [Y: a,F: a > a,A: set_a] :
( ( member_a @ Y @ ( image_a_a @ F @ A ) )
=> ( ( F @ ( hilbert_inv_into_a_a @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_559_f__inv__into__f,axiom,
! [Y: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Y @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( hilber3633877196798814958at_nat @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_560_f__inv__into__f,axiom,
! [Y: nat,F: a > nat,A: set_a] :
( ( member_nat @ Y @ ( image_a_nat @ F @ A ) )
=> ( ( F @ ( hilber7986931655781312002_a_nat @ A @ F @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_561_inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( hilber3633877196798814958at_nat @ A @ F @ Y )
= X ) ) ) ) ).
% inv_into_f_eq
thf(fact_562_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_563_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= Y )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_564_surj__f__inv__f,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_565_surj__f__inv__f,axiom,
! [F: nat > a,Y: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( F @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_566_surj__f__inv__f,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( F @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_567_surj__f__inv__f,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_568_surj__iff__all,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X2: nat] :
( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_569_surj__iff__all,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [X2: a] :
( ( F @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_570_surj__iff__all,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [X2: nat] :
( ( F @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_571_surj__iff__all,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [X2: a] :
( ( F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_572_image__f__inv__f,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_573_image__f__inv__f,axiom,
! [F: nat > a,A: set_a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( image_nat_a @ F @ ( image_a_nat @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_574_image__f__inv__f,axiom,
! [F: a > nat,A: set_nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( image_a_nat @ F @ ( image_nat_a @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_575_image__f__inv__f,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ F @ ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_576_surj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_577_surj__imp__inv__eq,axiom,
! [F: nat > a,G: a > nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_578_surj__imp__inv__eq,axiom,
! [F: a > nat,G: nat > a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber7986931655781312002_a_nat @ top_top_set_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_579_surj__imp__inv__eq,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into_a_a @ top_top_set_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_580_image__inv__into__cancel,axiom,
! [F: nat > a,A: set_nat,A6: set_a,B6: set_a] :
( ( ( image_nat_a @ F @ A )
= A6 )
=> ( ( ord_less_eq_set_a @ B6 @ A6 )
=> ( ( image_nat_a @ F @ ( image_a_nat @ ( hilber2795491120104822624_nat_a @ A @ F ) @ B6 ) )
= B6 ) ) ) ).
% image_inv_into_cancel
thf(fact_581_image__inv__into__cancel,axiom,
! [F: nat > nat,A: set_nat,A6: set_nat,B6: set_nat] :
( ( ( image_nat_nat @ F @ A )
= A6 )
=> ( ( ord_less_eq_set_nat @ B6 @ A6 )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ B6 ) )
= B6 ) ) ) ).
% image_inv_into_cancel
thf(fact_582_image__inv__into__cancel,axiom,
! [F: a > a,A: set_a,A6: set_a,B6: set_a] :
( ( ( image_a_a @ F @ A )
= A6 )
=> ( ( ord_less_eq_set_a @ B6 @ A6 )
=> ( ( image_a_a @ F @ ( image_a_a @ ( hilbert_inv_into_a_a @ A @ F ) @ B6 ) )
= B6 ) ) ) ).
% image_inv_into_cancel
thf(fact_583_image__inv__into__cancel,axiom,
! [F: a > nat,A: set_a,A6: set_nat,B6: set_nat] :
( ( ( image_a_nat @ F @ A )
= A6 )
=> ( ( ord_less_eq_set_nat @ B6 @ A6 )
=> ( ( image_a_nat @ F @ ( image_nat_a @ ( hilber7986931655781312002_a_nat @ A @ F ) @ B6 ) )
= B6 ) ) ) ).
% image_inv_into_cancel
thf(fact_584_inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_585_inv__f__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= Y )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y )
= X ) ) ) ).
% inv_f_eq
thf(fact_586_inj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [X3: nat] :
( ( F @ ( G @ X3 ) )
= X3 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_587_inj__transfer,axiom,
! [F: nat > a,P: nat > $o,X: nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ( P @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_588_inj__transfer,axiom,
! [F: nat > nat,P: nat > $o,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( P @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_589_inj__transfer,axiom,
! [F: a > a,P: a > $o,X: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ ( image_a_a @ F @ top_top_set_a ) )
=> ( P @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_590_inj__transfer,axiom,
! [F: a > nat,P: a > $o,X: a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( image_a_nat @ F @ top_top_set_a ) )
=> ( P @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_591_image__inv__f__f,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( image_a_nat @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F ) @ ( image_nat_a @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_592_image__inv__f__f,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_593_image__inv__f__f,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( image_nat_a @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F ) @ ( image_a_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_594_image__inv__f__f,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ ( image_a_a @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_595_inj__imp__surj__inv,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_596_inj__imp__surj__inv,axiom,
! [F: nat > a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( image_a_nat @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_597_inj__imp__surj__inv,axiom,
! [F: a > nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( image_nat_a @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F ) @ top_top_set_nat )
= top_top_set_a ) ) ).
% inj_imp_surj_inv
thf(fact_598_inj__imp__surj__inv,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a )
= top_top_set_a ) ) ).
% inj_imp_surj_inv
thf(fact_599_surj__imp__inj__inv,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_600_surj__imp__inj__inv,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( inj_on_a_nat @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F ) @ top_top_set_a ) ) ).
% surj_imp_inj_inv
thf(fact_601_surj__imp__inj__inv,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( inj_on_nat_a @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_602_surj__imp__inj__inv,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( inj_on_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a ) ) ).
% surj_imp_inj_inv
thf(fact_603_inj__on__inv__into,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( inj_on_a_nat @ ( hilber2795491120104822624_nat_a @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_604_inj__on__inv__into,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( inj_on_a_a @ ( hilbert_inv_into_a_a @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_605_inj__on__inv__into,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( inj_on_nat_a @ ( hilber7986931655781312002_a_nat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_606_inj__on__inv__into,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_607_inv__into__comp,axiom,
! [F: a > a,G: nat > a,A: set_nat,X: a] :
( ( inj_on_a_a @ F @ ( image_nat_a @ G @ A ) )
=> ( ( inj_on_nat_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_nat_a @ G @ A ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A @ ( comp_a_a_nat @ F @ G ) @ X )
= ( comp_a_nat_a @ ( hilber2795491120104822624_nat_a @ A @ G ) @ ( hilbert_inv_into_a_a @ ( image_nat_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_608_inv__into__comp,axiom,
! [F: a > a,G: a > a,A: set_a,X: a] :
( ( inj_on_a_a @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( hilbert_inv_into_a_a @ A @ ( comp_a_a_a @ F @ G ) @ X )
= ( comp_a_a_a @ ( hilbert_inv_into_a_a @ A @ G ) @ ( hilbert_inv_into_a_a @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_609_inv__into__comp,axiom,
! [F: nat > a,G: a > nat,A: set_a,X: a] :
( ( inj_on_nat_a @ F @ ( image_a_nat @ G @ A ) )
=> ( ( inj_on_a_nat @ G @ A )
=> ( ( member_a @ X @ ( image_nat_a @ F @ ( image_a_nat @ G @ A ) ) )
=> ( ( hilbert_inv_into_a_a @ A @ ( comp_nat_a_a @ F @ G ) @ X )
= ( comp_nat_a_a @ ( hilber7986931655781312002_a_nat @ A @ G ) @ ( hilber2795491120104822624_nat_a @ ( image_a_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_610_inv__into__comp,axiom,
! [F: nat > a,G: nat > nat,A: set_nat,X: a] :
( ( inj_on_nat_a @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_a @ X @ ( image_nat_a @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A @ ( comp_nat_a_nat @ F @ G ) @ X )
= ( comp_nat_nat_a @ ( hilber3633877196798814958at_nat @ A @ G ) @ ( hilber2795491120104822624_nat_a @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_611_inv__into__comp,axiom,
! [F: a > nat,G: nat > a,A: set_nat,X: nat] :
( ( inj_on_a_nat @ F @ ( image_nat_a @ G @ A ) )
=> ( ( inj_on_nat_a @ G @ A )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ ( image_nat_a @ G @ A ) ) )
=> ( ( hilber3633877196798814958at_nat @ A @ ( comp_a_nat_nat @ F @ G ) @ X )
= ( comp_a_nat_nat @ ( hilber2795491120104822624_nat_a @ A @ G ) @ ( hilber7986931655781312002_a_nat @ ( image_nat_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_612_inv__into__comp,axiom,
! [F: a > nat,G: a > a,A: set_a,X: nat] :
( ( inj_on_a_nat @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_nat @ X @ ( image_a_nat @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( hilber7986931655781312002_a_nat @ A @ ( comp_a_nat_a @ F @ G ) @ X )
= ( comp_a_a_nat @ ( hilbert_inv_into_a_a @ A @ G ) @ ( hilber7986931655781312002_a_nat @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_613_inv__into__comp,axiom,
! [F: nat > nat,G: a > nat,A: set_a,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_a_nat @ G @ A ) )
=> ( ( inj_on_a_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_a_nat @ G @ A ) ) )
=> ( ( hilber7986931655781312002_a_nat @ A @ ( comp_nat_nat_a @ F @ G ) @ X )
= ( comp_nat_a_nat @ ( hilber7986931655781312002_a_nat @ A @ G ) @ ( hilber3633877196798814958at_nat @ ( image_a_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_614_inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( hilber3633877196798814958at_nat @ A @ ( comp_nat_nat_nat @ F @ G ) @ X )
= ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ A @ G ) @ ( hilber3633877196798814958at_nat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_615_inj__on__the__inv__into,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( inj_on_a_nat @ ( the_inv_into_nat_a @ A @ F ) @ ( image_nat_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_616_inj__on__the__inv__into,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_617_inj__on__the__inv__into,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_nat_a @ ( the_inv_into_a_nat @ A @ F ) @ ( image_a_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_618_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_619_f__the__inv__into__f,axiom,
! [F: nat > a,A: set_nat,Y: a] :
( ( inj_on_nat_a @ F @ A )
=> ( ( member_a @ Y @ ( image_nat_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_a @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_620_f__the__inv__into__f,axiom,
! [F: a > a,A: set_a,Y: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ Y @ ( image_a_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_a @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_621_f__the__inv__into__f,axiom,
! [F: a > nat,A: set_a,Y: nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_nat @ Y @ ( image_a_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_622_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_623_the__inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_624_surj__iff,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ( comp_nat_nat_nat @ F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) )
= id_nat ) ) ).
% surj_iff
thf(fact_625_surj__iff,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ( comp_nat_a_a @ F @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F ) )
= id_a ) ) ).
% surj_iff
thf(fact_626_surj__iff,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ( comp_a_nat_nat @ F @ ( hilber7986931655781312002_a_nat @ top_top_set_a @ F ) )
= id_nat ) ) ).
% surj_iff
thf(fact_627_surj__iff,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ( comp_a_a_a @ F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) )
= id_a ) ) ).
% surj_iff
thf(fact_628_inj__iff,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% inj_iff
thf(fact_629_the__inv__f__o__f__id,axiom,
! [F: nat > nat,Z: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ top_top_set_nat @ F ) @ F @ Z )
= ( id_nat @ Z ) ) ) ).
% the_inv_f_o_f_id
thf(fact_630_bijection_Oinv__comp__left,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% bijection.inv_comp_left
thf(fact_631_bijection_Oinv__comp__left,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( ( comp_a_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ F )
= id_a ) ) ).
% bijection.inv_comp_left
thf(fact_632_bijection_Oinv__comp__right,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( comp_nat_nat_nat @ F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) )
= id_nat ) ) ).
% bijection.inv_comp_right
thf(fact_633_bijection_Oinv__comp__right,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( ( comp_a_a_a @ F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) )
= id_a ) ) ).
% bijection.inv_comp_right
thf(fact_634_bijection_Oinj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% bijection.inj_inv
thf(fact_635_bijection_Oinj__inv,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( inj_on_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a ) ) ).
% bijection.inj_inv
thf(fact_636_bijection_Osurj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_637_bijection_Osurj__inv,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a )
= top_top_set_a ) ) ).
% bijection.surj_inv
thf(fact_638_bijection_Osurj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_639_bijection_Osurj,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a ) ) ).
% bijection.surj
thf(fact_640_bijection_Oinj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_641_bijection_Oinj,axiom,
! [F: a > a] :
( ( hilbert_bijection_a @ F )
=> ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% bijection.inj
thf(fact_642_bijection_Oeq__invI,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_643_bijection_Oeq__invI,axiom,
! [F: a > a,A2: a,B: a] :
( ( hilbert_bijection_a @ F )
=> ( ( ( hilbert_inv_into_a_a @ top_top_set_a @ F @ A2 )
= ( hilbert_inv_into_a_a @ top_top_set_a @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_644_bijection_Oinv__left,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_645_bijection_Oinv__left,axiom,
! [F: a > a,A2: a] :
( ( hilbert_bijection_a @ F )
=> ( ( hilbert_inv_into_a_a @ top_top_set_a @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_646_bijection_Oinv__right,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_647_bijection_Oinv__right,axiom,
! [F: a > a,A2: a] :
( ( hilbert_bijection_a @ F )
=> ( ( F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_648_bijection_Oeq__inv__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_649_bijection_Oeq__inv__iff,axiom,
! [F: a > a,A2: a,B: a] :
( ( hilbert_bijection_a @ F )
=> ( ( ( hilbert_inv_into_a_a @ top_top_set_a @ F @ A2 )
= ( hilbert_inv_into_a_a @ top_top_set_a @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_650_bijection_Oinv__left__eq__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_651_bijection_Oinv__left__eq__iff,axiom,
! [F: a > a,A2: a,B: a] :
( ( hilbert_bijection_a @ F )
=> ( ( ( hilbert_inv_into_a_a @ top_top_set_a @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_652_bijection_Oinv__right__eq__iff,axiom,
! [F: nat > nat,B: nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( B
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_653_bijection_Oinv__right__eq__iff,axiom,
! [F: a > a,B: a,A2: a] :
( ( hilbert_bijection_a @ F )
=> ( ( B
= ( hilbert_inv_into_a_a @ top_top_set_a @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_654_Stream_Osmember__def,axiom,
( smember_nat
= ( ^ [X2: nat,S: stream_nat] : ( member_nat @ X2 @ ( sset_nat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_655_Stream_Osmember__def,axiom,
( smember_a
= ( ^ [X2: a,S: stream_a] : ( member_a @ X2 @ ( sset_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_656_streams__UNIV,axiom,
( ( streams_nat @ top_top_set_nat )
= top_to7548458143485696966am_nat ) ).
% streams_UNIV
thf(fact_657_streams__UNIV,axiom,
( ( streams_a @ top_top_set_a )
= top_top_set_stream_a ) ).
% streams_UNIV
thf(fact_658_inj__on__image__Fpow,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_659_inj__on__image__Fpow,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_660_inj__on__image__Fpow,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_661_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_662_smap__streams,axiom,
! [S2: stream_a,A: set_a,F: a > a,B2: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( member_stream_a @ ( smap_a_a @ F @ S2 ) @ ( streams_a @ B2 ) ) ) ) ).
% smap_streams
thf(fact_663_smap__streams,axiom,
! [S2: stream_a,A: set_a,F: a > nat,B2: set_nat] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( member_stream_nat @ ( smap_a_nat @ F @ S2 ) @ ( streams_nat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_664_smap__streams,axiom,
! [S2: stream_nat,A: set_nat,F: nat > a,B2: set_a] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( member_stream_a @ ( smap_nat_a @ F @ S2 ) @ ( streams_a @ B2 ) ) ) ) ).
% smap_streams
thf(fact_665_smap__streams,axiom,
! [S2: stream_nat,A: set_nat,F: nat > nat,B2: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( member_stream_nat @ ( smap_nat_nat @ F @ S2 ) @ ( streams_nat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_666_streams__iff__snth,axiom,
! [S2: stream_a,X5: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ X5 ) )
= ( ! [N3: nat] : ( member_a @ ( snth_a @ S2 @ N3 ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_667_streams__iff__snth,axiom,
! [S2: stream_nat,X5: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ X5 ) )
= ( ! [N3: nat] : ( member_nat @ ( snth_nat @ S2 @ N3 ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_668_snth__in,axiom,
! [S2: stream_a,X5: set_a,N2: nat] :
( ( member_stream_a @ S2 @ ( streams_a @ X5 ) )
=> ( member_a @ ( snth_a @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_669_snth__in,axiom,
! [S2: stream_nat,X5: set_nat,N2: nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ X5 ) )
=> ( member_nat @ ( snth_nat @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_670_streams__iff__sset,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
= ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_671_streams__sset,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_672_sset__streams,axiom,
! [S2: stream_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A )
=> ( member_stream_a @ S2 @ ( streams_a @ A ) ) ) ).
% sset_streams
thf(fact_673_image__Fpow__mono,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_674_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_675_image__Fpow__mono,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_676_image__Fpow__mono,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_677_inj__on__image__Pow,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_678_inj__on__image__Pow,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_679_inj__on__image__Pow,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_680_inj__on__image__Pow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_681_inj__on__image,axiom,
! [F: nat > a,A: set_set_nat] :
( ( inj_on_nat_a @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on_set_nat_set_a @ ( image_nat_a @ F ) @ A ) ) ).
% inj_on_image
thf(fact_682_inj__on__image,axiom,
! [F: a > a,A: set_set_a] :
( ( inj_on_a_a @ F @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on_set_a_set_a @ ( image_a_a @ F ) @ A ) ) ).
% inj_on_image
thf(fact_683_inj__on__image,axiom,
! [F: a > nat,A: set_set_a] :
( ( inj_on_a_nat @ F @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_684_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_685_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_686_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_687_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_688_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_689_Union__iff,axiom,
! [A: a,C: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ C )
& ( member_a @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_690_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_691_UnionI,axiom,
! [X5: set_a,C: set_set_a,A: a] :
( ( member_set_a @ X5 @ C )
=> ( ( member_a @ A @ X5 )
=> ( member_a @ A @ ( comple2307003609928055243_set_a @ C ) ) ) ) ).
% UnionI
thf(fact_692_UnionI,axiom,
! [X5: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_693_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_694_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_695_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_696_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_697_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_698_Pow__UNIV,axiom,
( ( pow_a @ top_top_set_a )
= top_top_set_set_a ) ).
% Pow_UNIV
thf(fact_699_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_700_Sup__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Sup_UNIV
thf(fact_701_SUP__id__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_id_eq
thf(fact_702_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_703_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_704_Union__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Union_UNIV
thf(fact_705_UnionE,axiom,
! [A: a,C: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C ) )
=> ~ ! [X6: set_a] :
( ( member_a @ A @ X6 )
=> ~ ( member_set_a @ X6 @ C ) ) ) ).
% UnionE
thf(fact_706_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_707_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_708_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_709_SUP__cong,axiom,
! [A: set_a,B2: set_a,C: a > nat,D: a > nat] :
( ( A = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_710_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_711_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( R2 @ X3 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_712_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_713_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_714_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_715_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_716_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_717_Suc__le__D,axiom,
! [N2: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_718_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_719_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_720_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_721_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_722_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_723_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_724_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_725_image__Pow__surj,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ( image_nat_a @ F @ A )
= B2 )
=> ( ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( pow_nat @ A ) )
= ( pow_a @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_726_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B2 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_727_image__Pow__surj,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ( image_a_a @ F @ A )
= B2 )
=> ( ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) )
= ( pow_a @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_728_image__Pow__surj,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ( image_a_nat @ F @ A )
= B2 )
=> ( ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_729_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_730_SUP__image,axiom,
! [G: nat > nat,F: a > nat,A: set_a] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_a_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ ( comp_nat_nat_a @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_731_SUP__image,axiom,
! [G: a > nat,F: nat > a,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_a_nat @ G @ ( image_nat_a @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_732_SUP__image,axiom,
! [G: a > nat,F: a > a,A: set_a] :
( ( complete_Sup_Sup_nat @ ( image_a_nat @ G @ ( image_a_a @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ ( comp_a_nat_a @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_733_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_734_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_735_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_736_image__Pow__mono,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( pow_nat @ A ) ) @ ( pow_a @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_737_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_738_image__Pow__mono,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) ) @ ( pow_a @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_739_image__Pow__mono,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_740_sset__smerge,axiom,
! [Ss: stream_stream_a] :
( ( sset_a @ ( smerge_a @ Ss ) )
= ( comple2307003609928055243_set_a @ ( image_stream_a_set_a @ sset_a @ ( sset_stream_a @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_741_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_742_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S5: set_a_o,X2: a] : ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_743_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S5: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_744_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_745_infinite__countable__subset,axiom,
! [S4: set_a] :
( ~ ( finite_finite_a @ S4 )
=> ? [F4: nat > a] :
( ( inj_on_nat_a @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ top_top_set_nat ) @ S4 ) ) ) ).
% infinite_countable_subset
thf(fact_746_infinite__countable__subset,axiom,
! [S4: set_nat] :
( ~ ( finite_finite_nat @ S4 )
=> ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ top_top_set_nat ) @ S4 ) ) ) ).
% infinite_countable_subset
thf(fact_747_infinite__iff__countable__subset,axiom,
! [S4: set_a] :
( ( ~ ( finite_finite_a @ S4 ) )
= ( ? [F2: nat > a] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ S4 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_748_infinite__iff__countable__subset,axiom,
! [S4: set_nat] :
( ( ~ ( finite_finite_nat @ S4 ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S4 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_749_finite__imageI,axiom,
! [F5: set_a,H: a > a] :
( ( finite_finite_a @ F5 )
=> ( finite_finite_a @ ( image_a_a @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_750_finite__imageI,axiom,
! [F5: set_a,H: a > nat] :
( ( finite_finite_a @ F5 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_751_finite__imageI,axiom,
! [F5: set_nat,H: nat > a] :
( ( finite_finite_nat @ F5 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_752_finite__imageI,axiom,
! [F5: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F5 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F5 ) ) ) ).
% finite_imageI
thf(fact_753_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_754_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_755_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_756_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_757_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_758_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_759_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_760_finite__compl,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( uminus_uminus_set_a @ A ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_compl
thf(fact_761_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_762_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A3: nat] :
~ ( member_nat @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_763_ex__new__if__finite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A )
=> ? [A3: a] :
~ ( member_a @ A3 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_764_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_765_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_766_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_767_finite__prod,axiom,
( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_768_finite__prod,axiom,
( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_769_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_770_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_771_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_772_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_773_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_774_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_775_Diff__infinite__finite,axiom,
! [T3: set_nat,S4: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S4 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_776_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_777_infinite__super,axiom,
! [S4: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ T3 )
=> ( ~ ( finite_finite_nat @ S4 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_778_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_779_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_780_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_781_finite__subset__Union,axiom,
! [A: set_nat,B7: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B7 ) )
=> ~ ! [F6: set_set_nat] :
( ( finite1152437895449049373et_nat @ F6 )
=> ( ( ord_le6893508408891458716et_nat @ F6 @ B7 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_782_le__cSup__finite,axiom,
! [X5: set_nat,X: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_783_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_784_all__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_785_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A ) )
=> ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_786_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_787_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_788_ex__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_789_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A )
& ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_790_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_791_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B2
= ( image_a_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_792_finite__subset__image,axiom,
! [B2: set_a,F: nat > a,A: set_nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_nat_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_793_finite__subset__image,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B2
= ( image_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_794_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_795_finite__surj,axiom,
! [A: set_a,B2: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_796_finite__surj,axiom,
! [A: set_a,B2: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_797_finite__surj,axiom,
! [A: set_nat,B2: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_798_finite__surj,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_799_finite__image__iff,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_800_finite__image__iff,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_801_finite__image__iff,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_802_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_803_finite__imageD,axiom,
! [F: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_804_finite__imageD,axiom,
! [F: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ( ( inj_on_nat_a @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_805_finite__imageD,axiom,
! [F: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ( ( inj_on_a_nat @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_806_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_807_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_808_finite__UNIV__surj__inj,axiom,
! [F: a > a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( inj_on_a_a @ F @ top_top_set_a ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_809_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_810_finite__UNIV__inj__surj,axiom,
! [F: a > a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_811_finite__surj__inj,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ A ) )
=> ( inj_on_a_a @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_812_finite__surj__inj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_813_inj__on__finite,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ) ).
% inj_on_finite
thf(fact_814_inj__on__finite,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( inj_on_nat_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_815_inj__on__finite,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_a @ A ) ) ) ) ).
% inj_on_finite
thf(fact_816_inj__on__finite,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_817_endo__inj__surj,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ A )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( image_a_a @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_818_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_819_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_820_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_821_inj__on__iff__card__le,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ? [F2: a > a] :
( ( inj_on_a_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_822_inj__on__iff__card__le,axiom,
! [A: set_a,B2: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_823_inj__on__iff__card__le,axiom,
! [A: set_nat,B2: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ? [F2: nat > a] :
( ( inj_on_nat_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_824_inj__on__iff__card__le,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_825_card__inj__on__le,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( inj_on_nat_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_826_card__inj__on__le,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_827_card__inj__on__le,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_828_card__inj__on__le,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_829_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_830_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_a] :
( ( finite_finite_a @ top_top_set_a )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ top_top_set_a ) )
=> ( A = top_top_set_a ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_831_card__subset__eq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_832_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_833_card__image,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( finite_card_a @ ( image_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_834_card__image,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( finite_card_a @ ( image_a_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_835_card__image,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_836_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_837_card__image__le,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_838_card__image__le,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_839_card__image__le,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_840_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_841_card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_842_card__seteq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_843_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C )
& ( ( finite_card_nat @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_844_obtain__subset__with__card__n,axiom,
! [N2: nat,S4: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S4 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S4 )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_845_finite__if__finite__subsets__card__bdd,axiom,
! [F5: set_nat,C: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F5 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C ) ) )
=> ( ( finite_finite_nat @ F5 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F5 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_846_eq__card__imp__inj__on,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ ( image_a_a @ F @ A ) )
= ( finite_card_a @ A ) )
=> ( inj_on_a_a @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_847_eq__card__imp__inj__on,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) )
=> ( inj_on_a_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_848_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_a @ ( image_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_a @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_849_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_850_inj__on__iff__eq__card,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_a @ F @ A )
= ( ( finite_card_a @ ( image_a_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_851_inj__on__iff__eq__card,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_nat @ F @ A )
= ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_852_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_a @ F @ A )
= ( ( finite_card_a @ ( image_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_853_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F @ A )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_854_card__le__sym__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_855_surj__card__le,axiom,
! [A: set_a,B2: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_856_surj__card__le,axiom,
! [A: set_a,B2: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_857_surj__card__le,axiom,
! [A: set_nat,B2: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_858_surj__card__le,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_859_card__bij__eq,axiom,
! [F: a > a,A: set_a,B2: set_a,G: a > a] :
( ( inj_on_a_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
=> ( ( inj_on_a_a @ G @ B2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G @ B2 ) @ A )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( finite_card_a @ A )
= ( finite_card_a @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_860_card__bij__eq,axiom,
! [F: a > nat,A: set_a,B2: set_nat,G: nat > a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_a @ G @ B2 )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G @ B2 ) @ A )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_a @ A )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_861_card__bij__eq,axiom,
! [F: nat > a,A: set_nat,B2: set_a,G: a > nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ( inj_on_a_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G @ B2 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( finite_card_nat @ A )
= ( finite_card_a @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_862_card__bij__eq,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B2 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_863_surjective__iff__injective__gen,axiom,
! [S4: set_a,T3: set_a,F: a > a] :
( ( finite_finite_a @ S4 )
=> ( ( finite_finite_a @ T3 )
=> ( ( ( finite_card_a @ S4 )
= ( finite_card_a @ T3 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ S4 ) @ T3 )
=> ( ( ! [X2: a] :
( ( member_a @ X2 @ T3 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ S4 )
& ( ( F @ Y2 )
= X2 ) ) ) )
= ( inj_on_a_a @ F @ S4 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_864_surjective__iff__injective__gen,axiom,
! [S4: set_a,T3: set_nat,F: a > nat] :
( ( finite_finite_a @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_a @ S4 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ S4 ) @ T3 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ S4 )
& ( ( F @ Y2 )
= X2 ) ) ) )
= ( inj_on_a_nat @ F @ S4 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_865_surjective__iff__injective__gen,axiom,
! [S4: set_nat,T3: set_a,F: nat > a] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_a @ T3 )
=> ( ( ( finite_card_nat @ S4 )
= ( finite_card_a @ T3 ) )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ S4 ) @ T3 )
=> ( ( ! [X2: a] :
( ( member_a @ X2 @ T3 )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ S4 )
& ( ( F @ Y2 )
= X2 ) ) ) )
= ( inj_on_nat_a @ F @ S4 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_866_surjective__iff__injective__gen,axiom,
! [S4: set_nat,T3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S4 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S4 ) @ T3 )
=> ( ( ! [X2: nat] :
( ( member_nat @ X2 @ T3 )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ S4 )
& ( ( F @ Y2 )
= X2 ) ) ) )
= ( inj_on_nat_nat @ F @ S4 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_867_card__Diff__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_868_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_869_card__le__inj,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) )
=> ? [F4: a > a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F4 @ A ) @ B2 )
& ( inj_on_a_a @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_870_card__le__inj,axiom,
! [A: set_a,B2: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: a > nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F4 @ A ) @ B2 )
& ( inj_on_a_nat @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_871_card__le__inj,axiom,
! [A: set_nat,B2: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) )
=> ? [F4: nat > a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F4 @ A ) @ B2 )
& ( inj_on_nat_a @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_872_card__le__inj,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ? [F4: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A ) @ B2 )
& ( inj_on_nat_nat @ F4 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_873_type__definition_OAbs__image,axiom,
! [Rep: nat > nat,Abs: nat > nat,A: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A )
=> ( ( image_nat_nat @ Abs @ A )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_874_type__definition_OAbs__image,axiom,
! [Rep: nat > a,Abs: a > nat,A: set_a] :
( ( type_d2627918313818726784_nat_a @ Rep @ Abs @ A )
=> ( ( image_a_nat @ Abs @ A )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_875_type__definition_OAbs__image,axiom,
! [Rep: a > nat,Abs: nat > a,A: set_nat] :
( ( type_d7819358849495216162_a_nat @ Rep @ Abs @ A )
=> ( ( image_nat_a @ Abs @ A )
= top_top_set_a ) ) ).
% type_definition.Abs_image
thf(fact_876_type__definition_OAbs__image,axiom,
! [Rep: a > a,Abs: a > a,A: set_a] :
( ( type_definition_a_a @ Rep @ Abs @ A )
=> ( ( image_a_a @ Abs @ A )
= top_top_set_a ) ) ).
% type_definition.Abs_image
thf(fact_877_type__definition_ORep__range,axiom,
! [Rep: nat > a,Abs: a > nat,A: set_a] :
( ( type_d2627918313818726784_nat_a @ Rep @ Abs @ A )
=> ( ( image_nat_a @ Rep @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_878_type__definition_ORep__range,axiom,
! [Rep: nat > nat,Abs: nat > nat,A: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A )
=> ( ( image_nat_nat @ Rep @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_879_type__definition_ORep__range,axiom,
! [Rep: a > a,Abs: a > a,A: set_a] :
( ( type_definition_a_a @ Rep @ Abs @ A )
=> ( ( image_a_a @ Rep @ top_top_set_a )
= A ) ) ).
% type_definition.Rep_range
thf(fact_880_type__definition_ORep__range,axiom,
! [Rep: a > nat,Abs: nat > a,A: set_nat] :
( ( type_d7819358849495216162_a_nat @ Rep @ Abs @ A )
=> ( ( image_a_nat @ Rep @ top_top_set_a )
= A ) ) ).
% type_definition.Rep_range
thf(fact_881_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_882_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_883_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B3: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_884_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R @ A3 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_885_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_886_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_887_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_888_finite__fun__UNIVD1,axiom,
( ( finite_finite_a_nat @ top_top_set_a_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_fun_UNIVD1
thf(fact_889_finite__fun__UNIVD1,axiom,
( ( finite_finite_nat_a @ top_top_set_nat_a )
=> ( ( ( finite_card_a @ top_top_set_a )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_890_finite__fun__UNIVD1,axiom,
( ( finite_finite_a_a @ top_top_set_a_a )
=> ( ( ( finite_card_a @ top_top_set_a )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_fun_UNIVD1
thf(fact_891_card__vimage__inj,axiom,
! [F: nat > a,A: set_a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_892_card__vimage__inj,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_893_card__vimage__inj,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ top_top_set_a ) )
=> ( ( finite_card_a @ ( vimage_a_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_894_card__vimage__inj,axiom,
! [F: a > nat,A: set_nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ A @ ( image_a_nat @ F @ top_top_set_a ) )
=> ( ( finite_card_a @ ( vimage_a_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_895_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_896_vimageI,axiom,
! [F: a > a,A2: a,B: a,B2: set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_a @ B @ B2 )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_897_vimageI,axiom,
! [F: nat > a,A2: nat,B: a,B2: set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_a @ B @ B2 )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_898_vimageI,axiom,
! [F: a > nat,A2: a,B: nat,B2: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member_a @ A2 @ ( vimage_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_899_vimageI,axiom,
! [F: nat > nat,A2: nat,B: nat,B2: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_900_vimage__eq,axiom,
! [A2: a,F: a > a,B2: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B2 ) )
= ( member_a @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_901_vimage__eq,axiom,
! [A2: a,F: a > nat,B2: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_902_vimage__eq,axiom,
! [A2: nat,F: nat > a,B2: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ B2 ) )
= ( member_a @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_903_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_904_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_905_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_906_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_907_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_908_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_909_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_910_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_911_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_912_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_913_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_914_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_915_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_916_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_917_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_918_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_919_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_920_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_921_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_922_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_923_vimage__UNIV,axiom,
! [F: a > nat] :
( ( vimage_a_nat @ F @ top_top_set_nat )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_924_vimage__UNIV,axiom,
! [F: nat > a] :
( ( vimage_nat_a @ F @ top_top_set_a )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_925_vimage__UNIV,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ top_top_set_a )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_926_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_927_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_928_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_929_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_930_image__add__0,axiom,
! [S4: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S4 )
= S4 ) ).
% image_add_0
thf(fact_931_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_932_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_933_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_934_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_935_vimageD,axiom,
! [A2: a,F: a > a,A: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_936_vimageD,axiom,
! [A2: a,F: a > nat,A: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_937_vimageD,axiom,
! [A2: nat,F: nat > a,A: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_938_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_939_vimageE,axiom,
! [A2: a,F: a > a,B2: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B2 ) )
=> ( member_a @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_940_vimageE,axiom,
! [A2: a,F: a > nat,B2: set_nat] :
( ( member_a @ A2 @ ( vimage_a_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_941_vimageE,axiom,
! [A2: nat,F: nat > a,B2: set_a] :
( ( member_nat @ A2 @ ( vimage_nat_a @ F @ B2 ) )
=> ( member_a @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_942_vimageE,axiom,
! [A2: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_943_vimageI2,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_944_vimageI2,axiom,
! [F: nat > a,A2: nat,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_945_vimageI2,axiom,
! [F: a > nat,A2: a,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_946_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_947_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_948_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_949_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_950_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_951_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_952_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_953_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_954_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_955_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M6: nat] :
( N2
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_956_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_957_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_958_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_959_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_960_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_961_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_962_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_963_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_964_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_965_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_966_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_967_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_968_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_969_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_970_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_971_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_972_surj__image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_973_surj__image__vimage__eq,axiom,
! [F: nat > a,A: set_a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( image_nat_a @ F @ ( vimage_nat_a @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_974_surj__image__vimage__eq,axiom,
! [F: a > nat,A: set_nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( image_a_nat @ F @ ( vimage_a_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_975_surj__image__vimage__eq,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ F @ ( vimage_a_a @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_976_image__subset__iff__subset__vimage,axiom,
! [F: nat > a,A: set_nat,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_a @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_977_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_978_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 )
= ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_979_image__subset__iff__subset__vimage,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
= ( ord_less_eq_set_a @ A @ ( vimage_a_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_980_image__vimage__subset,axiom,
! [F: nat > a,A: set_a] : ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( vimage_nat_a @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_981_image__vimage__subset,axiom,
! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_982_image__vimage__subset,axiom,
! [F: a > a,A: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( vimage_a_a @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_983_image__vimage__subset,axiom,
! [F: a > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( vimage_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_984_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_985_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_986_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_987_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_988_add__increasing2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_989_add__decreasing2,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_990_add__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_991_add__decreasing,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_992_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_993_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_994_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_995_finite__vimageD,axiom,
! [H: nat > nat,F5: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F5 ) )
=> ( ( ( image_nat_nat @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F5 ) ) ) ).
% finite_vimageD
thf(fact_996_finite__vimageD,axiom,
! [H: nat > a,F5: set_a] :
( ( finite_finite_nat @ ( vimage_nat_a @ H @ F5 ) )
=> ( ( ( image_nat_a @ H @ top_top_set_nat )
= top_top_set_a )
=> ( finite_finite_a @ F5 ) ) ) ).
% finite_vimageD
thf(fact_997_finite__vimageD,axiom,
! [H: a > nat,F5: set_nat] :
( ( finite_finite_a @ ( vimage_a_nat @ H @ F5 ) )
=> ( ( ( image_a_nat @ H @ top_top_set_a )
= top_top_set_nat )
=> ( finite_finite_nat @ F5 ) ) ) ).
% finite_vimageD
thf(fact_998_finite__vimageD,axiom,
! [H: a > a,F5: set_a] :
( ( finite_finite_a @ ( vimage_a_a @ H @ F5 ) )
=> ( ( ( image_a_a @ H @ top_top_set_a )
= top_top_set_a )
=> ( finite_finite_a @ F5 ) ) ) ).
% finite_vimageD
thf(fact_999_vimage__subsetD,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B2 ) @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1000_vimage__subsetD,axiom,
! [F: nat > a,B2: set_a,A: set_nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_a @ F @ B2 ) @ A )
=> ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1001_vimage__subsetD,axiom,
! [F: a > nat,B2: set_nat,A: set_a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ord_less_eq_set_a @ ( vimage_a_nat @ F @ B2 ) @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1002_vimage__subsetD,axiom,
! [F: a > a,B2: set_a,A: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B2 ) @ A )
=> ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_1003_inj__vimage__image__eq,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( vimage_nat_a @ F @ ( image_nat_a @ F @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1004_inj__vimage__image__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( vimage_nat_nat @ F @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1005_inj__vimage__image__eq,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( vimage_a_a @ F @ ( image_a_a @ F @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1006_inj__vimage__image__eq,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( vimage_a_nat @ F @ ( image_a_nat @ F @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1007_finite__vimageI,axiom,
! [F5: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F5 )
=> ( ( inj_on_nat_nat @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage_nat_nat @ H @ F5 ) ) ) ) ).
% finite_vimageI
thf(fact_1008_finite__vimageI,axiom,
! [F5: set_nat,H: a > nat] :
( ( finite_finite_nat @ F5 )
=> ( ( inj_on_a_nat @ H @ top_top_set_a )
=> ( finite_finite_a @ ( vimage_a_nat @ H @ F5 ) ) ) ) ).
% finite_vimageI
thf(fact_1009_finite__vimageD_H,axiom,
! [F: nat > a,A: set_a] :
( ( finite_finite_nat @ ( vimage_nat_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ( finite_finite_a @ A ) ) ) ).
% finite_vimageD'
thf(fact_1010_finite__vimageD_H,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_1011_finite__vimageD_H,axiom,
! [F: a > a,A: set_a] :
( ( finite_finite_a @ ( vimage_a_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ top_top_set_a ) )
=> ( finite_finite_a @ A ) ) ) ).
% finite_vimageD'
thf(fact_1012_finite__vimageD_H,axiom,
! [F: a > nat,A: set_nat] :
( ( finite_finite_a @ ( vimage_a_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_a_nat @ F @ top_top_set_a ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_1013_vimage__subsetI,axiom,
! [F: nat > a,B2: set_a,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_a @ F @ B2 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1014_vimage__subsetI,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B2 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1015_vimage__subsetI,axiom,
! [F: a > a,B2: set_a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B2 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1016_vimage__subsetI,axiom,
! [F: a > nat,B2: set_nat,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( ord_less_eq_set_a @ ( vimage_a_nat @ F @ B2 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1017_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1018_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1019_stream__smap__nats,axiom,
! [S2: stream_nat] :
( S2
= ( smap_nat_nat @ ( snth_nat @ S2 ) @ ( siterate_nat @ suc @ zero_zero_nat ) ) ) ).
% stream_smap_nats
thf(fact_1020_sconst__streams,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ( member_stream_a @ ( siterate_a @ id_a @ X ) @ ( streams_a @ A ) ) ) ).
% sconst_streams
thf(fact_1021_sconst__streams,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( member_stream_nat @ ( siterate_nat @ id_nat @ X ) @ ( streams_nat @ A ) ) ) ).
% sconst_streams
thf(fact_1022_card__range__greater__zero,axiom,
! [F: nat > a] :
( ( finite_finite_a @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1023_card__range__greater__zero,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_1024_card__range__greater__zero,axiom,
! [F: a > a] :
( ( finite_finite_a @ ( image_a_a @ F @ top_top_set_a ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( image_a_a @ F @ top_top_set_a ) ) ) ) ).
% card_range_greater_zero
thf(fact_1025_card__range__greater__zero,axiom,
! [F: a > nat] :
( ( finite_finite_nat @ ( image_a_nat @ F @ top_top_set_a ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_a_nat @ F @ top_top_set_a ) ) ) ) ).
% card_range_greater_zero
thf(fact_1026_card__vimage__inj__on__le,axiom,
! [F: nat > nat,D: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ D )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ D ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1027_IntI,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1028_IntI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1029_Int__iff,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_1030_Int__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_1031_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1032_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_1033_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_1034_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1035_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_1036_sup_Obounded__iff,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1037_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1038_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1039_inf__top_Oright__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ top_top_set_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1040_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B ) )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1041_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_a,B: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A2 @ B ) )
= ( ( A2 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1042_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1043_inf__top_Oleft__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1044_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1045_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1046_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1047_top__eq__inf__iff,axiom,
! [X: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X @ Y ) )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1048_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1049_inf__eq__top__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= top_top_set_a )
= ( ( X = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1050_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_1051_inf__top__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ top_top_set_a )
= X ) ).
% inf_top_right
thf(fact_1052_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_1053_inf__top__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X )
= X ) ).
% inf_top_left
thf(fact_1054_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_1055_boolean__algebra_Odisj__one__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% boolean_algebra.disj_one_right
thf(fact_1056_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_1057_boolean__algebra_Odisj__one__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_one_left
thf(fact_1058_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_1059_sup__top__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% sup_top_right
thf(fact_1060_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_1061_sup__top__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ top_top_set_a @ X )
= top_top_set_a ) ).
% sup_top_left
thf(fact_1062_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1063_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1064_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1065_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1066_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1067_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1068_Int__UNIV,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1069_Int__UNIV,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1070_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1071_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1072_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1073_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1074_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1075_sup__compl__top__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( sup_sup_set_nat @ X @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_1076_sup__compl__top__left1,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( sup_sup_set_a @ X @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left1
thf(fact_1077_sup__compl__top__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_1078_sup__compl__top__left2,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left2
thf(fact_1079_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1080_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1081_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1082_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1083_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1084_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1085_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1086_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1087_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1088_image__vimage__eq,axiom,
! [F: nat > a,A: set_a] :
( ( image_nat_a @ F @ ( vimage_nat_a @ F @ A ) )
= ( inf_inf_set_a @ A @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1089_image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_1090_image__vimage__eq,axiom,
! [F: a > a,A: set_a] :
( ( image_a_a @ F @ ( vimage_a_a @ F @ A ) )
= ( inf_inf_set_a @ A @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1091_image__vimage__eq,axiom,
! [F: a > nat,A: set_nat] :
( ( image_a_nat @ F @ ( vimage_a_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_a_nat @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1092_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1093_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1094_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1095_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1096_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1097_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1098_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1099_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_1100_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1101_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1102_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1103_sup__cancel__left2,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( sup_sup_set_nat @ X @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_1104_sup__cancel__left2,axiom,
! [X: set_a,A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ A2 ) @ ( sup_sup_set_a @ X @ B ) )
= top_top_set_a ) ).
% sup_cancel_left2
thf(fact_1105_sup__cancel__left1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ A2 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_1106_sup__cancel__left1,axiom,
! [X: set_a,A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ A2 ) @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ B ) )
= top_top_set_a ) ).
% sup_cancel_left1
thf(fact_1107_image__Un,axiom,
! [F: nat > a,A: set_nat,B2: set_nat] :
( ( image_nat_a @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1108_image__Un,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1109_image__Un,axiom,
! [F: a > a,A: set_a,B2: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1110_image__Un,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( image_a_nat @ F @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1111_inj__on__Int,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ( inj_on_nat_nat @ F @ A )
| ( inj_on_nat_nat @ F @ B2 ) )
=> ( inj_on_nat_nat @ F @ ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% inj_on_Int
thf(fact_1112_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_1113_verit__comp__simplify1_I3_J,axiom,
! [B10: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B10 @ A7 ) )
= ( ord_less_nat @ A7 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1114_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1115_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1116_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1117_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1118_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1119_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1120_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_1121_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_1122_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_1123_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1124_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1125_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_1126_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_1127_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_1128_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1129_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1130_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_1131_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1132_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1133_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1134_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1135_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1136_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1137_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_1138_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1139_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1140_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1141_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1142_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1143_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1144_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1145_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1146_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_1147_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_1148_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1149_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1150_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1151_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1152_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1153_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1154_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1155_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_1156_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1157_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1158_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1159_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1160_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1161_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1162_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_1163_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1164_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1165_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_1166_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1167_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1168_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1169_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1170_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1171_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1172_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1173_le__infE,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_1174_le__infI,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1175_inf__mono,axiom,
! [A2: nat,C2: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1176_le__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_1177_le__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_1178_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1179_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1180_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1181_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1182_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1183_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1184_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1185_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1186_inf_OboundedE,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1187_inf_OboundedI,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1188_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1189_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1190_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1191_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1192_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1193_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1194_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_1195_inf_OcoboundedI2,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_1196_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1197_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1198_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1199_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1200_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1201_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1202_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1203_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1204_sup_Omono,axiom,
! [C2: nat,A2: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1205_sup__mono,axiom,
! [A2: nat,C2: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1206_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_1207_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_1208_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1209_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1210_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1211_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1212_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1213_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1214_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1215_sup_OboundedE,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1216_sup_OboundedI,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1217_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_1218_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1219_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1220_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1221_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_1222_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1223_sup_OcoboundedI2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1224_Int__Collect__mono,axiom,
! [A: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1225_Int__Collect__mono,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1226_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1227_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1228_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1229_add__strict__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1230_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1231_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1232_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_1233_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_1234_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1235_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1236_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_1237_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1238_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1239_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1240_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1241_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_1242_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1243_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1244_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_1245_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_1246_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_1247_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_1248_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1249_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1250_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_1251_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1252_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1253_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1254_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_1255_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_1256_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1257_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1258_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1259_IntE,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1260_IntE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1261_IntD1,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_1262_IntD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_1263_IntD2,axiom,
! [C2: a,A: set_a,B2: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1264_IntD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1265_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1266_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1267_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_1268_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_1269_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_1270_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
% Conjectures (2)
thf(conj_0,hypothesis,
( ( image_nat_a @ f @ top_top_set_nat )
= top_top_set_a ) ).
thf(conj_1,conjecture,
( ( sset_a @ ( fair_fair_stream_a @ f ) )
= top_top_set_a ) ).
%------------------------------------------------------------------------------