TPTP Problem File: SLH0304^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0011_Completeness/prob_00120_004142__12151062_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1779 ( 603 unt; 499 typ; 0 def)
% Number of atoms : 3343 (1138 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9985 ( 300 ~; 34 |; 192 &;8001 @)
% ( 0 <=>;1458 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 67 ( 66 usr)
% Number of type conns : 1862 (1862 >; 0 *; 0 +; 0 <<)
% Number of symbols : 436 ( 433 usr; 29 con; 0-5 aty)
% Number of variables : 3713 ( 355 ^;3238 !; 120 ?;3713 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:28:11.358
%------------------------------------------------------------------------------
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bNF_cs4850752376292380463m_rule: set_Pr1008144964186165195m_rule > ( produc1754732471204517991ist_fm > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc1754732471204517991ist_fm ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > $o ).
thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
bNF_cs1207223144858359663e_rule: set_Pr1008144964186165195m_rule > ( produc164195504107695125m_rule > rule ) > ( produc780299800278082923e_rule > rule ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > ( produc164195504107695125m_rule > produc780299800278082923e_rule ) > $o ).
thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
bNF_cs8798692456506898627m_rule: set_Pr6543236023296505803e_rule > ( produc4630866025709511057ist_fm > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( produc780299800278082923e_rule > produc4630866025709511057ist_fm ) > ( produc780299800278082923e_rule > produc164195504107695125m_rule ) > $o ).
thf(sy_c_BNF__Def_OfstOp_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_fs6361025107161210196ist_fm: ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_BNF__Def_OfstOp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
bNF_fs3491168633039030583m_rule: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > produc164195504107695125m_rule > produc1754732471204517991ist_fm ).
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bNF_fs2143400664727635963e_rule: ( produc1996495991257130529ist_fm > rule > $o ) > ( rule > rule > $o ) > produc164195504107695125m_rule > produc164195504107695125m_rule ).
thf(sy_c_BNF__Def_OfstOp_001t__Syntax__Ofm_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Syntax__Ofm_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
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bNF_fs3887825027587321911ist_fm: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > produc4630866025709511057ist_fm > produc4630866025709511057ist_fm ).
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bNF_fs9177076566076742319m_rule: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > produc780299800278082923e_rule > produc4630866025709511057ist_fm ).
thf(sy_c_BNF__Def_Opick__middlep_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_pi3539088406881798022ist_fm: ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > list_fm > list_fm > list_fm ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
bNF_pi3438825987790746729m_rule: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule_001t__Syntax__Orule,type,
bNF_pi580457772512444745e_rule: ( produc1996495991257130529ist_fm > rule > $o ) > ( rule > rule > $o ) > produc1996495991257130529ist_fm > rule > rule ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Syntax__Ofm_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
bNF_pi3939735538430251307atural: ( fm > fm > $o ) > ( fm > produc7822875418678951345atural > $o ) > fm > produc7822875418678951345atural > fm ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
bNF_pi3835482382339038057ist_fm: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
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bNF_pi4963049444341131953ist_fm: ( rule > rule > $o ) > ( rule > produc1996495991257130529ist_fm > $o ) > rule > produc1996495991257130529ist_fm > rule ).
thf(sy_c_BNF__Def_OsndOp_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_sn5292590578203848342ist_fm: ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ).
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bNF_sn6617447237779055481m_rule: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > produc164195504107695125m_rule > produc164195504107695125m_rule ).
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bNF_sn6445580637211383353e_rule: ( produc1996495991257130529ist_fm > rule > $o ) > ( rule > rule > $o ) > produc164195504107695125m_rule > produc780299800278082923e_rule ).
thf(sy_c_BNF__Def_OsndOp_001t__Syntax__Ofm_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
bNF_sndOp_fm_fm_fm: ( fm > fm > $o ) > ( fm > fm > $o ) > product_prod_fm_fm > product_prod_fm_fm ).
thf(sy_c_BNF__Def_OsndOp_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
bNF_sn4255884501705713901m_rule: ( rule > produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > produc780299800278082923e_rule > produc164195504107695125m_rule ).
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bNF_sn1604800272185294753ist_fm: ( rule > rule > $o ) > ( rule > produc1996495991257130529ist_fm > $o ) > produc4630866025709511057ist_fm > produc4630866025709511057ist_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Syntax__Ofm,type,
bNF_Gr3717417623296551025ift_fm: set_list_fm > fm > set_list_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Syntax__Ofm,type,
bNF_Greatest_Succ_fm: set_list_fm > list_fm > set_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_Gr1661195949116424188ist_fm: set_list_fm > ( list_fm > list_fm ) > ( list_fm > list_fm ) > set_Pr7058068377845519745ist_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__Syntax__Ofm_J_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
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bNF_Gr869466243678018025ist_fm: set_Pr4463079037648049377_fm_fm > ( product_prod_fm_fm > list_fm ) > ( product_prod_fm_fm > list_fm ) > set_Pr7058068377845519745ist_fm ).
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bNF_Gr2224021619804040521_fm_fm: set_Pr4463079037648049377_fm_fm > ( product_prod_fm_fm > fm ) > ( product_prod_fm_fm > fm ) > set_Pr4463079037648049377_fm_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__Set__Oset_It__Syntax__Ofm_J_J_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
bNF_Gr2844036924399170616_fm_fm: set_set_set_fm > ( set_set_fm > fm ) > ( set_set_fm > fm ) > set_Pr4463079037648049377_fm_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_Gr6790864367800852130ist_fm: set_set_fm > ( set_fm > list_fm ) > ( set_fm > list_fm ) > set_Pr7058068377845519745ist_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__Syntax__Ofm_J_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ofm_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
bNF_Gr2680958839322950764ist_fm: set_fm > ( fm > list_fm ) > ( fm > list_fm ) > set_Pr7058068377845519745ist_fm ).
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bNF_Gr4411798878337052732m_rule: set_fm > ( fm > produc1996495991257130529ist_fm ) > ( fm > rule ) > set_Pr1008144964186165195m_rule ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ofm_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
bNF_Gr1203782223388445388_fm_fm: set_fm > ( fm > fm ) > ( fm > fm ) > set_Pr4463079037648049377_fm_fm ).
thf(sy_c_Basic__BNFs_Ofsts_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
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thf(sy_c_Basic__BNFs_Ofsts_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
basic_fsts_fm_fm: product_prod_fm_fm > set_fm ).
thf(sy_c_Basic__BNFs_Ofstsp_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
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thf(sy_c_Basic__BNFs_Osnds_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
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thf(sy_c_Completeness_OHintikka,type,
hintikka: set_fm > set_fm > $o ).
thf(sy_c_Completeness_OtreeA,type,
treeA: stream727092118206550309m_rule > set_fm ).
thf(sy_c_Completeness_OtreeB,type,
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thf(sy_c_Finite__Set_OFpow_001t__Syntax__Ofm,type,
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thf(sy_c_Fun__Def_Oin__rel_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Ofm_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__List__Olist_It__Syntax__Ofm_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Set__Oset_It__Syntax__Ofm_J,type,
if_set_fm: $o > set_fm > set_fm > set_fm ).
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__Set__Oset_It__Syntax__Ofm_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Syntax__Ofm_J,type,
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thf(sy_c_List_Obind_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
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thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Syntax__Ofm_J,type,
shd_list_fm: stream_list_fm > list_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
shd_Pr772355297128350925ist_fm: stream8299795917829157543ist_fm > produc1996495991257130529ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
shd_Pr4587271278095355155ist_fm: stream3409308193418444653ist_fm > produc1754732471204517991ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
shd_Pr7235097944458474089m_rule: stream727092118206550309m_rule > produc164195504107695125m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
shd_Pr5328274044443569301ist_fm: stream8936951515322355153ist_fm > produc4387668504790219841ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
shd_Pr2340755956080993207m_rule: stream6494289010434245521m_rule > produc3793571666044074251m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Syntax__Orule_J,type,
shd_Pr4631506658928855911e_rule: stream2312013117288958913e_rule > produc6288491646048892603e_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
shd_Pr8144422112483312429_fm_fm: stream1375025385734440967_fm_fm > product_prod_fm_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
shd_Pr2478396429205514213ist_fm: stream1960312990768957601ist_fm > produc4630866025709511057ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
shd_Pr3564872132220007m_rule: stream1878305877988316353m_rule > produc1660549859252256699m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J,type,
shd_Pr4790473843695609551m_rule: stream3856641574645492619m_rule > produc2329059809757588603m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
shd_Pr4573454438745646103e_rule: stream3588992565182678257e_rule > produc780299800278082923e_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Syntax__Ofm,type,
shd_fm: stream_fm > fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Syntax__Orule,type,
shd_rule: stream_rule > rule ).
thf(sy_c_Stream_Ostream__all_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
stream720971363120138347ist_fm: ( produc1996495991257130529ist_fm > $o ) > stream8299795917829157543ist_fm > $o ).
thf(sy_c_Stream_Ostream__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
stream456966007795584843m_rule: ( produc164195504107695125m_rule > $o ) > stream727092118206550309m_rule > $o ).
thf(sy_c_Stream_Ostream__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
stream919847744520025429m_rule: ( produc3793571666044074251m_rule > $o ) > stream6494289010434245521m_rule > $o ).
thf(sy_c_Stream_Ostream__all_001t__Syntax__Orule,type,
stream_all_rule: ( rule > $o ) > stream_rule > $o ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
szip_list_fm_list_fm: stream_list_fm > stream_list_fm > stream8299795917829157543ist_fm ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
szip_P6003557782763710490ist_fm: stream8299795917829157543ist_fm > stream8299795917829157543ist_fm > stream3409308193418444653ist_fm ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
szip_P5379742314337339356m_rule: stream8299795917829157543ist_fm > stream727092118206550309m_rule > stream642806101564963573m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
szip_P1977448745965526924m_rule: stream8299795917829157543ist_fm > stream_rule > stream727092118206550309m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
szip_P5077708527527104496ist_fm: stream727092118206550309m_rule > stream8299795917829157543ist_fm > stream8936951515322355153ist_fm ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
szip_P2499414959592755846m_rule: stream727092118206550309m_rule > stream727092118206550309m_rule > stream6494289010434245521m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule,type,
szip_P553291425819358518e_rule: stream727092118206550309m_rule > stream_rule > stream2312013117288958913e_rule ).
thf(sy_c_Stream_Oszip_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
szip_fm_fm: stream_fm > stream_fm > stream1375025385734440967_fm_fm ).
thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
szip_r8549736553299883584ist_fm: stream_rule > stream8299795917829157543ist_fm > stream1960312990768957601ist_fm ).
thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
szip_r2254114822203693110m_rule: stream_rule > stream727092118206550309m_rule > stream1878305877988316353m_rule ).
thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Syntax__Orule,type,
szip_rule_rule: stream_rule > stream_rule > stream3588992565182678257e_rule ).
thf(sy_c_Syntax_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_Zorn_Ochains_001t__Syntax__Ofm,type,
chains_fm: set_set_fm > set_set_set_fm ).
thf(sy_c_member_001t__List__Olist_It__Syntax__Ofm_J,type,
member_list_fm2: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
member8280416476884352090atural: produc7822875418678951345atural > set_Pr7564037697178849041atural > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
member8102475879199740618ist_fm: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
member4220325220686508332m_rule: produc164195504107695125m_rule > set_Pr1008144964186165195m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
member2434505771962676916m_rule: produc3793571666044074251m_rule > set_Pr5090152306159784043m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
member5418343652702035821atural: produc7777387212919034070atural > set_Pr8331943448356867980atural > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
member8474499337054950954_fm_fm: product_prod_fm_fm > set_Pr4463079037648049377_fm_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
member8686995742288324264ist_fm: produc4630866025709511057ist_fm > set_Pr2241365836748572487ist_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Syntax__Ofm_J_J,type,
member_set_set_fm2: set_set_fm > set_set_set_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Syntax__Ofm_J,type,
member_set_fm2: set_fm > set_set_fm > $o ).
thf(sy_c_member_001t__Syntax__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__Syntax__Orule,type,
member_rule2: rule > set_rule > $o ).
thf(sy_v_P____,type,
p: nat ).
thf(sy_v_steps,type,
steps: stream727092118206550309m_rule ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_ts____,type,
ts: list_tm ).
% Relevant facts (1271)
thf(fact_0_treeA__snth,axiom,
! [P: fm,Steps: stream727092118206550309m_rule] :
( ( member_fm2 @ P @ ( treeA @ Steps ) )
=> ? [N: nat] : ( member_fm2 @ P @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ) ) ) ) ).
% treeA_snth
thf(fact_1__092_060open_062_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_092_060in_062_AtreeA_Asteps_092_060close_062,axiom,
member_fm2 @ ( pre @ p @ ts ) @ ( treeA @ steps ) ).
% \<open>\<^bold>\<ddagger> P ts \<in> treeA steps\<close>
thf(fact_2_fm_Oinject_I1_J,axiom,
! [X21: nat,X22: list_tm,Y21: nat,Y22: list_tm] :
( ( ( pre @ X21 @ X22 )
= ( pre @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fm.inject(1)
thf(fact_3_treeB__snth,axiom,
! [P: fm,Steps: stream727092118206550309m_rule] :
( ( member_fm2 @ P @ ( treeB @ Steps ) )
=> ? [N: nat] : ( member_fm2 @ P @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ) ) ) ) ).
% treeB_snth
thf(fact_4_sdrop__szip,axiom,
! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
( ( sdrop_7224736112439592940m_rule @ N2 @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
= ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_5_sdrop__szip,axiom,
! [N2: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
( ( sdrop_7373388980841566196m_rule @ N2 @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
= ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_6_sdrop__szip,axiom,
! [N2: nat,S1: stream_rule,S2: stream_rule] :
( ( sdrop_2199339672054592340e_rule @ N2 @ ( szip_rule_rule @ S1 @ S2 ) )
= ( szip_rule_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_7_sdrop__szip,axiom,
! [N2: nat,S1: stream_list_fm,S2: stream_list_fm] :
( ( sdrop_4442373711808556042ist_fm @ N2 @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
= ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N2 @ S1 ) @ ( sdrop_list_fm @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_8_sdrop__szip,axiom,
! [N2: nat,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
( ( sdrop_2468034597186633064ist_fm @ N2 @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
= ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_9_sdrop__szip,axiom,
! [N2: nat,S1: stream727092118206550309m_rule,S2: stream_rule] :
( ( sdrop_6392754500180887844e_rule @ N2 @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
= ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
! [N2: nat,S1: stream_rule,S2: stream727092118206550309m_rule] :
( ( sdrop_1764812713384251940m_rule @ N2 @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
= ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_11_sdrop__szip,axiom,
! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
( ( sdrop_1261538545438237392ist_fm @ N2 @ ( szip_P6003557782763710490ist_fm @ S1 @ S2 ) )
= ( szip_P6003557782763710490ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_12_sdrop__szip,axiom,
! [N2: nat,S1: stream727092118206550309m_rule,S2: stream8299795917829157543ist_fm] :
( ( sdrop_3792585760419738520ist_fm @ N2 @ ( szip_P5077708527527104496ist_fm @ S1 @ S2 ) )
= ( szip_P5077708527527104496ist_fm @ ( sdrop_7224736112439592940m_rule @ N2 @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
! [N2: nat,S1: stream8299795917829157543ist_fm,S2: stream727092118206550309m_rule] :
( ( sdrop_9094843686636319420m_rule @ N2 @ ( szip_P5379742314337339356m_rule @ S1 @ S2 ) )
= ( szip_P5379742314337339356m_rule @ ( sdrop_4442373711808556042ist_fm @ N2 @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_14_Hintikka_OBasic,axiom,
! [A: set_fm,B: set_fm,P2: nat,Ts: list_tm] :
( ( hintikka @ A @ B )
=> ( ( member_fm2 @ ( pre @ P2 @ Ts ) @ A )
=> ~ ( member_fm2 @ ( pre @ P2 @ Ts ) @ B ) ) ) ).
% Hintikka.Basic
thf(fact_15_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream3856641574645492619m_rule] :
( ( shd_Pr4790473843695609551m_rule @ ( sdrop_4762382306744956754m_rule @ N2 @ S ) )
= ( snth_P4313833507243213527m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_16_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream1878305877988316353m_rule] :
( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N2 @ S ) )
= ( snth_P7967347330282200671m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_17_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream1960312990768957601ist_fm] :
( ( shd_Pr2478396429205514213ist_fm @ ( sdrop_2468034597186633064ist_fm @ N2 @ S ) )
= ( snth_P6179250367719980013ist_fm @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_18_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream3588992565182678257e_rule] :
( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ S ) )
= ( snth_P8487267234472522255e_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_19_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream_list_fm] :
( ( shd_list_fm @ ( sdrop_list_fm @ N2 @ S ) )
= ( snth_list_fm @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_20_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream8299795917829157543ist_fm] :
( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ S ) )
= ( snth_P2624749845746815685ist_fm @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_21_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream_rule] :
( ( shd_rule @ ( sdrop_rule @ N2 @ S ) )
= ( snth_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_22_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream6494289010434245521m_rule] :
( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N2 @ S ) )
= ( snth_P8950227884416379311m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_23_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream727092118206550309m_rule] :
( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ S ) )
= ( snth_P1712579846118164081m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_24_treeA__sdrop,axiom,
! [N2: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeA @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) @ ( treeA @ Steps ) ) ).
% treeA_sdrop
thf(fact_25_in__set__member,axiom,
! [X: product_prod_fm_fm,Xs: list_P6934817935912052487_fm_fm] :
( ( member8474499337054950954_fm_fm @ X @ ( set_Pr6624747810323803798_fm_fm @ Xs ) )
= ( member636769053065275292_fm_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_26_in__set__member,axiom,
! [X: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
( ( member8102475879199740618ist_fm @ X @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
= ( member8995235396497187388ist_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_27_in__set__member,axiom,
! [X: set_fm,Xs: list_set_fm] :
( ( member_set_fm2 @ X @ ( set_set_fm2 @ Xs ) )
= ( member_set_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_28_in__set__member,axiom,
! [X: set_set_fm,Xs: list_set_set_fm] :
( ( member_set_set_fm2 @ X @ ( set_set_set_fm2 @ Xs ) )
= ( member_set_set_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_29_in__set__member,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( member_list_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_30_in__set__member,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_31_fstsp_Ocases,axiom,
! [P: produc7777387212919034070atural,A2: fm] :
( ( basic_2707087092747759734atural @ P @ A2 )
=> ( A2
= ( produc7697328821926353820atural @ P ) ) ) ).
% fstsp.cases
thf(fact_32_fstsp_Ocases,axiom,
! [P: produc1996495991257130529ist_fm,A2: list_fm] :
( ( basic_1563062696195910635ist_fm @ P @ A2 )
=> ( A2
= ( produc1501393135466168645ist_fm @ P ) ) ) ).
% fstsp.cases
thf(fact_33_fstsp_Ocases,axiom,
! [P: produc164195504107695125m_rule,A2: produc1996495991257130529ist_fm] :
( ( basic_5389291090992669949m_rule @ P @ A2 )
=> ( A2
= ( produc6879501374131015971m_rule @ P ) ) ) ).
% fstsp.cases
thf(fact_34_fstsp_Osimps,axiom,
( basic_2707087092747759734atural
= ( ^ [P3: produc7777387212919034070atural,A3: fm] :
( A3
= ( produc7697328821926353820atural @ P3 ) ) ) ) ).
% fstsp.simps
thf(fact_35_fstsp_Osimps,axiom,
( basic_1563062696195910635ist_fm
= ( ^ [P3: produc1996495991257130529ist_fm,A3: list_fm] :
( A3
= ( produc1501393135466168645ist_fm @ P3 ) ) ) ) ).
% fstsp.simps
thf(fact_36_fstsp_Osimps,axiom,
( basic_5389291090992669949m_rule
= ( ^ [P3: produc164195504107695125m_rule,A3: produc1996495991257130529ist_fm] :
( A3
= ( produc6879501374131015971m_rule @ P3 ) ) ) ) ).
% fstsp.simps
thf(fact_37_fstsp_Ointros,axiom,
! [P: produc7777387212919034070atural] : ( basic_2707087092747759734atural @ P @ ( produc7697328821926353820atural @ P ) ) ).
% fstsp.intros
thf(fact_38_fstsp_Ointros,axiom,
! [P: produc1996495991257130529ist_fm] : ( basic_1563062696195910635ist_fm @ P @ ( produc1501393135466168645ist_fm @ P ) ) ).
% fstsp.intros
thf(fact_39_fstsp_Ointros,axiom,
! [P: produc164195504107695125m_rule] : ( basic_5389291090992669949m_rule @ P @ ( produc6879501374131015971m_rule @ P ) ) ).
% fstsp.intros
thf(fact_40_subset__code_I1_J,axiom,
! [Xs: list_P6934817935912052487_fm_fm,B: set_Pr4463079037648049377_fm_fm] :
( ( ord_le7532831136298237057_fm_fm @ ( set_Pr6624747810323803798_fm_fm @ Xs ) @ B )
= ( ! [X2: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X2 @ ( set_Pr6624747810323803798_fm_fm @ Xs ) )
=> ( member8474499337054950954_fm_fm @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_41_subset__code_I1_J,axiom,
! [Xs: list_set_set_fm,B: set_set_set_fm] :
( ( ord_le8219319623975992658set_fm @ ( set_set_set_fm2 @ Xs ) @ B )
= ( ! [X2: set_set_fm] :
( ( member_set_set_fm2 @ X2 @ ( set_set_set_fm2 @ Xs ) )
=> ( member_set_set_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_42_subset__code_I1_J,axiom,
! [Xs: list_P5616295576739893671ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xs ) @ B )
= ( ! [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
=> ( member8102475879199740618ist_fm @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_43_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ ( set_list_fm2 @ Xs ) @ B )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_44_subset__code_I1_J,axiom,
! [Xs: list_set_fm,B: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ ( set_set_fm2 @ Xs ) @ B )
= ( ! [X2: set_fm] :
( ( member_set_fm2 @ X2 @ ( set_set_fm2 @ Xs ) )
=> ( member_set_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_45_subset__code_I1_J,axiom,
! [Xs: list_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_46_treeB__sdrop,axiom,
! [N2: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeB @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) @ ( treeB @ Steps ) ) ).
% treeB_sdrop
thf(fact_47_subsetI,axiom,
! [A: set_Pr4463079037648049377_fm_fm,B: set_Pr4463079037648049377_fm_fm] :
( ! [X3: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X3 @ A )
=> ( member8474499337054950954_fm_fm @ X3 @ B ) )
=> ( ord_le7532831136298237057_fm_fm @ A @ B ) ) ).
% subsetI
thf(fact_48_subsetI,axiom,
! [A: set_set_set_fm,B: set_set_set_fm] :
( ! [X3: set_set_fm] :
( ( member_set_set_fm2 @ X3 @ A )
=> ( member_set_set_fm2 @ X3 @ B ) )
=> ( ord_le8219319623975992658set_fm @ A @ B ) ) ).
% subsetI
thf(fact_49_subsetI,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
=> ( member8102475879199740618ist_fm @ X3 @ B ) )
=> ( ord_le2055072402069232929ist_fm @ A @ B ) ) ).
% subsetI
thf(fact_50_subsetI,axiom,
! [A: set_list_fm,B: set_list_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A )
=> ( member_list_fm2 @ X3 @ B ) )
=> ( ord_le3173885194822255862ist_fm @ A @ B ) ) ).
% subsetI
thf(fact_51_subsetI,axiom,
! [A: set_set_fm,B: set_set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ A )
=> ( member_set_fm2 @ X3 @ B ) )
=> ( ord_le1461404734466536732set_fm @ A @ B ) ) ).
% subsetI
thf(fact_52_subsetI,axiom,
! [A: set_fm,B: set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member_fm2 @ X3 @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% subsetI
thf(fact_53_subset__antisym,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ( ( ord_le2055072402069232929ist_fm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_54_subset__antisym,axiom,
! [A: set_list_fm,B: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ( ord_le3173885194822255862ist_fm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_55_subset__antisym,axiom,
! [A: set_set_fm,B: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ( ord_le1461404734466536732set_fm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_56_subset__antisym,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_57_order__refl,axiom,
! [X: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ X @ X ) ).
% order_refl
thf(fact_58_order__refl,axiom,
! [X: set_list_fm] : ( ord_le3173885194822255862ist_fm @ X @ X ) ).
% order_refl
thf(fact_59_order__refl,axiom,
! [X: set_set_fm] : ( ord_le1461404734466536732set_fm @ X @ X ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X: $o > set_fm] : ( ord_less_eq_o_set_fm @ X @ X ) ).
% order_refl
thf(fact_61_order__refl,axiom,
! [X: $o > nat] : ( ord_less_eq_o_nat @ X @ X ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X: set_fm] : ( ord_less_eq_set_fm @ X @ X ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_64_dual__order_Orefl,axiom,
! [A2: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_65_dual__order_Orefl,axiom,
! [A2: set_list_fm] : ( ord_le3173885194822255862ist_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_66_dual__order_Orefl,axiom,
! [A2: set_set_fm] : ( ord_le1461404734466536732set_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A2: $o > set_fm] : ( ord_less_eq_o_set_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_71_prod__eq__iff,axiom,
( ( ^ [Y: produc7777387212919034070atural,Z: produc7777387212919034070atural] : ( Y = Z ) )
= ( ^ [S3: produc7777387212919034070atural,T: produc7777387212919034070atural] :
( ( ( produc7697328821926353820atural @ S3 )
= ( produc7697328821926353820atural @ T ) )
& ( ( produc7769300925896546778atural @ S3 )
= ( produc7769300925896546778atural @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_72_prod__eq__iff,axiom,
( ( ^ [Y: produc4630866025709511057ist_fm,Z: produc4630866025709511057ist_fm] : ( Y = Z ) )
= ( ^ [S3: produc4630866025709511057ist_fm,T: produc4630866025709511057ist_fm] :
( ( ( produc4228417144610596823ist_fm @ S3 )
= ( produc4228417144610596823ist_fm @ T ) )
& ( ( produc4514744107061996309ist_fm @ S3 )
= ( produc4514744107061996309ist_fm @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_73_prod__eq__iff,axiom,
( ( ^ [Y: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y = Z ) )
= ( ^ [S3: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
( ( ( produc6879501374131015971m_rule @ S3 )
= ( produc6879501374131015971m_rule @ T ) )
& ( ( produc7165828336582415457m_rule @ S3 )
= ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_74_prod__eq__iff,axiom,
( ( ^ [Y: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y = Z ) )
= ( ^ [S3: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
( ( ( produc1501393135466168645ist_fm @ S3 )
= ( produc1501393135466168645ist_fm @ T ) )
& ( ( produc4588648349897876871ist_fm @ S3 )
= ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_75_exE__realizer_H,axiom,
! [P2: produc7822875418678951345atural > fm > $o,P: produc7777387212919034070atural] :
( ( P2 @ ( produc7769300925896546778atural @ P ) @ ( produc7697328821926353820atural @ P ) )
=> ~ ! [X3: fm,Y2: produc7822875418678951345atural] :
~ ( P2 @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_76_exE__realizer_H,axiom,
! [P2: produc1996495991257130529ist_fm > rule > $o,P: produc4630866025709511057ist_fm] :
( ( P2 @ ( produc4514744107061996309ist_fm @ P ) @ ( produc4228417144610596823ist_fm @ P ) )
=> ~ ! [X3: rule,Y2: produc1996495991257130529ist_fm] :
~ ( P2 @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_77_exE__realizer_H,axiom,
! [P2: rule > produc1996495991257130529ist_fm > $o,P: produc164195504107695125m_rule] :
( ( P2 @ ( produc7165828336582415457m_rule @ P ) @ ( produc6879501374131015971m_rule @ P ) )
=> ~ ! [X3: produc1996495991257130529ist_fm,Y2: rule] :
~ ( P2 @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_78_exE__realizer_H,axiom,
! [P2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
( ( P2 @ ( produc4588648349897876871ist_fm @ P ) @ ( produc1501393135466168645ist_fm @ P ) )
=> ~ ! [X3: list_fm,Y2: list_fm] :
~ ( P2 @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_79_prod__eqI,axiom,
! [P: produc7777387212919034070atural,Q: produc7777387212919034070atural] :
( ( ( produc7697328821926353820atural @ P )
= ( produc7697328821926353820atural @ Q ) )
=> ( ( ( produc7769300925896546778atural @ P )
= ( produc7769300925896546778atural @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_80_prod__eqI,axiom,
! [P: produc4630866025709511057ist_fm,Q: produc4630866025709511057ist_fm] :
( ( ( produc4228417144610596823ist_fm @ P )
= ( produc4228417144610596823ist_fm @ Q ) )
=> ( ( ( produc4514744107061996309ist_fm @ P )
= ( produc4514744107061996309ist_fm @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_81_prod__eqI,axiom,
! [P: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
( ( ( produc6879501374131015971m_rule @ P )
= ( produc6879501374131015971m_rule @ Q ) )
=> ( ( ( produc7165828336582415457m_rule @ P )
= ( produc7165828336582415457m_rule @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_82_prod__eqI,axiom,
! [P: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
( ( ( produc1501393135466168645ist_fm @ P )
= ( produc1501393135466168645ist_fm @ Q ) )
=> ( ( ( produc4588648349897876871ist_fm @ P )
= ( produc4588648349897876871ist_fm @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_83_prod_Oexpand,axiom,
! [Prod: produc7777387212919034070atural,Prod2: produc7777387212919034070atural] :
( ( ( ( produc7697328821926353820atural @ Prod )
= ( produc7697328821926353820atural @ Prod2 ) )
& ( ( produc7769300925896546778atural @ Prod )
= ( produc7769300925896546778atural @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_84_prod_Oexpand,axiom,
! [Prod: produc4630866025709511057ist_fm,Prod2: produc4630866025709511057ist_fm] :
( ( ( ( produc4228417144610596823ist_fm @ Prod )
= ( produc4228417144610596823ist_fm @ Prod2 ) )
& ( ( produc4514744107061996309ist_fm @ Prod )
= ( produc4514744107061996309ist_fm @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_85_prod_Oexpand,axiom,
! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
( ( ( ( produc6879501374131015971m_rule @ Prod )
= ( produc6879501374131015971m_rule @ Prod2 ) )
& ( ( produc7165828336582415457m_rule @ Prod )
= ( produc7165828336582415457m_rule @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_86_prod_Oexpand,axiom,
! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
( ( ( ( produc1501393135466168645ist_fm @ Prod )
= ( produc1501393135466168645ist_fm @ Prod2 ) )
& ( ( produc4588648349897876871ist_fm @ Prod )
= ( produc4588648349897876871ist_fm @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_87_szip_Osimps_I1_J,axiom,
! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
= ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_88_szip_Osimps_I1_J,axiom,
! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
= ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_89_szip_Osimps_I1_J,axiom,
! [S1: stream_fm,S2: stream_fm] :
( ( shd_Pr8144422112483312429_fm_fm @ ( szip_fm_fm @ S1 @ S2 ) )
= ( product_Pair_fm_fm @ ( shd_fm @ S1 ) @ ( shd_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_90_szip_Osimps_I1_J,axiom,
! [S1: stream_rule,S2: stream_rule] :
( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
= ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_91_szip_Osimps_I1_J,axiom,
! [S1: stream_list_fm,S2: stream_list_fm] :
( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
= ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_92_szip_Osimps_I1_J,axiom,
! [S1: stream_rule,S2: stream8299795917829157543ist_fm] :
( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
= ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_93_szip_Osimps_I1_J,axiom,
! [S1: stream727092118206550309m_rule,S2: stream_rule] :
( ( shd_Pr4631506658928855911e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
= ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_94_szip_Osimps_I1_J,axiom,
! [S1: stream_rule,S2: stream727092118206550309m_rule] :
( ( shd_Pr3564872132220007m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
= ( produc6084188410630241011m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_95_szip_Osimps_I1_J,axiom,
! [S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
( ( shd_Pr4587271278095355155ist_fm @ ( szip_P6003557782763710490ist_fm @ S1 @ S2 ) )
= ( produc2592314732767339223ist_fm @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_96_szip_Osimps_I1_J,axiom,
! [S1: stream727092118206550309m_rule,S2: stream8299795917829157543ist_fm] :
( ( shd_Pr5328274044443569301ist_fm @ ( szip_P5077708527527104496ist_fm @ S1 @ S2 ) )
= ( produc4643107087651910131ist_fm @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_97_subset__code_I2_J,axiom,
! [A: set_Pr4463079037648049377_fm_fm,Ys: list_P6934817935912052487_fm_fm] :
( ( ord_le7532831136298237057_fm_fm @ A @ ( coset_180832915674267764_fm_fm @ Ys ) )
= ( ! [X2: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X2 @ ( set_Pr6624747810323803798_fm_fm @ Ys ) )
=> ~ ( member8474499337054950954_fm_fm @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_98_subset__code_I2_J,axiom,
! [A: set_set_set_fm,Ys: list_set_set_fm] :
( ( ord_le8219319623975992658set_fm @ A @ ( coset_set_set_fm @ Ys ) )
= ( ! [X2: set_set_fm] :
( ( member_set_set_fm2 @ X2 @ ( set_set_set_fm2 @ Ys ) )
=> ~ ( member_set_set_fm2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_99_subset__code_I2_J,axiom,
! [A: set_Pr7058068377845519745ist_fm,Ys: list_P5616295576739893671ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ ( coset_820689808388072212ist_fm @ Ys ) )
= ( ! [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ ( set_Pr8767716839810916150ist_fm @ Ys ) )
=> ~ ( member8102475879199740618ist_fm @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_100_subset__code_I2_J,axiom,
! [A: set_list_fm,Ys: list_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ ( coset_list_fm @ Ys ) )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Ys ) )
=> ~ ( member_list_fm2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_101_subset__code_I2_J,axiom,
! [A: set_set_fm,Ys: list_set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ ( coset_set_fm @ Ys ) )
= ( ! [X2: set_fm] :
( ( member_set_fm2 @ X2 @ ( set_set_fm2 @ Ys ) )
=> ~ ( member_set_fm2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_102_subset__code_I2_J,axiom,
! [A: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A @ ( coset_fm @ Ys ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_103_in__mono,axiom,
! [A: set_Pr4463079037648049377_fm_fm,B: set_Pr4463079037648049377_fm_fm,X: product_prod_fm_fm] :
( ( ord_le7532831136298237057_fm_fm @ A @ B )
=> ( ( member8474499337054950954_fm_fm @ X @ A )
=> ( member8474499337054950954_fm_fm @ X @ B ) ) ) ).
% in_mono
thf(fact_104_in__mono,axiom,
! [A: set_set_set_fm,B: set_set_set_fm,X: set_set_fm] :
( ( ord_le8219319623975992658set_fm @ A @ B )
=> ( ( member_set_set_fm2 @ X @ A )
=> ( member_set_set_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_105_in__mono,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,X: produc1996495991257130529ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ( ( member8102475879199740618ist_fm @ X @ A )
=> ( member8102475879199740618ist_fm @ X @ B ) ) ) ).
% in_mono
thf(fact_106_in__mono,axiom,
! [A: set_list_fm,B: set_list_fm,X: list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ( member_list_fm2 @ X @ A )
=> ( member_list_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_107_in__mono,axiom,
! [A: set_set_fm,B: set_set_fm,X: set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ( member_set_fm2 @ X @ A )
=> ( member_set_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_108_in__mono,axiom,
! [A: set_fm,B: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm2 @ X @ A )
=> ( member_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_109_subsetD,axiom,
! [A: set_Pr4463079037648049377_fm_fm,B: set_Pr4463079037648049377_fm_fm,C: product_prod_fm_fm] :
( ( ord_le7532831136298237057_fm_fm @ A @ B )
=> ( ( member8474499337054950954_fm_fm @ C @ A )
=> ( member8474499337054950954_fm_fm @ C @ B ) ) ) ).
% subsetD
thf(fact_110_subsetD,axiom,
! [A: set_set_set_fm,B: set_set_set_fm,C: set_set_fm] :
( ( ord_le8219319623975992658set_fm @ A @ B )
=> ( ( member_set_set_fm2 @ C @ A )
=> ( member_set_set_fm2 @ C @ B ) ) ) ).
% subsetD
thf(fact_111_subsetD,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ( ( member8102475879199740618ist_fm @ C @ A )
=> ( member8102475879199740618ist_fm @ C @ B ) ) ) ).
% subsetD
thf(fact_112_subsetD,axiom,
! [A: set_list_fm,B: set_list_fm,C: list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ( member_list_fm2 @ C @ A )
=> ( member_list_fm2 @ C @ B ) ) ) ).
% subsetD
thf(fact_113_subsetD,axiom,
! [A: set_set_fm,B: set_set_fm,C: set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ( member_set_fm2 @ C @ A )
=> ( member_set_fm2 @ C @ B ) ) ) ).
% subsetD
thf(fact_114_subsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% subsetD
thf(fact_115_old_Oprod_Oinject,axiom,
! [A2: produc1996495991257130529ist_fm,B2: rule,A4: produc1996495991257130529ist_fm,B3: rule] :
( ( ( produc491467635432902671m_rule @ A2 @ B2 )
= ( produc491467635432902671m_rule @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_116_old_Oprod_Oinject,axiom,
! [A2: produc164195504107695125m_rule,B2: produc164195504107695125m_rule,A4: produc164195504107695125m_rule,B3: produc164195504107695125m_rule] :
( ( ( produc5927390650430071747m_rule @ A2 @ B2 )
= ( produc5927390650430071747m_rule @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_117_old_Oprod_Oinject,axiom,
! [A2: list_fm,B2: list_fm,A4: list_fm,B3: list_fm] :
( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
= ( produc381145313068854617ist_fm @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_118_old_Oprod_Oinject,axiom,
! [A2: rule,B2: produc1996495991257130529ist_fm,A4: rule,B3: produc1996495991257130529ist_fm] :
( ( ( produc7063755442767259331ist_fm @ A2 @ B2 )
= ( produc7063755442767259331ist_fm @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_119_old_Oprod_Oinject,axiom,
! [A2: fm,B2: fm,A4: fm,B3: fm] :
( ( ( product_Pair_fm_fm @ A2 @ B2 )
= ( product_Pair_fm_fm @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_120_prod_Oinject,axiom,
! [X1: produc1996495991257130529ist_fm,X23: rule,Y1: produc1996495991257130529ist_fm,Y23: rule] :
( ( ( produc491467635432902671m_rule @ X1 @ X23 )
= ( produc491467635432902671m_rule @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_121_prod_Oinject,axiom,
! [X1: produc164195504107695125m_rule,X23: produc164195504107695125m_rule,Y1: produc164195504107695125m_rule,Y23: produc164195504107695125m_rule] :
( ( ( produc5927390650430071747m_rule @ X1 @ X23 )
= ( produc5927390650430071747m_rule @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_122_prod_Oinject,axiom,
! [X1: list_fm,X23: list_fm,Y1: list_fm,Y23: list_fm] :
( ( ( produc381145313068854617ist_fm @ X1 @ X23 )
= ( produc381145313068854617ist_fm @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_123_prod_Oinject,axiom,
! [X1: rule,X23: produc1996495991257130529ist_fm,Y1: rule,Y23: produc1996495991257130529ist_fm] :
( ( ( produc7063755442767259331ist_fm @ X1 @ X23 )
= ( produc7063755442767259331ist_fm @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_124_prod_Oinject,axiom,
! [X1: fm,X23: fm,Y1: fm,Y23: fm] :
( ( ( product_Pair_fm_fm @ X1 @ X23 )
= ( product_Pair_fm_fm @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_125_prod_Ocollapse,axiom,
! [Prod: produc3793571666044074251m_rule] :
( ( produc5927390650430071747m_rule @ ( produc4837027324325561263m_rule @ Prod ) @ ( produc5389215402712513009m_rule @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_126_prod_Ocollapse,axiom,
! [Prod: product_prod_fm_fm] :
( ( product_Pair_fm_fm @ ( product_fst_fm_fm @ Prod ) @ ( product_snd_fm_fm @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_127_prod_Ocollapse,axiom,
! [Prod: produc7777387212919034070atural] :
( ( produc912813528046151816atural @ ( produc7697328821926353820atural @ Prod ) @ ( produc7769300925896546778atural @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_128_prod_Ocollapse,axiom,
! [Prod: produc4630866025709511057ist_fm] :
( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_129_prod_Ocollapse,axiom,
! [Prod: produc164195504107695125m_rule] :
( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_130_prod_Ocollapse,axiom,
! [Prod: produc1996495991257130529ist_fm] :
( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_131_snth__szip,axiom,
! [S1: stream8299795917829157543ist_fm,S2: stream_rule,N2: nat] :
( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) @ N2 )
= ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_132_snth__szip,axiom,
! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N2: nat] :
( ( snth_P8950227884416379311m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) @ N2 )
= ( produc5927390650430071747m_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N2 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_133_snth__szip,axiom,
! [S1: stream_rule,S2: stream_rule,N2: nat] :
( ( snth_P8487267234472522255e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N2 )
= ( produc1769597910952761123e_rule @ ( snth_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_134_snth__szip,axiom,
! [S1: stream_fm,S2: stream_fm,N2: nat] :
( ( snth_P8653008792276846885_fm_fm @ ( szip_fm_fm @ S1 @ S2 ) @ N2 )
= ( product_Pair_fm_fm @ ( snth_fm @ S1 @ N2 ) @ ( snth_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_135_snth__szip,axiom,
! [S1: stream_list_fm,S2: stream_list_fm,N2: nat] :
( ( snth_P2624749845746815685ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) @ N2 )
= ( produc381145313068854617ist_fm @ ( snth_list_fm @ S1 @ N2 ) @ ( snth_list_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_136_snth__szip,axiom,
! [S1: stream_rule,S2: stream8299795917829157543ist_fm,N2: nat] :
( ( snth_P6179250367719980013ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) @ N2 )
= ( produc7063755442767259331ist_fm @ ( snth_rule @ S1 @ N2 ) @ ( snth_P2624749845746815685ist_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_137_snth__szip,axiom,
! [S1: stream727092118206550309m_rule,S2: stream_rule,N2: nat] :
( ( snth_P3371917080224060767e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) @ N2 )
= ( produc4383365014245906419e_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_138_snth__szip,axiom,
! [S1: stream_rule,S2: stream727092118206550309m_rule,N2: nat] :
( ( snth_P7967347330282200671m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) @ N2 )
= ( produc6084188410630241011m_rule @ ( snth_rule @ S1 @ N2 ) @ ( snth_P1712579846118164081m_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_139_snth__szip,axiom,
! [S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm,N2: nat] :
( ( snth_P6007491322845746955ist_fm @ ( szip_P6003557782763710490ist_fm @ S1 @ S2 ) @ N2 )
= ( produc2592314732767339223ist_fm @ ( snth_P2624749845746815685ist_fm @ S1 @ N2 ) @ ( snth_P2624749845746815685ist_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_140_snth__szip,axiom,
! [S1: stream727092118206550309m_rule,S2: stream8299795917829157543ist_fm,N2: nat] :
( ( snth_P1870038792014611613ist_fm @ ( szip_P5077708527527104496ist_fm @ S1 @ S2 ) @ N2 )
= ( produc4643107087651910131ist_fm @ ( snth_P1712579846118164081m_rule @ S1 @ N2 ) @ ( snth_P2624749845746815685ist_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_141_prod__induct3,axiom,
! [P2: produc3793571666044074251m_rule > $o,X: produc3793571666044074251m_rule] :
( ! [A5: produc164195504107695125m_rule,B4: produc1996495991257130529ist_fm,C2: rule] : ( P2 @ ( produc5927390650430071747m_rule @ A5 @ ( produc491467635432902671m_rule @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_142_prod__induct3,axiom,
! [P2: produc4630866025709511057ist_fm > $o,X: produc4630866025709511057ist_fm] :
( ! [A5: rule,B4: list_fm,C2: list_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A5 @ ( produc381145313068854617ist_fm @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_143_prod__cases3,axiom,
! [Y3: produc3793571666044074251m_rule] :
~ ! [A5: produc164195504107695125m_rule,B4: produc1996495991257130529ist_fm,C2: rule] :
( Y3
!= ( produc5927390650430071747m_rule @ A5 @ ( produc491467635432902671m_rule @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_144_prod__cases3,axiom,
! [Y3: produc4630866025709511057ist_fm] :
~ ! [A5: rule,B4: list_fm,C2: list_fm] :
( Y3
!= ( produc7063755442767259331ist_fm @ A5 @ ( produc381145313068854617ist_fm @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_145_Pair__inject,axiom,
! [A2: produc1996495991257130529ist_fm,B2: rule,A4: produc1996495991257130529ist_fm,B3: rule] :
( ( ( produc491467635432902671m_rule @ A2 @ B2 )
= ( produc491467635432902671m_rule @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_146_Pair__inject,axiom,
! [A2: produc164195504107695125m_rule,B2: produc164195504107695125m_rule,A4: produc164195504107695125m_rule,B3: produc164195504107695125m_rule] :
( ( ( produc5927390650430071747m_rule @ A2 @ B2 )
= ( produc5927390650430071747m_rule @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_147_Pair__inject,axiom,
! [A2: list_fm,B2: list_fm,A4: list_fm,B3: list_fm] :
( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
= ( produc381145313068854617ist_fm @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_148_Pair__inject,axiom,
! [A2: rule,B2: produc1996495991257130529ist_fm,A4: rule,B3: produc1996495991257130529ist_fm] :
( ( ( produc7063755442767259331ist_fm @ A2 @ B2 )
= ( produc7063755442767259331ist_fm @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_149_Pair__inject,axiom,
! [A2: fm,B2: fm,A4: fm,B3: fm] :
( ( ( product_Pair_fm_fm @ A2 @ B2 )
= ( product_Pair_fm_fm @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_150_prod__cases,axiom,
! [P2: produc164195504107695125m_rule > $o,P: produc164195504107695125m_rule] :
( ! [A5: produc1996495991257130529ist_fm,B4: rule] : ( P2 @ ( produc491467635432902671m_rule @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_151_prod__cases,axiom,
! [P2: produc3793571666044074251m_rule > $o,P: produc3793571666044074251m_rule] :
( ! [A5: produc164195504107695125m_rule,B4: produc164195504107695125m_rule] : ( P2 @ ( produc5927390650430071747m_rule @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_152_prod__cases,axiom,
! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm] :
( ! [A5: list_fm,B4: list_fm] : ( P2 @ ( produc381145313068854617ist_fm @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_153_prod__cases,axiom,
! [P2: produc4630866025709511057ist_fm > $o,P: produc4630866025709511057ist_fm] :
( ! [A5: rule,B4: produc1996495991257130529ist_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_154_prod__cases,axiom,
! [P2: product_prod_fm_fm > $o,P: product_prod_fm_fm] :
( ! [A5: fm,B4: fm] : ( P2 @ ( product_Pair_fm_fm @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_155_surj__pair,axiom,
! [P: produc164195504107695125m_rule] :
? [X3: produc1996495991257130529ist_fm,Y2: rule] :
( P
= ( produc491467635432902671m_rule @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_156_surj__pair,axiom,
! [P: produc3793571666044074251m_rule] :
? [X3: produc164195504107695125m_rule,Y2: produc164195504107695125m_rule] :
( P
= ( produc5927390650430071747m_rule @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_157_surj__pair,axiom,
! [P: produc1996495991257130529ist_fm] :
? [X3: list_fm,Y2: list_fm] :
( P
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_158_surj__pair,axiom,
! [P: produc4630866025709511057ist_fm] :
? [X3: rule,Y2: produc1996495991257130529ist_fm] :
( P
= ( produc7063755442767259331ist_fm @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_159_surj__pair,axiom,
! [P: product_prod_fm_fm] :
? [X3: fm,Y2: fm] :
( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_160_mem__Collect__eq,axiom,
! [A2: product_prod_fm_fm,P2: product_prod_fm_fm > $o] :
( ( member8474499337054950954_fm_fm @ A2 @ ( collec4425076577877259116_fm_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A2: list_fm,P2: list_fm > $o] :
( ( member_list_fm2 @ A2 @ ( collect_list_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A2: set_fm,P2: set_fm > $o] :
( ( member_set_fm2 @ A2 @ ( collect_set_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
! [A2: set_set_fm,P2: set_set_fm > $o] :
( ( member_set_set_fm2 @ A2 @ ( collect_set_set_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
! [A2: produc1996495991257130529ist_fm,P2: produc1996495991257130529ist_fm > $o] :
( ( member8102475879199740618ist_fm @ A2 @ ( collec7231575776666803724ist_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A2: produc164195504107695125m_rule,P2: produc164195504107695125m_rule > $o] :
( ( member4220325220686508332m_rule @ A2 @ ( collec2069506698588392810m_rule @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A2: fm,P2: fm > $o] :
( ( member_fm2 @ A2 @ ( collect_fm @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A: set_Pr4463079037648049377_fm_fm] :
( ( collec4425076577877259116_fm_fm
@ ^ [X2: product_prod_fm_fm] : ( member8474499337054950954_fm_fm @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A: set_list_fm] :
( ( collect_list_fm
@ ^ [X2: list_fm] : ( member_list_fm2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A: set_set_fm] :
( ( collect_set_fm
@ ^ [X2: set_fm] : ( member_set_fm2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A: set_set_set_fm] :
( ( collect_set_set_fm
@ ^ [X2: set_set_fm] : ( member_set_set_fm2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
! [A: set_Pr7058068377845519745ist_fm] :
( ( collec7231575776666803724ist_fm
@ ^ [X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_172_Collect__mem__eq,axiom,
! [A: set_Pr1008144964186165195m_rule] :
( ( collec2069506698588392810m_rule
@ ^ [X2: produc164195504107695125m_rule] : ( member4220325220686508332m_rule @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_174_Collect__cong,axiom,
! [P2: fm > $o,Q2: fm > $o] :
( ! [X3: fm] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_fm @ P2 )
= ( collect_fm @ Q2 ) ) ) ).
% Collect_cong
thf(fact_175_Collect__cong,axiom,
! [P2: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec7231575776666803724ist_fm @ P2 )
= ( collec7231575776666803724ist_fm @ Q2 ) ) ) ).
% Collect_cong
thf(fact_176_Collect__cong,axiom,
! [P2: produc164195504107695125m_rule > $o,Q2: produc164195504107695125m_rule > $o] :
( ! [X3: produc164195504107695125m_rule] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec2069506698588392810m_rule @ P2 )
= ( collec2069506698588392810m_rule @ Q2 ) ) ) ).
% Collect_cong
thf(fact_177_old_Oprod_Oexhaust,axiom,
! [Y3: produc164195504107695125m_rule] :
~ ! [A5: produc1996495991257130529ist_fm,B4: rule] :
( Y3
!= ( produc491467635432902671m_rule @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_178_old_Oprod_Oexhaust,axiom,
! [Y3: produc3793571666044074251m_rule] :
~ ! [A5: produc164195504107695125m_rule,B4: produc164195504107695125m_rule] :
( Y3
!= ( produc5927390650430071747m_rule @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_179_old_Oprod_Oexhaust,axiom,
! [Y3: produc1996495991257130529ist_fm] :
~ ! [A5: list_fm,B4: list_fm] :
( Y3
!= ( produc381145313068854617ist_fm @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_180_old_Oprod_Oexhaust,axiom,
! [Y3: produc4630866025709511057ist_fm] :
~ ! [A5: rule,B4: produc1996495991257130529ist_fm] :
( Y3
!= ( produc7063755442767259331ist_fm @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_181_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_fm_fm] :
~ ! [A5: fm,B4: fm] :
( Y3
!= ( product_Pair_fm_fm @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_182_fst__eqD,axiom,
! [X: produc164195504107695125m_rule,Y3: produc164195504107695125m_rule,A2: produc164195504107695125m_rule] :
( ( ( produc4837027324325561263m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_183_fst__eqD,axiom,
! [X: rule,Y3: produc1996495991257130529ist_fm,A2: rule] :
( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_184_fst__eqD,axiom,
! [X: fm,Y3: fm,A2: fm] :
( ( ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_185_fst__eqD,axiom,
! [X: fm,Y3: produc7822875418678951345atural,A2: fm] :
( ( ( produc7697328821926353820atural @ ( produc912813528046151816atural @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_186_fst__eqD,axiom,
! [X: list_fm,Y3: list_fm,A2: list_fm] :
( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_187_fst__eqD,axiom,
! [X: produc1996495991257130529ist_fm,Y3: rule,A2: produc1996495991257130529ist_fm] :
( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_188_fst__conv,axiom,
! [X1: produc164195504107695125m_rule,X23: produc164195504107695125m_rule] :
( ( produc4837027324325561263m_rule @ ( produc5927390650430071747m_rule @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_189_fst__conv,axiom,
! [X1: rule,X23: produc1996495991257130529ist_fm] :
( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_190_fst__conv,axiom,
! [X1: fm,X23: fm] :
( ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_191_fst__conv,axiom,
! [X1: fm,X23: produc7822875418678951345atural] :
( ( produc7697328821926353820atural @ ( produc912813528046151816atural @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_192_fst__conv,axiom,
! [X1: list_fm,X23: list_fm] :
( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_193_fst__conv,axiom,
! [X1: produc1996495991257130529ist_fm,X23: rule] :
( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_194_snd__conv,axiom,
! [X1: produc164195504107695125m_rule,X23: produc164195504107695125m_rule] :
( ( produc5389215402712513009m_rule @ ( produc5927390650430071747m_rule @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_195_snd__conv,axiom,
! [X1: fm,X23: fm] :
( ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_196_snd__conv,axiom,
! [X1: produc1996495991257130529ist_fm,X23: rule] :
( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_197_snd__conv,axiom,
! [X1: rule,X23: produc1996495991257130529ist_fm] :
( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_198_snd__conv,axiom,
! [X1: list_fm,X23: list_fm] :
( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_199_snd__eqD,axiom,
! [X: produc164195504107695125m_rule,Y3: produc164195504107695125m_rule,A2: produc164195504107695125m_rule] :
( ( ( produc5389215402712513009m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_200_snd__eqD,axiom,
! [X: fm,Y3: fm,A2: fm] :
( ( ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_201_snd__eqD,axiom,
! [X: produc1996495991257130529ist_fm,Y3: rule,A2: rule] :
( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_202_snd__eqD,axiom,
! [X: rule,Y3: produc1996495991257130529ist_fm,A2: produc1996495991257130529ist_fm] :
( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_203_snd__eqD,axiom,
! [X: list_fm,Y3: list_fm,A2: list_fm] :
( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) )
= A2 )
=> ( Y3 = A2 ) ) ).
% snd_eqD
thf(fact_204_surjective__pairing,axiom,
! [T2: produc3793571666044074251m_rule] :
( T2
= ( produc5927390650430071747m_rule @ ( produc4837027324325561263m_rule @ T2 ) @ ( produc5389215402712513009m_rule @ T2 ) ) ) ).
% surjective_pairing
thf(fact_205_surjective__pairing,axiom,
! [T2: product_prod_fm_fm] :
( T2
= ( product_Pair_fm_fm @ ( product_fst_fm_fm @ T2 ) @ ( product_snd_fm_fm @ T2 ) ) ) ).
% surjective_pairing
thf(fact_206_surjective__pairing,axiom,
! [T2: produc7777387212919034070atural] :
( T2
= ( produc912813528046151816atural @ ( produc7697328821926353820atural @ T2 ) @ ( produc7769300925896546778atural @ T2 ) ) ) ).
% surjective_pairing
thf(fact_207_surjective__pairing,axiom,
! [T2: produc4630866025709511057ist_fm] :
( T2
= ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T2 ) @ ( produc4514744107061996309ist_fm @ T2 ) ) ) ).
% surjective_pairing
thf(fact_208_surjective__pairing,axiom,
! [T2: produc164195504107695125m_rule] :
( T2
= ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).
% surjective_pairing
thf(fact_209_surjective__pairing,axiom,
! [T2: produc1996495991257130529ist_fm] :
( T2
= ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).
% surjective_pairing
thf(fact_210_conjI__realizer,axiom,
! [P2: produc164195504107695125m_rule > $o,P: produc164195504107695125m_rule,Q2: produc164195504107695125m_rule > $o,Q: produc164195504107695125m_rule] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc4837027324325561263m_rule @ ( produc5927390650430071747m_rule @ P @ Q ) ) )
& ( Q2 @ ( produc5389215402712513009m_rule @ ( produc5927390650430071747m_rule @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_211_conjI__realizer,axiom,
! [P2: fm > $o,P: fm,Q2: fm > $o,Q: fm] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst_fm_fm @ ( product_Pair_fm_fm @ P @ Q ) ) )
& ( Q2 @ ( product_snd_fm_fm @ ( product_Pair_fm_fm @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_212_conjI__realizer,axiom,
! [P2: fm > $o,P: fm,Q2: produc7822875418678951345atural > $o,Q: produc7822875418678951345atural] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc7697328821926353820atural @ ( produc912813528046151816atural @ P @ Q ) ) )
& ( Q2 @ ( produc7769300925896546778atural @ ( produc912813528046151816atural @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_213_conjI__realizer,axiom,
! [P2: rule > $o,P: rule,Q2: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) )
& ( Q2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_214_conjI__realizer,axiom,
! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q2: rule > $o,Q: rule] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) )
& ( Q2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_215_conjI__realizer,axiom,
! [P2: list_fm > $o,P: list_fm,Q2: list_fm > $o,Q: list_fm] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) )
& ( Q2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_216_exI__realizer,axiom,
! [P2: produc164195504107695125m_rule > produc164195504107695125m_rule > $o,Y3: produc164195504107695125m_rule,X: produc164195504107695125m_rule] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( produc5389215402712513009m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) ) @ ( produc4837027324325561263m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_217_exI__realizer,axiom,
! [P2: fm > fm > $o,Y3: fm,X: fm] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( product_snd_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) ) @ ( product_fst_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_218_exI__realizer,axiom,
! [P2: produc7822875418678951345atural > fm > $o,Y3: produc7822875418678951345atural,X: fm] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( produc7769300925896546778atural @ ( produc912813528046151816atural @ X @ Y3 ) ) @ ( produc7697328821926353820atural @ ( produc912813528046151816atural @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_219_exI__realizer,axiom,
! [P2: produc1996495991257130529ist_fm > rule > $o,Y3: produc1996495991257130529ist_fm,X: rule] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_220_exI__realizer,axiom,
! [P2: rule > produc1996495991257130529ist_fm > $o,Y3: rule,X: produc1996495991257130529ist_fm] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_221_exI__realizer,axiom,
! [P2: list_fm > list_fm > $o,Y3: list_fm,X: list_fm] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_222_prod_Oexhaust__sel,axiom,
! [Prod: produc3793571666044074251m_rule] :
( Prod
= ( produc5927390650430071747m_rule @ ( produc4837027324325561263m_rule @ Prod ) @ ( produc5389215402712513009m_rule @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_223_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_fm_fm] :
( Prod
= ( product_Pair_fm_fm @ ( product_fst_fm_fm @ Prod ) @ ( product_snd_fm_fm @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_224_prod_Oexhaust__sel,axiom,
! [Prod: produc7777387212919034070atural] :
( Prod
= ( produc912813528046151816atural @ ( produc7697328821926353820atural @ Prod ) @ ( produc7769300925896546778atural @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_225_prod_Oexhaust__sel,axiom,
! [Prod: produc4630866025709511057ist_fm] :
( Prod
= ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_226_prod_Oexhaust__sel,axiom,
! [Prod: produc164195504107695125m_rule] :
( Prod
= ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_227_prod_Oexhaust__sel,axiom,
! [Prod: produc1996495991257130529ist_fm] :
( Prod
= ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_228_order__antisym__conv,axiom,
! [Y3: set_Pr7058068377845519745ist_fm,X: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ Y3 @ X )
=> ( ( ord_le2055072402069232929ist_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_229_order__antisym__conv,axiom,
! [Y3: set_list_fm,X: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ Y3 @ X )
=> ( ( ord_le3173885194822255862ist_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_230_order__antisym__conv,axiom,
! [Y3: set_set_fm,X: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ Y3 @ X )
=> ( ( ord_le1461404734466536732set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_231_order__antisym__conv,axiom,
! [Y3: $o > set_fm,X: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ Y3 @ X )
=> ( ( ord_less_eq_o_set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_232_order__antisym__conv,axiom,
! [Y3: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y3 @ X )
=> ( ( ord_less_eq_o_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_233_order__antisym__conv,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( ( ord_less_eq_set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_234_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_235_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_236_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_237_ord__le__eq__subst,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_238_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_239_ord__le__eq__subst,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_240_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_list_fm,C: set_list_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3173885194822255862ist_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_241_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_fm,C: set_set_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1461404734466536732set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_242_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_243_ord__le__eq__subst,axiom,
! [A2: set_list_fm,B2: set_list_fm,F: set_list_fm > nat,C: nat] :
( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_list_fm,Y2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A2: set_set_fm,B2: set_set_fm,F: set_set_fm > nat,C: nat] :
( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_fm,Y2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__le__eq__subst,axiom,
! [A2: $o > nat,B2: $o > nat,F: ( $o > nat ) > nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_247_ord__eq__le__subst,axiom,
! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_248_ord__eq__le__subst,axiom,
! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_249_ord__eq__le__subst,axiom,
! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_250_ord__eq__le__subst,axiom,
! [A2: set_list_fm,F: nat > set_list_fm,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3173885194822255862ist_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_251_ord__eq__le__subst,axiom,
! [A2: set_set_fm,F: nat > set_set_fm,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1461404734466536732set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A2: $o > nat,F: nat > $o > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A2: nat,F: set_list_fm > nat,B2: set_list_fm,C: set_list_fm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3173885194822255862ist_fm @ B2 @ C )
=> ( ! [X3: set_list_fm,Y2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_fm > nat,B2: set_set_fm,C: set_set_fm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1461404734466536732set_fm @ B2 @ C )
=> ( ! [X3: set_set_fm,Y2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B2: $o > nat,C: $o > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_nat @ B2 @ C )
=> ( ! [X3: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_257_order__eq__refl,axiom,
! [X: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( X = Y3 )
=> ( ord_le2055072402069232929ist_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_258_order__eq__refl,axiom,
! [X: set_list_fm,Y3: set_list_fm] :
( ( X = Y3 )
=> ( ord_le3173885194822255862ist_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_259_order__eq__refl,axiom,
! [X: set_set_fm,Y3: set_set_fm] :
( ( X = Y3 )
=> ( ord_le1461404734466536732set_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_260_order__eq__refl,axiom,
! [X: $o > set_fm,Y3: $o > set_fm] :
( ( X = Y3 )
=> ( ord_less_eq_o_set_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_261_order__eq__refl,axiom,
! [X: $o > nat,Y3: $o > nat] :
( ( X = Y3 )
=> ( ord_less_eq_o_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_262_order__eq__refl,axiom,
! [X: set_fm,Y3: set_fm] :
( ( X = Y3 )
=> ( ord_less_eq_set_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_263_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_264_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_265_order__subst2,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_266_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_267_order__subst2,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_268_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_list_fm,C: set_list_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le3173885194822255862ist_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3173885194822255862ist_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_269_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_fm,C: set_set_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le1461404734466536732set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1461404734466536732set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_270_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_271_order__subst2,axiom,
! [A2: set_list_fm,B2: set_list_fm,F: set_list_fm > nat,C: nat] :
( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_list_fm,Y2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_272_order__subst2,axiom,
! [A2: set_set_fm,B2: set_set_fm,F: set_set_fm > nat,C: nat] :
( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_fm,Y2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_273_order__subst2,axiom,
! [A2: $o > nat,B2: $o > nat,F: ( $o > nat ) > nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_274_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_275_order__subst1,axiom,
! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_276_order__subst1,axiom,
! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_277_order__subst1,axiom,
! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_278_order__subst1,axiom,
! [A2: nat,F: set_list_fm > nat,B2: set_list_fm,C: set_list_fm] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3173885194822255862ist_fm @ B2 @ C )
=> ( ! [X3: set_list_fm,Y2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_279_order__subst1,axiom,
! [A2: nat,F: set_set_fm > nat,B2: set_set_fm,C: set_set_fm] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1461404734466536732set_fm @ B2 @ C )
=> ( ! [X3: set_set_fm,Y2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_280_order__subst1,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B2: $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_o_nat @ B2 @ C )
=> ( ! [X3: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_281_order__subst1,axiom,
! [A2: set_list_fm,F: nat > set_list_fm,B2: nat,C: nat] :
( ( ord_le3173885194822255862ist_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3173885194822255862ist_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_282_order__subst1,axiom,
! [A2: set_set_fm,F: nat > set_set_fm,B2: nat,C: nat] :
( ( ord_le1461404734466536732set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1461404734466536732set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_283_order__subst1,axiom,
! [A2: $o > nat,F: nat > $o > nat,B2: nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_284_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Pr7058068377845519745ist_fm,Z: set_Pr7058068377845519745ist_fm] : ( Y = Z ) )
= ( ^ [A3: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A3 @ B5 )
& ( ord_le2055072402069232929ist_fm @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_285_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_list_fm,Z: set_list_fm] : ( Y = Z ) )
= ( ^ [A3: set_list_fm,B5: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A3 @ B5 )
& ( ord_le3173885194822255862ist_fm @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_286_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_set_fm,Z: set_set_fm] : ( Y = Z ) )
= ( ^ [A3: set_set_fm,B5: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A3 @ B5 )
& ( ord_le1461404734466536732set_fm @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_287_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: $o > set_fm,Z: $o > set_fm] : ( Y = Z ) )
= ( ^ [A3: $o > set_fm,B5: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ A3 @ B5 )
& ( ord_less_eq_o_set_fm @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_288_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: $o > nat,Z: $o > nat] : ( Y = Z ) )
= ( ^ [A3: $o > nat,B5: $o > nat] :
( ( ord_less_eq_o_nat @ A3 @ B5 )
& ( ord_less_eq_o_nat @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_289_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
= ( ^ [A3: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B5 )
& ( ord_less_eq_set_fm @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_290_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_291_le__fun__def,axiom,
( ord_less_eq_o_set_fm
= ( ^ [F2: $o > set_fm,G: $o > set_fm] :
! [X2: $o] : ( ord_less_eq_set_fm @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_292_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G: $o > nat] :
! [X2: $o] : ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_293_le__funI,axiom,
! [F: $o > set_fm,G2: $o > set_fm] :
( ! [X3: $o] : ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq_o_set_fm @ F @ G2 ) ) ).
% le_funI
thf(fact_294_le__funI,axiom,
! [F: $o > nat,G2: $o > nat] :
( ! [X3: $o] : ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq_o_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_295_le__funE,axiom,
! [F: $o > set_fm,G2: $o > set_fm,X: $o] :
( ( ord_less_eq_o_set_fm @ F @ G2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_296_le__funE,axiom,
! [F: $o > nat,G2: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_297_le__funD,axiom,
! [F: $o > set_fm,G2: $o > set_fm,X: $o] :
( ( ord_less_eq_o_set_fm @ F @ G2 )
=> ( ord_less_eq_set_fm @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_298_le__funD,axiom,
! [F: $o > nat,G2: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_299_antisym,axiom,
! [A2: set_Pr7058068377845519745ist_fm,B2: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A2 @ B2 )
=> ( ( ord_le2055072402069232929ist_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_300_antisym,axiom,
! [A2: set_list_fm,B2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( ( ord_le3173885194822255862ist_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_301_antisym,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( ( ord_le1461404734466536732set_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_302_antisym,axiom,
! [A2: $o > set_fm,B2: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_o_set_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_303_antisym,axiom,
! [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_304_antisym,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_305_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_306_dual__order_Otrans,axiom,
! [B2: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ B2 @ A2 )
=> ( ( ord_le2055072402069232929ist_fm @ C @ B2 )
=> ( ord_le2055072402069232929ist_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_307_dual__order_Otrans,axiom,
! [B2: set_list_fm,A2: set_list_fm,C: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ B2 @ A2 )
=> ( ( ord_le3173885194822255862ist_fm @ C @ B2 )
=> ( ord_le3173885194822255862ist_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_308_dual__order_Otrans,axiom,
! [B2: set_set_fm,A2: set_set_fm,C: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ B2 @ A2 )
=> ( ( ord_le1461404734466536732set_fm @ C @ B2 )
=> ( ord_le1461404734466536732set_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_309_dual__order_Otrans,axiom,
! [B2: $o > set_fm,A2: $o > set_fm,C: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_o_set_fm @ C @ B2 )
=> ( ord_less_eq_o_set_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_310_dual__order_Otrans,axiom,
! [B2: $o > nat,A2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( ord_less_eq_o_nat @ C @ B2 )
=> ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_311_dual__order_Otrans,axiom,
! [B2: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_set_fm @ C @ B2 )
=> ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_312_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_313_dual__order_Oantisym,axiom,
! [B2: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ B2 @ A2 )
=> ( ( ord_le2055072402069232929ist_fm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_314_dual__order_Oantisym,axiom,
! [B2: set_list_fm,A2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ B2 @ A2 )
=> ( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_315_dual__order_Oantisym,axiom,
! [B2: set_set_fm,A2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ B2 @ A2 )
=> ( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_316_dual__order_Oantisym,axiom,
! [B2: $o > set_fm,A2: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_o_set_fm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_317_dual__order_Oantisym,axiom,
! [B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_318_dual__order_Oantisym,axiom,
! [B2: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_319_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_320_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Pr7058068377845519745ist_fm,Z: set_Pr7058068377845519745ist_fm] : ( Y = Z ) )
= ( ^ [A3: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ B5 @ A3 )
& ( ord_le2055072402069232929ist_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_321_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_list_fm,Z: set_list_fm] : ( Y = Z ) )
= ( ^ [A3: set_list_fm,B5: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ B5 @ A3 )
& ( ord_le3173885194822255862ist_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_322_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_set_fm,Z: set_set_fm] : ( Y = Z ) )
= ( ^ [A3: set_set_fm,B5: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ B5 @ A3 )
& ( ord_le1461404734466536732set_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_323_dual__order_Oeq__iff,axiom,
( ( ^ [Y: $o > set_fm,Z: $o > set_fm] : ( Y = Z ) )
= ( ^ [A3: $o > set_fm,B5: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ B5 @ A3 )
& ( ord_less_eq_o_set_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_324_dual__order_Oeq__iff,axiom,
( ( ^ [Y: $o > nat,Z: $o > nat] : ( Y = Z ) )
= ( ^ [A3: $o > nat,B5: $o > nat] :
( ( ord_less_eq_o_nat @ B5 @ A3 )
& ( ord_less_eq_o_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_325_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
= ( ^ [A3: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A3 )
& ( ord_less_eq_set_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_326_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_327_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_328_order__trans,axiom,
! [X: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm,Z2: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ X @ Y3 )
=> ( ( ord_le2055072402069232929ist_fm @ Y3 @ Z2 )
=> ( ord_le2055072402069232929ist_fm @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_329_order__trans,axiom,
! [X: set_list_fm,Y3: set_list_fm,Z2: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X @ Y3 )
=> ( ( ord_le3173885194822255862ist_fm @ Y3 @ Z2 )
=> ( ord_le3173885194822255862ist_fm @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_330_order__trans,axiom,
! [X: set_set_fm,Y3: set_set_fm,Z2: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X @ Y3 )
=> ( ( ord_le1461404734466536732set_fm @ Y3 @ Z2 )
=> ( ord_le1461404734466536732set_fm @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_331_order__trans,axiom,
! [X: $o > set_fm,Y3: $o > set_fm,Z2: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_o_set_fm @ Y3 @ Z2 )
=> ( ord_less_eq_o_set_fm @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_332_order__trans,axiom,
! [X: $o > nat,Y3: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ Z2 )
=> ( ord_less_eq_o_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_333_order__trans,axiom,
! [X: set_fm,Y3: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ Z2 )
=> ( ord_less_eq_set_fm @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_334_order__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_335_order_Otrans,axiom,
! [A2: set_Pr7058068377845519745ist_fm,B2: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A2 @ B2 )
=> ( ( ord_le2055072402069232929ist_fm @ B2 @ C )
=> ( ord_le2055072402069232929ist_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_336_order_Otrans,axiom,
! [A2: set_list_fm,B2: set_list_fm,C: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( ( ord_le3173885194822255862ist_fm @ B2 @ C )
=> ( ord_le3173885194822255862ist_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_337_order_Otrans,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( ( ord_le1461404734466536732set_fm @ B2 @ C )
=> ( ord_le1461404734466536732set_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_338_order_Otrans,axiom,
! [A2: $o > set_fm,B2: $o > set_fm,C: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_o_set_fm @ B2 @ C )
=> ( ord_less_eq_o_set_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_339_order_Otrans,axiom,
! [A2: $o > nat,B2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_340_order_Otrans,axiom,
! [A2: set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_341_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_342_order__antisym,axiom,
! [X: set_Pr7058068377845519745ist_fm,Y3: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ X @ Y3 )
=> ( ( ord_le2055072402069232929ist_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_343_order__antisym,axiom,
! [X: set_list_fm,Y3: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X @ Y3 )
=> ( ( ord_le3173885194822255862ist_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_344_order__antisym,axiom,
! [X: set_set_fm,Y3: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X @ Y3 )
=> ( ( ord_le1461404734466536732set_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_345_order__antisym,axiom,
! [X: $o > set_fm,Y3: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_o_set_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_346_order__antisym,axiom,
! [X: $o > nat,Y3: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_347_order__antisym,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_348_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_349_ord__le__eq__trans,axiom,
! [A2: set_Pr7058068377845519745ist_fm,B2: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le2055072402069232929ist_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_350_ord__le__eq__trans,axiom,
! [A2: set_list_fm,B2: set_list_fm,C: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3173885194822255862ist_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_351_ord__le__eq__trans,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1461404734466536732set_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_352_ord__le__eq__trans,axiom,
! [A2: $o > set_fm,B2: $o > set_fm,C: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_o_set_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_353_ord__le__eq__trans,axiom,
! [A2: $o > nat,B2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_354_ord__le__eq__trans,axiom,
! [A2: set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_355_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_356_ord__eq__le__trans,axiom,
! [A2: set_Pr7058068377845519745ist_fm,B2: set_Pr7058068377845519745ist_fm,C: set_Pr7058068377845519745ist_fm] :
( ( A2 = B2 )
=> ( ( ord_le2055072402069232929ist_fm @ B2 @ C )
=> ( ord_le2055072402069232929ist_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_357_ord__eq__le__trans,axiom,
! [A2: set_list_fm,B2: set_list_fm,C: set_list_fm] :
( ( A2 = B2 )
=> ( ( ord_le3173885194822255862ist_fm @ B2 @ C )
=> ( ord_le3173885194822255862ist_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_358_ord__eq__le__trans,axiom,
! [A2: set_set_fm,B2: set_set_fm,C: set_set_fm] :
( ( A2 = B2 )
=> ( ( ord_le1461404734466536732set_fm @ B2 @ C )
=> ( ord_le1461404734466536732set_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_359_ord__eq__le__trans,axiom,
! [A2: $o > set_fm,B2: $o > set_fm,C: $o > set_fm] :
( ( A2 = B2 )
=> ( ( ord_less_eq_o_set_fm @ B2 @ C )
=> ( ord_less_eq_o_set_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_360_ord__eq__le__trans,axiom,
! [A2: $o > nat,B2: $o > nat,C: $o > nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_361_ord__eq__le__trans,axiom,
! [A2: set_fm,B2: set_fm,C: set_fm] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_362_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_363_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Pr7058068377845519745ist_fm,Z: set_Pr7058068377845519745ist_fm] : ( Y = Z ) )
= ( ^ [X2: set_Pr7058068377845519745ist_fm,Y4: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ X2 @ Y4 )
& ( ord_le2055072402069232929ist_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_364_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_list_fm,Z: set_list_fm] : ( Y = Z ) )
= ( ^ [X2: set_list_fm,Y4: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ X2 @ Y4 )
& ( ord_le3173885194822255862ist_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_365_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_set_fm,Z: set_set_fm] : ( Y = Z ) )
= ( ^ [X2: set_set_fm,Y4: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ X2 @ Y4 )
& ( ord_le1461404734466536732set_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_366_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: $o > set_fm,Z: $o > set_fm] : ( Y = Z ) )
= ( ^ [X2: $o > set_fm,Y4: $o > set_fm] :
( ( ord_less_eq_o_set_fm @ X2 @ Y4 )
& ( ord_less_eq_o_set_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_367_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: $o > nat,Z: $o > nat] : ( Y = Z ) )
= ( ^ [X2: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y4 )
& ( ord_less_eq_o_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_368_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ( ord_less_eq_set_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_369_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_370_le__cases3,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_371_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_372_Collect__mono__iff,axiom,
! [P2: produc164195504107695125m_rule > $o,Q2: produc164195504107695125m_rule > $o] :
( ( ord_le5052251716133888043m_rule @ ( collec2069506698588392810m_rule @ P2 ) @ ( collec2069506698588392810m_rule @ Q2 ) )
= ( ! [X2: produc164195504107695125m_rule] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_373_Collect__mono__iff,axiom,
! [P2: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
( ( ord_le2055072402069232929ist_fm @ ( collec7231575776666803724ist_fm @ P2 ) @ ( collec7231575776666803724ist_fm @ Q2 ) )
= ( ! [X2: produc1996495991257130529ist_fm] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_374_Collect__mono__iff,axiom,
! [P2: list_fm > $o,Q2: list_fm > $o] :
( ( ord_le3173885194822255862ist_fm @ ( collect_list_fm @ P2 ) @ ( collect_list_fm @ Q2 ) )
= ( ! [X2: list_fm] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_375_Collect__mono__iff,axiom,
! [P2: set_fm > $o,Q2: set_fm > $o] :
( ( ord_le1461404734466536732set_fm @ ( collect_set_fm @ P2 ) @ ( collect_set_fm @ Q2 ) )
= ( ! [X2: set_fm] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_376_Collect__mono__iff,axiom,
! [P2: fm > $o,Q2: fm > $o] :
( ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) )
= ( ! [X2: fm] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_377_set__eq__subset,axiom,
( ( ^ [Y: set_Pr7058068377845519745ist_fm,Z: set_Pr7058068377845519745ist_fm] : ( Y = Z ) )
= ( ^ [A6: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A6 @ B6 )
& ( ord_le2055072402069232929ist_fm @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_378_set__eq__subset,axiom,
( ( ^ [Y: set_list_fm,Z: set_list_fm] : ( Y = Z ) )
= ( ^ [A6: set_list_fm,B6: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A6 @ B6 )
& ( ord_le3173885194822255862ist_fm @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_379_set__eq__subset,axiom,
( ( ^ [Y: set_set_fm,Z: set_set_fm] : ( Y = Z ) )
= ( ^ [A6: set_set_fm,B6: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A6 @ B6 )
& ( ord_le1461404734466536732set_fm @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_380_set__eq__subset,axiom,
( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
= ( ^ [A6: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B6 )
& ( ord_less_eq_set_fm @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_381_subset__trans,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C3: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ( ( ord_le2055072402069232929ist_fm @ B @ C3 )
=> ( ord_le2055072402069232929ist_fm @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_382_subset__trans,axiom,
! [A: set_list_fm,B: set_list_fm,C3: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ( ord_le3173885194822255862ist_fm @ B @ C3 )
=> ( ord_le3173885194822255862ist_fm @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_383_subset__trans,axiom,
! [A: set_set_fm,B: set_set_fm,C3: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ( ord_le1461404734466536732set_fm @ B @ C3 )
=> ( ord_le1461404734466536732set_fm @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_384_subset__trans,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C3 )
=> ( ord_less_eq_set_fm @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_385_Collect__mono,axiom,
! [P2: produc164195504107695125m_rule > $o,Q2: produc164195504107695125m_rule > $o] :
( ! [X3: produc164195504107695125m_rule] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le5052251716133888043m_rule @ ( collec2069506698588392810m_rule @ P2 ) @ ( collec2069506698588392810m_rule @ Q2 ) ) ) ).
% Collect_mono
thf(fact_386_Collect__mono,axiom,
! [P2: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
( ! [X3: produc1996495991257130529ist_fm] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le2055072402069232929ist_fm @ ( collec7231575776666803724ist_fm @ P2 ) @ ( collec7231575776666803724ist_fm @ Q2 ) ) ) ).
% Collect_mono
thf(fact_387_Collect__mono,axiom,
! [P2: list_fm > $o,Q2: list_fm > $o] :
( ! [X3: list_fm] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3173885194822255862ist_fm @ ( collect_list_fm @ P2 ) @ ( collect_list_fm @ Q2 ) ) ) ).
% Collect_mono
thf(fact_388_Collect__mono,axiom,
! [P2: set_fm > $o,Q2: set_fm > $o] :
( ! [X3: set_fm] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le1461404734466536732set_fm @ ( collect_set_fm @ P2 ) @ ( collect_set_fm @ Q2 ) ) ) ).
% Collect_mono
thf(fact_389_Collect__mono,axiom,
! [P2: fm > $o,Q2: fm > $o] :
( ! [X3: fm] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) ) ) ).
% Collect_mono
thf(fact_390_subset__refl,axiom,
! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ A ) ).
% subset_refl
thf(fact_391_subset__refl,axiom,
! [A: set_list_fm] : ( ord_le3173885194822255862ist_fm @ A @ A ) ).
% subset_refl
thf(fact_392_subset__refl,axiom,
! [A: set_set_fm] : ( ord_le1461404734466536732set_fm @ A @ A ) ).
% subset_refl
thf(fact_393_subset__refl,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).
% subset_refl
thf(fact_394_subset__iff,axiom,
( ord_le7532831136298237057_fm_fm
= ( ^ [A6: set_Pr4463079037648049377_fm_fm,B6: set_Pr4463079037648049377_fm_fm] :
! [T: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ T @ A6 )
=> ( member8474499337054950954_fm_fm @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_395_subset__iff,axiom,
( ord_le8219319623975992658set_fm
= ( ^ [A6: set_set_set_fm,B6: set_set_set_fm] :
! [T: set_set_fm] :
( ( member_set_set_fm2 @ T @ A6 )
=> ( member_set_set_fm2 @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_396_subset__iff,axiom,
( ord_le2055072402069232929ist_fm
= ( ^ [A6: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
! [T: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ T @ A6 )
=> ( member8102475879199740618ist_fm @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__iff,axiom,
( ord_le3173885194822255862ist_fm
= ( ^ [A6: set_list_fm,B6: set_list_fm] :
! [T: list_fm] :
( ( member_list_fm2 @ T @ A6 )
=> ( member_list_fm2 @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_398_subset__iff,axiom,
( ord_le1461404734466536732set_fm
= ( ^ [A6: set_set_fm,B6: set_set_fm] :
! [T: set_fm] :
( ( member_set_fm2 @ T @ A6 )
=> ( member_set_fm2 @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_399_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
! [T: fm] :
( ( member_fm2 @ T @ A6 )
=> ( member_fm2 @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_400_equalityD2,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( A = B )
=> ( ord_le2055072402069232929ist_fm @ B @ A ) ) ).
% equalityD2
thf(fact_401_equalityD2,axiom,
! [A: set_list_fm,B: set_list_fm] :
( ( A = B )
=> ( ord_le3173885194822255862ist_fm @ B @ A ) ) ).
% equalityD2
thf(fact_402_equalityD2,axiom,
! [A: set_set_fm,B: set_set_fm] :
( ( A = B )
=> ( ord_le1461404734466536732set_fm @ B @ A ) ) ).
% equalityD2
thf(fact_403_equalityD2,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ( ord_less_eq_set_fm @ B @ A ) ) ).
% equalityD2
thf(fact_404_equalityD1,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( A = B )
=> ( ord_le2055072402069232929ist_fm @ A @ B ) ) ).
% equalityD1
thf(fact_405_equalityD1,axiom,
! [A: set_list_fm,B: set_list_fm] :
( ( A = B )
=> ( ord_le3173885194822255862ist_fm @ A @ B ) ) ).
% equalityD1
thf(fact_406_equalityD1,axiom,
! [A: set_set_fm,B: set_set_fm] :
( ( A = B )
=> ( ord_le1461404734466536732set_fm @ A @ B ) ) ).
% equalityD1
thf(fact_407_equalityD1,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% equalityD1
thf(fact_408_subset__eq,axiom,
( ord_le7532831136298237057_fm_fm
= ( ^ [A6: set_Pr4463079037648049377_fm_fm,B6: set_Pr4463079037648049377_fm_fm] :
! [X2: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X2 @ A6 )
=> ( member8474499337054950954_fm_fm @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_409_subset__eq,axiom,
( ord_le8219319623975992658set_fm
= ( ^ [A6: set_set_set_fm,B6: set_set_set_fm] :
! [X2: set_set_fm] :
( ( member_set_set_fm2 @ X2 @ A6 )
=> ( member_set_set_fm2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_410_subset__eq,axiom,
( ord_le2055072402069232929ist_fm
= ( ^ [A6: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
! [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A6 )
=> ( member8102475879199740618ist_fm @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_411_subset__eq,axiom,
( ord_le3173885194822255862ist_fm
= ( ^ [A6: set_list_fm,B6: set_list_fm] :
! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A6 )
=> ( member_list_fm2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_412_subset__eq,axiom,
( ord_le1461404734466536732set_fm
= ( ^ [A6: set_set_fm,B6: set_set_fm] :
! [X2: set_fm] :
( ( member_set_fm2 @ X2 @ A6 )
=> ( member_set_fm2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_413_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
! [X2: fm] :
( ( member_fm2 @ X2 @ A6 )
=> ( member_fm2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_414_equalityE,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
( ( A = B )
=> ~ ( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ~ ( ord_le2055072402069232929ist_fm @ B @ A ) ) ) ).
% equalityE
thf(fact_415_equalityE,axiom,
! [A: set_list_fm,B: set_list_fm] :
( ( A = B )
=> ~ ( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ~ ( ord_le3173885194822255862ist_fm @ B @ A ) ) ) ).
% equalityE
thf(fact_416_equalityE,axiom,
! [A: set_set_fm,B: set_set_fm] :
( ( A = B )
=> ~ ( ( ord_le1461404734466536732set_fm @ A @ B )
=> ~ ( ord_le1461404734466536732set_fm @ B @ A ) ) ) ).
% equalityE
thf(fact_417_equalityE,axiom,
! [A: set_fm,B: set_fm] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_fm @ A @ B )
=> ~ ( ord_less_eq_set_fm @ B @ A ) ) ) ).
% equalityE
thf(fact_418_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: produc164195504107695125m_rule > produc164195504107695125m_rule > $o,X: produc164195504107695125m_rule,Y3: produc164195504107695125m_rule,A2: produc3793571666044074251m_rule] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( produc5927390650430071747m_rule @ X @ Y3 ) )
=> ( P2 @ ( produc4837027324325561263m_rule @ A2 ) @ ( produc5389215402712513009m_rule @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_419_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: fm > fm > $o,X: fm,Y3: fm,A2: product_prod_fm_fm] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( product_Pair_fm_fm @ X @ Y3 ) )
=> ( P2 @ ( product_fst_fm_fm @ A2 ) @ ( product_snd_fm_fm @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_420_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: fm > produc7822875418678951345atural > $o,X: fm,Y3: produc7822875418678951345atural,A2: produc7777387212919034070atural] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( produc912813528046151816atural @ X @ Y3 ) )
=> ( P2 @ ( produc7697328821926353820atural @ A2 ) @ ( produc7769300925896546778atural @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_421_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: rule > produc1996495991257130529ist_fm > $o,X: rule,Y3: produc1996495991257130529ist_fm,A2: produc4630866025709511057ist_fm] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( produc7063755442767259331ist_fm @ X @ Y3 ) )
=> ( P2 @ ( produc4228417144610596823ist_fm @ A2 ) @ ( produc4514744107061996309ist_fm @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_422_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: produc1996495991257130529ist_fm > rule > $o,X: produc1996495991257130529ist_fm,Y3: rule,A2: produc164195504107695125m_rule] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( produc491467635432902671m_rule @ X @ Y3 ) )
=> ( P2 @ ( produc6879501374131015971m_rule @ A2 ) @ ( produc7165828336582415457m_rule @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_423_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: list_fm > list_fm > $o,X: list_fm,Y3: list_fm,A2: produc1996495991257130529ist_fm] :
( ( P2 @ X @ Y3 )
=> ( ( A2
= ( produc381145313068854617ist_fm @ X @ Y3 ) )
=> ( P2 @ ( produc1501393135466168645ist_fm @ A2 ) @ ( produc4588648349897876871ist_fm @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_424_eq__snd__iff,axiom,
! [B2: produc164195504107695125m_rule,P: produc3793571666044074251m_rule] :
( ( B2
= ( produc5389215402712513009m_rule @ P ) )
= ( ? [A3: produc164195504107695125m_rule] :
( P
= ( produc5927390650430071747m_rule @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_425_eq__snd__iff,axiom,
! [B2: fm,P: product_prod_fm_fm] :
( ( B2
= ( product_snd_fm_fm @ P ) )
= ( ? [A3: fm] :
( P
= ( product_Pair_fm_fm @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_426_eq__snd__iff,axiom,
! [B2: rule,P: produc164195504107695125m_rule] :
( ( B2
= ( produc7165828336582415457m_rule @ P ) )
= ( ? [A3: produc1996495991257130529ist_fm] :
( P
= ( produc491467635432902671m_rule @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_427_eq__snd__iff,axiom,
! [B2: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
( ( B2
= ( produc4514744107061996309ist_fm @ P ) )
= ( ? [A3: rule] :
( P
= ( produc7063755442767259331ist_fm @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_428_eq__snd__iff,axiom,
! [B2: list_fm,P: produc1996495991257130529ist_fm] :
( ( B2
= ( produc4588648349897876871ist_fm @ P ) )
= ( ? [A3: list_fm] :
( P
= ( produc381145313068854617ist_fm @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_429_sndI,axiom,
! [X: produc3793571666044074251m_rule,Y3: produc164195504107695125m_rule,Z2: produc164195504107695125m_rule] :
( ( X
= ( produc5927390650430071747m_rule @ Y3 @ Z2 ) )
=> ( ( produc5389215402712513009m_rule @ X )
= Z2 ) ) ).
% sndI
thf(fact_430_sndI,axiom,
! [X: product_prod_fm_fm,Y3: fm,Z2: fm] :
( ( X
= ( product_Pair_fm_fm @ Y3 @ Z2 ) )
=> ( ( product_snd_fm_fm @ X )
= Z2 ) ) ).
% sndI
thf(fact_431_sndI,axiom,
! [X: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
( ( X
= ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
=> ( ( produc7165828336582415457m_rule @ X )
= Z2 ) ) ).
% sndI
thf(fact_432_sndI,axiom,
! [X: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
( ( X
= ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
=> ( ( produc4514744107061996309ist_fm @ X )
= Z2 ) ) ).
% sndI
thf(fact_433_sndI,axiom,
! [X: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
( ( X
= ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
=> ( ( produc4588648349897876871ist_fm @ X )
= Z2 ) ) ).
% sndI
thf(fact_434_fstI,axiom,
! [X: produc3793571666044074251m_rule,Y3: produc164195504107695125m_rule,Z2: produc164195504107695125m_rule] :
( ( X
= ( produc5927390650430071747m_rule @ Y3 @ Z2 ) )
=> ( ( produc4837027324325561263m_rule @ X )
= Y3 ) ) ).
% fstI
thf(fact_435_fstI,axiom,
! [X: produc4630866025709511057ist_fm,Y3: rule,Z2: produc1996495991257130529ist_fm] :
( ( X
= ( produc7063755442767259331ist_fm @ Y3 @ Z2 ) )
=> ( ( produc4228417144610596823ist_fm @ X )
= Y3 ) ) ).
% fstI
thf(fact_436_fstI,axiom,
! [X: product_prod_fm_fm,Y3: fm,Z2: fm] :
( ( X
= ( product_Pair_fm_fm @ Y3 @ Z2 ) )
=> ( ( product_fst_fm_fm @ X )
= Y3 ) ) ).
% fstI
thf(fact_437_fstI,axiom,
! [X: produc7777387212919034070atural,Y3: fm,Z2: produc7822875418678951345atural] :
( ( X
= ( produc912813528046151816atural @ Y3 @ Z2 ) )
=> ( ( produc7697328821926353820atural @ X )
= Y3 ) ) ).
% fstI
thf(fact_438_fstI,axiom,
! [X: produc1996495991257130529ist_fm,Y3: list_fm,Z2: list_fm] :
( ( X
= ( produc381145313068854617ist_fm @ Y3 @ Z2 ) )
=> ( ( produc1501393135466168645ist_fm @ X )
= Y3 ) ) ).
% fstI
thf(fact_439_fstI,axiom,
! [X: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z2: rule] :
( ( X
= ( produc491467635432902671m_rule @ Y3 @ Z2 ) )
=> ( ( produc6879501374131015971m_rule @ X )
= Y3 ) ) ).
% fstI
thf(fact_440_eq__fst__iff,axiom,
! [A2: produc164195504107695125m_rule,P: produc3793571666044074251m_rule] :
( ( A2
= ( produc4837027324325561263m_rule @ P ) )
= ( ? [B5: produc164195504107695125m_rule] :
( P
= ( produc5927390650430071747m_rule @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_441_eq__fst__iff,axiom,
! [A2: rule,P: produc4630866025709511057ist_fm] :
( ( A2
= ( produc4228417144610596823ist_fm @ P ) )
= ( ? [B5: produc1996495991257130529ist_fm] :
( P
= ( produc7063755442767259331ist_fm @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_442_eq__fst__iff,axiom,
! [A2: fm,P: product_prod_fm_fm] :
( ( A2
= ( product_fst_fm_fm @ P ) )
= ( ? [B5: fm] :
( P
= ( product_Pair_fm_fm @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_443_eq__fst__iff,axiom,
! [A2: fm,P: produc7777387212919034070atural] :
( ( A2
= ( produc7697328821926353820atural @ P ) )
= ( ? [B5: produc7822875418678951345atural] :
( P
= ( produc912813528046151816atural @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_444_eq__fst__iff,axiom,
! [A2: list_fm,P: produc1996495991257130529ist_fm] :
( ( A2
= ( produc1501393135466168645ist_fm @ P ) )
= ( ? [B5: list_fm] :
( P
= ( produc381145313068854617ist_fm @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_445_eq__fst__iff,axiom,
! [A2: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
( ( A2
= ( produc6879501374131015971m_rule @ P ) )
= ( ? [B5: rule] :
( P
= ( produc491467635432902671m_rule @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_446_prod_Oswap__def,axiom,
( produc1276282870026899811m_rule
= ( ^ [P3: produc3793571666044074251m_rule] : ( produc5927390650430071747m_rule @ ( produc5389215402712513009m_rule @ P3 ) @ ( produc4837027324325561263m_rule @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_447_prod_Oswap__def,axiom,
( product_swap_fm_fm
= ( ^ [P3: product_prod_fm_fm] : ( product_Pair_fm_fm @ ( product_snd_fm_fm @ P3 ) @ ( product_fst_fm_fm @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_448_prod_Oswap__def,axiom,
( produc4876742319169725160atural
= ( ^ [P3: produc7777387212919034070atural] : ( produc8796667856227859626ral_fm @ ( produc7769300925896546778atural @ P3 ) @ ( produc7697328821926353820atural @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_449_prod_Oswap__def,axiom,
( produc9213896635423732003ist_fm
= ( ^ [P3: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P3 ) @ ( produc4228417144610596823ist_fm @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_450_prod_Oswap__def,axiom,
( produc2641608828089375343m_rule
= ( ^ [P3: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P3 ) @ ( produc6879501374131015971m_rule @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_451_prod_Oswap__def,axiom,
( produc7936585717479832313ist_fm
= ( ^ [P3: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P3 ) @ ( produc1501393135466168645ist_fm @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_452_snd__swap,axiom,
! [X: produc7777387212919034070atural] :
( ( produc6429783217223478780ral_fm @ ( produc4876742319169725160atural @ X ) )
= ( produc7697328821926353820atural @ X ) ) ).
% snd_swap
thf(fact_453_snd__swap,axiom,
! [X: produc4630866025709511057ist_fm] :
( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X ) )
= ( produc4228417144610596823ist_fm @ X ) ) ).
% snd_swap
thf(fact_454_snd__swap,axiom,
! [X: produc164195504107695125m_rule] :
( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X ) )
= ( produc6879501374131015971m_rule @ X ) ) ).
% snd_swap
thf(fact_455_snd__swap,axiom,
! [X: produc1996495991257130529ist_fm] :
( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X ) )
= ( produc1501393135466168645ist_fm @ X ) ) ).
% snd_swap
thf(fact_456_fst__swap,axiom,
! [X: produc3094967117286489776ral_fm] :
( ( produc7697328821926353820atural @ ( produc3537224610496657162ral_fm @ X ) )
= ( produc6429783217223478780ral_fm @ X ) ) ).
% fst_swap
thf(fact_457_fst__swap,axiom,
! [X: produc164195504107695125m_rule] :
( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X ) )
= ( produc7165828336582415457m_rule @ X ) ) ).
% fst_swap
thf(fact_458_fst__swap,axiom,
! [X: produc4630866025709511057ist_fm] :
( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X ) )
= ( produc4514744107061996309ist_fm @ X ) ) ).
% fst_swap
thf(fact_459_fst__swap,axiom,
! [X: produc1996495991257130529ist_fm] :
( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X ) )
= ( produc4588648349897876871ist_fm @ X ) ) ).
% fst_swap
thf(fact_460_relChain__def,axiom,
( bNF_Ca7336959984753315782fm_nat
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > nat] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_461_relChain__def,axiom,
( bNF_Ca4343761633051459196set_fm
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > set_fm] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_less_eq_set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_462_relChain__def,axiom,
( bNF_Ca5108556014793540054fm_nat
= ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > nat] :
! [I: list_fm,J: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
=> ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_463_relChain__def,axiom,
( bNF_Ca2777790331493722252set_fm
= ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > set_fm] :
! [I: list_fm,J: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
=> ( ord_less_eq_set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_464_relChain__def,axiom,
( bNF_Ca2946272924589226636ist_fm
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > set_list_fm] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_le3173885194822255862ist_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_465_relChain__def,axiom,
( bNF_Ca8600108726579631538set_fm
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > set_set_fm] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_le1461404734466536732set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_466_relChain__def,axiom,
( bNF_Ca8034671686211279459_o_nat
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > $o > nat] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_less_eq_o_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_467_relChain__def,axiom,
( bNF_Ca4506260793539140764ist_fm
= ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > set_list_fm] :
! [I: list_fm,J: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
=> ( ord_le3173885194822255862ist_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_468_relChain__def,axiom,
( bNF_Ca4478743228358967234set_fm
= ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > set_set_fm] :
! [I: list_fm,J: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
=> ( ord_le1461404734466536732set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_469_relChain__def,axiom,
( bNF_Ca5393516298241387801set_fm
= ( ^ [R: set_Pr4463079037648049377_fm_fm,As: fm > $o > set_fm] :
! [I: fm,J: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ I @ J ) @ R )
=> ( ord_less_eq_o_set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_470_swap__swap,axiom,
! [P: produc1996495991257130529ist_fm] :
( ( produc7936585717479832313ist_fm @ ( produc7936585717479832313ist_fm @ P ) )
= P ) ).
% swap_swap
thf(fact_471_swap__swap,axiom,
! [P: produc4630866025709511057ist_fm] :
( ( produc2641608828089375343m_rule @ ( produc9213896635423732003ist_fm @ P ) )
= P ) ).
% swap_swap
thf(fact_472_swap__swap,axiom,
! [P: produc164195504107695125m_rule] :
( ( produc9213896635423732003ist_fm @ ( produc2641608828089375343m_rule @ P ) )
= P ) ).
% swap_swap
thf(fact_473_swap__simp,axiom,
! [X: produc164195504107695125m_rule,Y3: produc164195504107695125m_rule] :
( ( produc1276282870026899811m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) )
= ( produc5927390650430071747m_rule @ Y3 @ X ) ) ).
% swap_simp
thf(fact_474_swap__simp,axiom,
! [X: fm,Y3: fm] :
( ( product_swap_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) )
= ( product_Pair_fm_fm @ Y3 @ X ) ) ).
% swap_simp
thf(fact_475_swap__simp,axiom,
! [X: list_fm,Y3: list_fm] :
( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) )
= ( produc381145313068854617ist_fm @ Y3 @ X ) ) ).
% swap_simp
thf(fact_476_swap__simp,axiom,
! [X: produc1996495991257130529ist_fm,Y3: rule] :
( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) )
= ( produc7063755442767259331ist_fm @ Y3 @ X ) ) ).
% swap_simp
thf(fact_477_swap__simp,axiom,
! [X: rule,Y3: produc1996495991257130529ist_fm] :
( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) )
= ( produc491467635432902671m_rule @ Y3 @ X ) ) ).
% swap_simp
thf(fact_478_fstOp__def,axiom,
( bNF_fs7376498899940486621atural
= ( ^ [P4: fm > fm > $o,Q3: fm > produc7822875418678951345atural > $o,Ac: produc7777387212919034070atural] : ( product_Pair_fm_fm @ ( produc7697328821926353820atural @ Ac ) @ ( bNF_pi3939735538430251307atural @ P4 @ Q3 @ ( produc7697328821926353820atural @ Ac ) @ ( produc7769300925896546778atural @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_479_fstOp__def,axiom,
( bNF_fs6361025107161210196ist_fm
= ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Ac ) @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_480_fstOp__def,axiom,
( bNF_fs2143400664727635963e_rule
= ( ^ [P4: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Ac ) @ ( bNF_pi580457772512444745e_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_481_fstOp__def,axiom,
( bNF_fs3887825027587321911ist_fm
= ( ^ [P4: rule > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Ac: produc4630866025709511057ist_fm] : ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Ac ) @ ( bNF_pi3835482382339038057ist_fm @ P4 @ Q3 @ ( produc4228417144610596823ist_fm @ Ac ) @ ( produc4514744107061996309ist_fm @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_482_sndOp__def,axiom,
( bNF_sn5292590578203848342ist_fm
= ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_483_sndOp__def,axiom,
( bNF_sn6617447237779055481m_rule
= ( ^ [P4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( bNF_pi3438825987790746729m_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_484_sndOp__def,axiom,
( bNF_sn1604800272185294753ist_fm
= ( ^ [P4: rule > rule > $o,Q3: rule > produc1996495991257130529ist_fm > $o,Ac: produc4630866025709511057ist_fm] : ( produc7063755442767259331ist_fm @ ( bNF_pi4963049444341131953ist_fm @ P4 @ Q3 @ ( produc4228417144610596823ist_fm @ Ac ) @ ( produc4514744107061996309ist_fm @ Ac ) ) @ ( produc4514744107061996309ist_fm @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_485_subset__code_I3_J,axiom,
~ ( ord_le3684904531561281547m_rule @ ( coset_5243742680598159614m_rule @ nil_Pr3052741188495898641m_rule ) @ ( set_Pr2379943022980910880m_rule @ nil_Pr3052741188495898641m_rule ) ) ).
% subset_code(3)
thf(fact_486_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_rule @ ( coset_rule @ nil_rule ) @ ( set_rule2 @ nil_rule ) ) ).
% subset_code(3)
thf(fact_487_subset__code_I3_J,axiom,
~ ( ord_le5052251716133888043m_rule @ ( coset_772924026395289186m_rule @ nil_Pr474642538098133391m_rule ) @ ( set_Pr8505323785428441536m_rule @ nil_Pr474642538098133391m_rule ) ) ).
% subset_code(3)
thf(fact_488_subset__code_I3_J,axiom,
~ ( ord_le2055072402069232929ist_fm @ ( coset_820689808388072212ist_fm @ nil_Pr6600767949821390631ist_fm ) @ ( set_Pr8767716839810916150ist_fm @ nil_Pr6600767949821390631ist_fm ) ) ).
% subset_code(3)
thf(fact_489_subset__code_I3_J,axiom,
~ ( ord_le3173885194822255862ist_fm @ ( coset_list_fm @ nil_list_fm ) @ ( set_list_fm2 @ nil_list_fm ) ) ).
% subset_code(3)
thf(fact_490_subset__code_I3_J,axiom,
~ ( ord_le1461404734466536732set_fm @ ( coset_set_fm @ nil_set_fm ) @ ( set_set_fm2 @ nil_set_fm ) ) ).
% subset_code(3)
thf(fact_491_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_492_image2__eqI,axiom,
! [B2: fm,F: fm > fm,X: fm,C: fm,G2: fm > fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B2 @ C ) @ ( bNF_Gr1203782223388445388_fm_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_493_image2__eqI,axiom,
! [B2: fm,F: list_fm > fm,X: list_fm,C: fm,G2: list_fm > fm,A: set_list_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_list_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B2 @ C ) @ ( bNF_Gr8523389835497889372_fm_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_494_image2__eqI,axiom,
! [B2: fm,F: set_fm > fm,X: set_fm,C: fm,G2: set_fm > fm,A: set_set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_set_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B2 @ C ) @ ( bNF_Gr2527083199176774402_fm_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_495_image2__eqI,axiom,
! [B2: list_fm,F: fm > list_fm,X: fm,C: list_fm,G2: fm > list_fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B2 @ C ) @ ( bNF_Gr2680958839322950764ist_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_496_image2__eqI,axiom,
! [B2: fm,F: product_prod_fm_fm > fm,X: product_prod_fm_fm,C: fm,G2: product_prod_fm_fm > fm,A: set_Pr4463079037648049377_fm_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member8474499337054950954_fm_fm @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B2 @ C ) @ ( bNF_Gr2224021619804040521_fm_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_497_image2__eqI,axiom,
! [B2: fm,F: set_set_fm > fm,X: set_set_fm,C: fm,G2: set_set_fm > fm,A: set_set_set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_set_set_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B2 @ C ) @ ( bNF_Gr2844036924399170616_fm_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_498_image2__eqI,axiom,
! [B2: list_fm,F: list_fm > list_fm,X: list_fm,C: list_fm,G2: list_fm > list_fm,A: set_list_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_list_fm2 @ X @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B2 @ C ) @ ( bNF_Gr1661195949116424188ist_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_499_image2__eqI,axiom,
! [B2: list_fm,F: set_fm > list_fm,X: set_fm,C: list_fm,G2: set_fm > list_fm,A: set_set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_set_fm2 @ X @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B2 @ C ) @ ( bNF_Gr6790864367800852130ist_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_500_image2__eqI,axiom,
! [B2: produc1996495991257130529ist_fm,F: fm > produc1996495991257130529ist_fm,X: fm,C: rule,G2: fm > rule,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ B2 @ C ) @ ( bNF_Gr4411798878337052732m_rule @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_501_image2__eqI,axiom,
! [B2: list_fm,F: product_prod_fm_fm > list_fm,X: product_prod_fm_fm,C: list_fm,G2: product_prod_fm_fm > list_fm,A: set_Pr4463079037648049377_fm_fm] :
( ( B2
= ( F @ X ) )
=> ( ( C
= ( G2 @ X ) )
=> ( ( member8474499337054950954_fm_fm @ X @ A )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B2 @ C ) @ ( bNF_Gr869466243678018025ist_fm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_502_GreatestI2__order,axiom,
! [P2: set_Pr7058068377845519745ist_fm > $o,X: set_Pr7058068377845519745ist_fm,Q2: set_Pr7058068377845519745ist_fm > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_Pr7058068377845519745ist_fm] :
( ( P2 @ Y2 )
=> ( ord_le2055072402069232929ist_fm @ Y2 @ X ) )
=> ( ! [X3: set_Pr7058068377845519745ist_fm] :
( ( P2 @ X3 )
=> ( ! [Y5: set_Pr7058068377845519745ist_fm] :
( ( P2 @ Y5 )
=> ( ord_le2055072402069232929ist_fm @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_3259246943450084570ist_fm @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_503_GreatestI2__order,axiom,
! [P2: set_list_fm > $o,X: set_list_fm,Q2: set_list_fm > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_list_fm] :
( ( P2 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ Y2 @ X ) )
=> ( ! [X3: set_list_fm] :
( ( P2 @ X3 )
=> ( ! [Y5: set_list_fm] :
( ( P2 @ Y5 )
=> ( ord_le3173885194822255862ist_fm @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_8607021373033991037ist_fm @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_504_GreatestI2__order,axiom,
! [P2: set_set_fm > $o,X: set_set_fm,Q2: set_set_fm > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_set_fm] :
( ( P2 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ Y2 @ X ) )
=> ( ! [X3: set_set_fm] :
( ( P2 @ X3 )
=> ( ! [Y5: set_set_fm] :
( ( P2 @ Y5 )
=> ( ord_le1461404734466536732set_fm @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_7720725141959773987set_fm @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_505_GreatestI2__order,axiom,
! [P2: ( $o > set_fm ) > $o,X: $o > set_fm,Q2: ( $o > set_fm ) > $o] :
( ( P2 @ X )
=> ( ! [Y2: $o > set_fm] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_set_fm @ Y2 @ X ) )
=> ( ! [X3: $o > set_fm] :
( ( P2 @ X3 )
=> ( ! [Y5: $o > set_fm] :
( ( P2 @ Y5 )
=> ( ord_less_eq_o_set_fm @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_7145282227375980264set_fm @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_506_GreatestI2__order,axiom,
! [P2: ( $o > nat ) > $o,X: $o > nat,Q2: ( $o > nat ) > $o] :
( ( P2 @ X )
=> ( ! [Y2: $o > nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X ) )
=> ( ! [X3: $o > nat] :
( ( P2 @ X3 )
=> ( ! [Y5: $o > nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_o_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_507_GreatestI2__order,axiom,
! [P2: set_fm > $o,X: set_fm,Q2: set_fm > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_fm] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_fm @ Y2 @ X ) )
=> ( ! [X3: set_fm] :
( ( P2 @ X3 )
=> ( ! [Y5: set_fm] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_fm @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_6179083242224974829set_fm @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_508_GreatestI2__order,axiom,
! [P2: nat > $o,X: nat,Q2: nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_509_Greatest__equality,axiom,
! [P2: set_Pr7058068377845519745ist_fm > $o,X: set_Pr7058068377845519745ist_fm] :
( ( P2 @ X )
=> ( ! [Y2: set_Pr7058068377845519745ist_fm] :
( ( P2 @ Y2 )
=> ( ord_le2055072402069232929ist_fm @ Y2 @ X ) )
=> ( ( order_3259246943450084570ist_fm @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_510_Greatest__equality,axiom,
! [P2: set_list_fm > $o,X: set_list_fm] :
( ( P2 @ X )
=> ( ! [Y2: set_list_fm] :
( ( P2 @ Y2 )
=> ( ord_le3173885194822255862ist_fm @ Y2 @ X ) )
=> ( ( order_8607021373033991037ist_fm @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_511_Greatest__equality,axiom,
! [P2: set_set_fm > $o,X: set_set_fm] :
( ( P2 @ X )
=> ( ! [Y2: set_set_fm] :
( ( P2 @ Y2 )
=> ( ord_le1461404734466536732set_fm @ Y2 @ X ) )
=> ( ( order_7720725141959773987set_fm @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_512_Greatest__equality,axiom,
! [P2: ( $o > set_fm ) > $o,X: $o > set_fm] :
( ( P2 @ X )
=> ( ! [Y2: $o > set_fm] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_set_fm @ Y2 @ X ) )
=> ( ( order_7145282227375980264set_fm @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_513_Greatest__equality,axiom,
! [P2: ( $o > nat ) > $o,X: $o > nat] :
( ( P2 @ X )
=> ( ! [Y2: $o > nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X ) )
=> ( ( order_Greatest_o_nat @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_514_Greatest__equality,axiom,
! [P2: set_fm > $o,X: set_fm] :
( ( P2 @ X )
=> ( ! [Y2: set_fm] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_fm @ Y2 @ X ) )
=> ( ( order_6179083242224974829set_fm @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_515_Greatest__equality,axiom,
! [P2: nat > $o,X: nat] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_516_ssubst__Pair__rhs,axiom,
! [R2: produc1996495991257130529ist_fm,S: rule,R3: set_Pr1008144964186165195m_rule,S4: rule] :
( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_517_ssubst__Pair__rhs,axiom,
! [R2: produc164195504107695125m_rule,S: produc164195504107695125m_rule,R3: set_Pr5090152306159784043m_rule,S4: produc164195504107695125m_rule] :
( ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ R2 @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ R2 @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_518_ssubst__Pair__rhs,axiom,
! [R2: list_fm,S: list_fm,R3: set_Pr7058068377845519745ist_fm,S4: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_519_ssubst__Pair__rhs,axiom,
! [R2: rule,S: produc1996495991257130529ist_fm,R3: set_Pr2241365836748572487ist_fm,S4: produc1996495991257130529ist_fm] :
( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_520_ssubst__Pair__rhs,axiom,
! [R2: fm,S: fm,R3: set_Pr4463079037648049377_fm_fm,S4: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ R2 @ S ) @ R3 )
=> ( ( S4 = S )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ R2 @ S4 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_521_subrelI,axiom,
! [R2: set_Pr1008144964186165195m_rule,S: set_Pr1008144964186165195m_rule] :
( ! [X3: produc1996495991257130529ist_fm,Y2: rule] :
( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y2 ) @ R2 )
=> ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y2 ) @ S ) )
=> ( ord_le5052251716133888043m_rule @ R2 @ S ) ) ).
% subrelI
thf(fact_522_subrelI,axiom,
! [R2: set_Pr5090152306159784043m_rule,S: set_Pr5090152306159784043m_rule] :
( ! [X3: produc164195504107695125m_rule,Y2: produc164195504107695125m_rule] :
( ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ X3 @ Y2 ) @ R2 )
=> ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ X3 @ Y2 ) @ S ) )
=> ( ord_le3684904531561281547m_rule @ R2 @ S ) ) ).
% subrelI
thf(fact_523_subrelI,axiom,
! [R2: set_Pr2241365836748572487ist_fm,S: set_Pr2241365836748572487ist_fm] :
( ! [X3: rule,Y2: produc1996495991257130529ist_fm] :
( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y2 ) @ R2 )
=> ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y2 ) @ S ) )
=> ( ord_le6285472588696295335ist_fm @ R2 @ S ) ) ).
% subrelI
thf(fact_524_subrelI,axiom,
! [R2: set_Pr4463079037648049377_fm_fm,S: set_Pr4463079037648049377_fm_fm] :
( ! [X3: fm,Y2: fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y2 ) @ R2 )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y2 ) @ S ) )
=> ( ord_le7532831136298237057_fm_fm @ R2 @ S ) ) ).
% subrelI
thf(fact_525_subrelI,axiom,
! [R2: set_Pr7058068377845519745ist_fm,S: set_Pr7058068377845519745ist_fm] :
( ! [X3: list_fm,Y2: list_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y2 ) @ R2 )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y2 ) @ S ) )
=> ( ord_le2055072402069232929ist_fm @ R2 @ S ) ) ).
% subrelI
thf(fact_526_prod_Osplit__sel,axiom,
! [P2: $o > $o,F: list_fm > list_fm > $o,Prod: produc1996495991257130529ist_fm] :
( ( P2 @ ( produc2011568048573023402t_fm_o @ F @ Prod ) )
= ( ( Prod
= ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
=> ( P2 @ ( F @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_527_prod_Osplit__sel,axiom,
! [P2: $o > $o,F: produc1996495991257130529ist_fm > rule > $o,Prod: produc164195504107695125m_rule] :
( ( P2 @ ( produc7369492558480804444rule_o @ F @ Prod ) )
= ( ( Prod
= ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) )
=> ( P2 @ ( F @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_528_curryI,axiom,
! [F: produc164195504107695125m_rule > $o,A2: produc1996495991257130529ist_fm,B2: rule] :
( ( F @ ( produc491467635432902671m_rule @ A2 @ B2 ) )
=> ( produc7868929780990704507rule_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_529_curryI,axiom,
! [F: produc3793571666044074251m_rule > $o,A2: produc164195504107695125m_rule,B2: produc164195504107695125m_rule] :
( ( F @ ( produc5927390650430071747m_rule @ A2 @ B2 ) )
=> ( produc1995819562154721121rule_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_530_curryI,axiom,
! [F: produc1996495991257130529ist_fm > $o,A2: list_fm,B2: list_fm] :
( ( F @ ( produc381145313068854617ist_fm @ A2 @ B2 ) )
=> ( produc4804894507120027467t_fm_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_531_curryI,axiom,
! [F: produc4630866025709511057ist_fm > $o,A2: rule,B2: produc1996495991257130529ist_fm] :
( ( F @ ( produc7063755442767259331ist_fm @ A2 @ B2 ) )
=> ( produc2127844983229472711t_fm_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_532_curryI,axiom,
! [F: product_prod_fm_fm > $o,A2: fm,B2: fm] :
( ( F @ ( product_Pair_fm_fm @ A2 @ B2 ) )
=> ( produc5605581514536698091m_fm_o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_533_case__prod__curry,axiom,
! [F: produc1996495991257130529ist_fm > $o] :
( ( produc2011568048573023402t_fm_o @ ( produc4804894507120027467t_fm_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_534_case__prod__curry,axiom,
! [F: produc164195504107695125m_rule > $o] :
( ( produc7369492558480804444rule_o @ ( produc7868929780990704507rule_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_535_curry__case__prod,axiom,
! [F: list_fm > list_fm > $o] :
( ( produc4804894507120027467t_fm_o @ ( produc2011568048573023402t_fm_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_536_curry__case__prod,axiom,
! [F: produc1996495991257130529ist_fm > rule > $o] :
( ( produc7868929780990704507rule_o @ ( produc7369492558480804444rule_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_537_Collect__case__prod__mono,axiom,
! [A: list_fm > list_fm > $o,B: list_fm > list_fm > $o] :
( ( ord_le1774903256965159270t_fm_o @ A @ B )
=> ( ord_le2055072402069232929ist_fm @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ A ) ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ B ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_538_Collect__case__prod__mono,axiom,
! [A: produc1996495991257130529ist_fm > rule > $o,B: produc1996495991257130529ist_fm > rule > $o] :
( ( ord_le4891813983447887534rule_o @ A @ B )
=> ( ord_le5052251716133888043m_rule @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ A ) ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ B ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_539_mem__case__prodE,axiom,
! [Z2: fm,C: fm > fm > set_fm,P: product_prod_fm_fm] :
( ( member_fm2 @ Z2 @ ( produc9176972916100549972set_fm @ C @ P ) )
=> ~ ! [X3: fm,Y2: fm] :
( ( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) )
=> ~ ( member_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_540_mem__case__prodE,axiom,
! [Z2: list_fm,C: fm > fm > set_list_fm,P: product_prod_fm_fm] :
( ( member_list_fm2 @ Z2 @ ( produc4770878748755606884ist_fm @ C @ P ) )
=> ~ ! [X3: fm,Y2: fm] :
( ( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) )
=> ~ ( member_list_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_541_mem__case__prodE,axiom,
! [Z2: set_fm,C: fm > fm > set_set_fm,P: product_prod_fm_fm] :
( ( member_set_fm2 @ Z2 @ ( produc74870073573622922set_fm @ C @ P ) )
=> ~ ! [X3: fm,Y2: fm] :
( ( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) )
=> ~ ( member_set_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_542_mem__case__prodE,axiom,
! [Z2: fm,C: list_fm > list_fm > set_fm,P: produc1996495991257130529ist_fm] :
( ( member_fm2 @ Z2 @ ( produc4802205653939627508set_fm @ C @ P ) )
=> ~ ! [X3: list_fm,Y2: list_fm] :
( ( P
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) )
=> ~ ( member_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_543_mem__case__prodE,axiom,
! [Z2: product_prod_fm_fm,C: fm > fm > set_Pr4463079037648049377_fm_fm,P: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ Z2 @ ( produc6401441230987168403_fm_fm @ C @ P ) )
=> ~ ! [X3: fm,Y2: fm] :
( ( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) )
=> ~ ( member8474499337054950954_fm_fm @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_544_mem__case__prodE,axiom,
! [Z2: set_set_fm,C: fm > fm > set_set_set_fm,P: product_prod_fm_fm] :
( ( member_set_set_fm2 @ Z2 @ ( produc3181786390841906368set_fm @ C @ P ) )
=> ~ ! [X3: fm,Y2: fm] :
( ( P
= ( product_Pair_fm_fm @ X3 @ Y2 ) )
=> ~ ( member_set_set_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_545_mem__case__prodE,axiom,
! [Z2: list_fm,C: list_fm > list_fm > set_list_fm,P: produc1996495991257130529ist_fm] :
( ( member_list_fm2 @ Z2 @ ( produc492036894160897540ist_fm @ C @ P ) )
=> ~ ! [X3: list_fm,Y2: list_fm] :
( ( P
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) )
=> ~ ( member_list_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_546_mem__case__prodE,axiom,
! [Z2: set_fm,C: list_fm > list_fm > set_set_fm,P: produc1996495991257130529ist_fm] :
( ( member_set_fm2 @ Z2 @ ( produc5469387797198216490set_fm @ C @ P ) )
=> ~ ! [X3: list_fm,Y2: list_fm] :
( ( P
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) )
=> ~ ( member_set_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_547_mem__case__prodE,axiom,
! [Z2: fm,C: produc1996495991257130529ist_fm > rule > set_fm,P: produc164195504107695125m_rule] :
( ( member_fm2 @ Z2 @ ( produc966423478193074626set_fm @ C @ P ) )
=> ~ ! [X3: produc1996495991257130529ist_fm,Y2: rule] :
( ( P
= ( produc491467635432902671m_rule @ X3 @ Y2 ) )
=> ~ ( member_fm2 @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_548_mem__case__prodE,axiom,
! [Z2: product_prod_fm_fm,C: list_fm > list_fm > set_Pr4463079037648049377_fm_fm,P: produc1996495991257130529ist_fm] :
( ( member8474499337054950954_fm_fm @ Z2 @ ( produc1074050000962238963_fm_fm @ C @ P ) )
=> ~ ! [X3: list_fm,Y2: list_fm] :
( ( P
= ( produc381145313068854617ist_fm @ X3 @ Y2 ) )
=> ~ ( member8474499337054950954_fm_fm @ Z2 @ ( C @ X3 @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_549_old_Oprod_Ocase,axiom,
! [F: list_fm > list_fm > $o,X1: list_fm,X23: list_fm] :
( ( produc2011568048573023402t_fm_o @ F @ ( produc381145313068854617ist_fm @ X1 @ X23 ) )
= ( F @ X1 @ X23 ) ) ).
% old.prod.case
thf(fact_550_old_Oprod_Ocase,axiom,
! [F: produc1996495991257130529ist_fm > rule > $o,X1: produc1996495991257130529ist_fm,X23: rule] :
( ( produc7369492558480804444rule_o @ F @ ( produc491467635432902671m_rule @ X1 @ X23 ) )
= ( F @ X1 @ X23 ) ) ).
% old.prod.case
thf(fact_551_curryE,axiom,
! [F: produc164195504107695125m_rule > $o,A2: produc1996495991257130529ist_fm,B2: rule] :
( ( produc7868929780990704507rule_o @ F @ A2 @ B2 )
=> ( F @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_552_curryE,axiom,
! [F: produc3793571666044074251m_rule > $o,A2: produc164195504107695125m_rule,B2: produc164195504107695125m_rule] :
( ( produc1995819562154721121rule_o @ F @ A2 @ B2 )
=> ( F @ ( produc5927390650430071747m_rule @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_553_curryE,axiom,
! [F: produc1996495991257130529ist_fm > $o,A2: list_fm,B2: list_fm] :
( ( produc4804894507120027467t_fm_o @ F @ A2 @ B2 )
=> ( F @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_554_curryE,axiom,
! [F: produc4630866025709511057ist_fm > $o,A2: rule,B2: produc1996495991257130529ist_fm] :
( ( produc2127844983229472711t_fm_o @ F @ A2 @ B2 )
=> ( F @ ( produc7063755442767259331ist_fm @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_555_curryE,axiom,
! [F: product_prod_fm_fm > $o,A2: fm,B2: fm] :
( ( produc5605581514536698091m_fm_o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair_fm_fm @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_556_curryD,axiom,
! [F: produc164195504107695125m_rule > $o,A2: produc1996495991257130529ist_fm,B2: rule] :
( ( produc7868929780990704507rule_o @ F @ A2 @ B2 )
=> ( F @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_557_curryD,axiom,
! [F: produc3793571666044074251m_rule > $o,A2: produc164195504107695125m_rule,B2: produc164195504107695125m_rule] :
( ( produc1995819562154721121rule_o @ F @ A2 @ B2 )
=> ( F @ ( produc5927390650430071747m_rule @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_558_curryD,axiom,
! [F: produc1996495991257130529ist_fm > $o,A2: list_fm,B2: list_fm] :
( ( produc4804894507120027467t_fm_o @ F @ A2 @ B2 )
=> ( F @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_559_curryD,axiom,
! [F: produc4630866025709511057ist_fm > $o,A2: rule,B2: produc1996495991257130529ist_fm] :
( ( produc2127844983229472711t_fm_o @ F @ A2 @ B2 )
=> ( F @ ( produc7063755442767259331ist_fm @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_560_curryD,axiom,
! [F: product_prod_fm_fm > $o,A2: fm,B2: fm] :
( ( produc5605581514536698091m_fm_o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair_fm_fm @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_561_Product__Type_OCollect__case__prodD,axiom,
! [X: product_prod_fm_fm,A: fm > fm > $o] :
( ( member8474499337054950954_fm_fm @ X @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ A ) ) )
=> ( A @ ( product_fst_fm_fm @ X ) @ ( product_snd_fm_fm @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_562_Product__Type_OCollect__case__prodD,axiom,
! [X: produc7777387212919034070atural,A: fm > produc7822875418678951345atural > $o] :
( ( member5418343652702035821atural @ X @ ( collec7665630151490057899atural @ ( produc3617581063489722275ural_o @ A ) ) )
=> ( A @ ( produc7697328821926353820atural @ X ) @ ( produc7769300925896546778atural @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_563_Product__Type_OCollect__case__prodD,axiom,
! [X: produc4630866025709511057ist_fm,A: rule > produc1996495991257130529ist_fm > $o] :
( ( member8686995742288324264ist_fm @ X @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ A ) ) )
=> ( A @ ( produc4228417144610596823ist_fm @ X ) @ ( produc4514744107061996309ist_fm @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_564_Product__Type_OCollect__case__prodD,axiom,
! [X: produc164195504107695125m_rule,A: produc1996495991257130529ist_fm > rule > $o] :
( ( member4220325220686508332m_rule @ X @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ A ) ) )
=> ( A @ ( produc6879501374131015971m_rule @ X ) @ ( produc7165828336582415457m_rule @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_565_Product__Type_OCollect__case__prodD,axiom,
! [X: produc1996495991257130529ist_fm,A: list_fm > list_fm > $o] :
( ( member8102475879199740618ist_fm @ X @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ A ) ) )
=> ( A @ ( produc1501393135466168645ist_fm @ X ) @ ( produc4588648349897876871ist_fm @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_566_member__rec_I2_J,axiom,
! [Y3: produc3793571666044074251m_rule] :
~ ( member5034220497408493606m_rule @ nil_Pr3052741188495898641m_rule @ Y3 ) ).
% member_rec(2)
thf(fact_567_member__rec_I2_J,axiom,
! [Y3: rule] :
~ ( member_rule @ nil_rule @ Y3 ) ).
% member_rec(2)
thf(fact_568_member__rec_I2_J,axiom,
! [Y3: produc1996495991257130529ist_fm] :
~ ( member8995235396497187388ist_fm @ nil_Pr6600767949821390631ist_fm @ Y3 ) ).
% member_rec(2)
thf(fact_569_member__rec_I2_J,axiom,
! [Y3: produc164195504107695125m_rule] :
~ ( member5003039597548830010m_rule @ nil_Pr474642538098133391m_rule @ Y3 ) ).
% member_rec(2)
thf(fact_570_member__rec_I2_J,axiom,
! [Y3: fm] :
~ ( member_fm @ nil_fm @ Y3 ) ).
% member_rec(2)
thf(fact_571_case__prod__beta,axiom,
( produc2011568048573023402t_fm_o
= ( ^ [F2: list_fm > list_fm > $o,P3: produc1996495991257130529ist_fm] : ( F2 @ ( produc1501393135466168645ist_fm @ P3 ) @ ( produc4588648349897876871ist_fm @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_572_case__prod__beta,axiom,
( produc7369492558480804444rule_o
= ( ^ [F2: produc1996495991257130529ist_fm > rule > $o,P3: produc164195504107695125m_rule] : ( F2 @ ( produc6879501374131015971m_rule @ P3 ) @ ( produc7165828336582415457m_rule @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_573_split__beta,axiom,
( produc2011568048573023402t_fm_o
= ( ^ [F2: list_fm > list_fm > $o,Prod3: produc1996495991257130529ist_fm] : ( F2 @ ( produc1501393135466168645ist_fm @ Prod3 ) @ ( produc4588648349897876871ist_fm @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_574_split__beta,axiom,
( produc7369492558480804444rule_o
= ( ^ [F2: produc1996495991257130529ist_fm > rule > $o,Prod3: produc164195504107695125m_rule] : ( F2 @ ( produc6879501374131015971m_rule @ Prod3 ) @ ( produc7165828336582415457m_rule @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_575_internal__case__prod__def,axiom,
produc7663040964597641044t_fm_o = produc2011568048573023402t_fm_o ).
% internal_case_prod_def
thf(fact_576_internal__case__prod__def,axiom,
produc5721985997398625586rule_o = produc7369492558480804444rule_o ).
% internal_case_prod_def
thf(fact_577_prod_Osplit__sel__asm,axiom,
! [P2: $o > $o,F: list_fm > list_fm > $o,Prod: produc1996495991257130529ist_fm] :
( ( P2 @ ( produc2011568048573023402t_fm_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_578_prod_Osplit__sel__asm,axiom,
! [P2: $o > $o,F: produc1996495991257130529ist_fm > rule > $o,Prod: produc164195504107695125m_rule] :
( ( P2 @ ( produc7369492558480804444rule_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) )
& ~ ( P2 @ ( F @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_579_split__cong,axiom,
! [Q: produc1996495991257130529ist_fm,F: list_fm > list_fm > $o,G2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
( ! [X3: list_fm,Y2: list_fm] :
( ( ( produc381145313068854617ist_fm @ X3 @ Y2 )
= Q )
=> ( ( F @ X3 @ Y2 )
= ( G2 @ X3 @ Y2 ) ) )
=> ( ( P = Q )
=> ( ( produc2011568048573023402t_fm_o @ F @ P )
= ( produc2011568048573023402t_fm_o @ G2 @ Q ) ) ) ) ).
% split_cong
thf(fact_580_split__cong,axiom,
! [Q: produc164195504107695125m_rule,F: produc1996495991257130529ist_fm > rule > $o,G2: produc1996495991257130529ist_fm > rule > $o,P: produc164195504107695125m_rule] :
( ! [X3: produc1996495991257130529ist_fm,Y2: rule] :
( ( ( produc491467635432902671m_rule @ X3 @ Y2 )
= Q )
=> ( ( F @ X3 @ Y2 )
= ( G2 @ X3 @ Y2 ) ) )
=> ( ( P = Q )
=> ( ( produc7369492558480804444rule_o @ F @ P )
= ( produc7369492558480804444rule_o @ G2 @ Q ) ) ) ) ).
% split_cong
thf(fact_581_list__ex1__simps_I1_J,axiom,
! [P2: produc3793571666044074251m_rule > $o] :
~ ( list_e6764831062926836948m_rule @ P2 @ nil_Pr3052741188495898641m_rule ) ).
% list_ex1_simps(1)
thf(fact_582_list__ex1__simps_I1_J,axiom,
! [P2: rule > $o] :
~ ( list_ex1_rule @ P2 @ nil_rule ) ).
% list_ex1_simps(1)
thf(fact_583_list__ex1__simps_I1_J,axiom,
! [P2: produc1996495991257130529ist_fm > $o] :
~ ( list_e1321013721834914026ist_fm @ P2 @ nil_Pr6600767949821390631ist_fm ) ).
% list_ex1_simps(1)
thf(fact_584_list__ex1__simps_I1_J,axiom,
! [P2: produc164195504107695125m_rule > $o] :
~ ( list_e771460942643608332m_rule @ P2 @ nil_Pr474642538098133391m_rule ) ).
% list_ex1_simps(1)
thf(fact_585_list__ex1__simps_I1_J,axiom,
! [P2: fm > $o] :
~ ( list_ex1_fm @ P2 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_586_csquare__fstOp__sndOp,axiom,
! [F: ( produc1996495991257130529ist_fm > rule > $o ) > produc164195504107695125m_rule > $o,P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] : ( bNF_cs4850752376292380463m_rule @ ( collec2069506698588392810m_rule @ ( F @ ( relcom439222529760170038m_rule @ P2 @ Q2 ) ) ) @ produc1639249265332130309ist_fm @ produc6879501374131015971m_rule @ ( bNF_fs3491168633039030583m_rule @ P2 @ Q2 ) @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_587_csquare__fstOp__sndOp,axiom,
! [F: ( produc1996495991257130529ist_fm > rule > $o ) > produc164195504107695125m_rule > $o,P2: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o] : ( bNF_cs1207223144858359663e_rule @ ( collec2069506698588392810m_rule @ ( F @ ( relcom6373949985761432764e_rule @ P2 @ Q2 ) ) ) @ produc7165828336582415457m_rule @ produc7264528503596239631e_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q2 ) @ ( bNF_sn6445580637211383353e_rule @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_588_csquare__fstOp__sndOp,axiom,
! [F: ( rule > rule > $o ) > produc780299800278082923e_rule > $o,P2: rule > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] : ( bNF_cs8798692456506898627m_rule @ ( collec5224624948449408342e_rule @ ( F @ ( relcom4184253850255763312m_rule @ P2 @ Q2 ) ) ) @ produc4514744107061996309ist_fm @ produc6879501374131015971m_rule @ ( bNF_fs9177076566076742319m_rule @ P2 @ Q2 ) @ ( bNF_sn4255884501705713901m_rule @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_589_csquare__fstOp__sndOp,axiom,
! [F: ( list_fm > list_fm > $o ) > produc1996495991257130529ist_fm > $o,P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] : ( bNF_cs3500245479755196598ist_fm @ ( collec7231575776666803724ist_fm @ ( F @ ( relcom6651562906911586515ist_fm @ P2 @ Q2 ) ) ) @ produc4588648349897876871ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q2 ) @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_590_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_591_bind__simps_I1_J,axiom,
! [F: fm > list_rule] :
( ( bind_fm_rule @ nil_fm @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_592_bind__simps_I1_J,axiom,
! [F: rule > list_fm] :
( ( bind_rule_fm @ nil_rule @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_593_bind__simps_I1_J,axiom,
! [F: rule > list_rule] :
( ( bind_rule_rule @ nil_rule @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_594_bind__simps_I1_J,axiom,
! [F: fm > list_P5616295576739893671ist_fm] :
( ( bind_f7324206428783605000ist_fm @ nil_fm @ F )
= nil_Pr6600767949821390631ist_fm ) ).
% bind_simps(1)
thf(fact_595_bind__simps_I1_J,axiom,
! [F: rule > list_P5616295576739893671ist_fm] :
( ( bind_r594112373614653651ist_fm @ nil_rule @ F )
= nil_Pr6600767949821390631ist_fm ) ).
% bind_simps(1)
thf(fact_596_bind__simps_I1_J,axiom,
! [F: produc1996495991257130529ist_fm > list_fm] :
( ( bind_P2655831839271694122_fm_fm @ nil_Pr6600767949821390631ist_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_597_bind__simps_I1_J,axiom,
! [F: produc1996495991257130529ist_fm > list_rule] :
( ( bind_P3245196603135072799m_rule @ nil_Pr6600767949821390631ist_fm @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_598_bind__simps_I1_J,axiom,
! [F: fm > list_P8332459875677254181m_rule] :
( ( bind_f2621841864267614062m_rule @ nil_fm @ F )
= nil_Pr474642538098133391m_rule ) ).
% bind_simps(1)
thf(fact_599_bind__simps_I1_J,axiom,
! [F: rule > list_P8332459875677254181m_rule] :
( ( bind_r5997413720929594595m_rule @ nil_rule @ F )
= nil_Pr474642538098133391m_rule ) ).
% bind_simps(1)
thf(fact_600_in__measures_I1_J,axiom,
! [X: produc164195504107695125m_rule,Y3: produc164195504107695125m_rule] :
~ ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) @ ( measur7285701395685234453m_rule @ nil_Pr746248845563852926le_nat ) ) ).
% in_measures(1)
thf(fact_601_in__measures_I1_J,axiom,
! [X: list_fm,Y3: list_fm] :
~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ ( measures_list_fm @ nil_list_fm_nat ) ) ).
% in_measures(1)
thf(fact_602_in__measures_I1_J,axiom,
! [X: fm,Y3: fm] :
~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ ( measures_fm @ nil_fm_nat ) ) ).
% in_measures(1)
thf(fact_603_le__rel__bool__arg__iff,axiom,
( ord_le7832897532537454638ist_fm
= ( ^ [X4: $o > set_Pr7058068377845519745ist_fm,Y6: $o > set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2055072402069232929ist_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_604_le__rel__bool__arg__iff,axiom,
( ord_le2648260146891443775ist_fm
= ( ^ [X4: $o > set_list_fm,Y6: $o > set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3173885194822255862ist_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_605_le__rel__bool__arg__iff,axiom,
( ord_le1533557716410907237set_fm
= ( ^ [X4: $o > set_set_fm,Y6: $o > set_set_fm] :
( ( ord_le1461404734466536732set_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1461404734466536732set_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_606_le__rel__bool__arg__iff,axiom,
( ord_le7111125303644257340set_fm
= ( ^ [X4: $o > $o > set_fm,Y6: $o > $o > set_fm] :
( ( ord_less_eq_o_set_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_set_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_607_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X4: $o > $o > nat,Y6: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_608_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_fm
= ( ^ [X4: $o > set_fm,Y6: $o > set_fm] :
( ( ord_less_eq_set_fm @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_fm @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_609_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_610_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_611_leq__OOI,axiom,
! [R3: list_fm > list_fm > $o] :
( ( R3
= ( ^ [Y: list_fm,Z: list_fm] : ( Y = Z ) ) )
=> ( ord_le1774903256965159270t_fm_o @ R3 @ ( relcom6651562906911586515ist_fm @ R3 @ R3 ) ) ) ).
% leq_OOI
thf(fact_612_relcompp__mono,axiom,
! [R4: list_fm > list_fm > $o,R2: list_fm > list_fm > $o,S4: list_fm > list_fm > $o,S: list_fm > list_fm > $o] :
( ( ord_le1774903256965159270t_fm_o @ R4 @ R2 )
=> ( ( ord_le1774903256965159270t_fm_o @ S4 @ S )
=> ( ord_le1774903256965159270t_fm_o @ ( relcom6651562906911586515ist_fm @ R4 @ S4 ) @ ( relcom6651562906911586515ist_fm @ R2 @ S ) ) ) ) ).
% relcompp_mono
thf(fact_613_relcompp__assoc,axiom,
! [R2: list_fm > list_fm > $o,S: list_fm > list_fm > $o,T2: list_fm > list_fm > $o] :
( ( relcom6651562906911586515ist_fm @ ( relcom6651562906911586515ist_fm @ R2 @ S ) @ T2 )
= ( relcom6651562906911586515ist_fm @ R2 @ ( relcom6651562906911586515ist_fm @ S @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_614_relcompp__apply,axiom,
( relcom6651562906911586515ist_fm
= ( ^ [R5: list_fm > list_fm > $o,S5: list_fm > list_fm > $o,A3: list_fm,C4: list_fm] :
? [B5: list_fm] :
( ( R5 @ A3 @ B5 )
& ( S5 @ B5 @ C4 ) ) ) ) ).
% relcompp_apply
thf(fact_615_relcomppI,axiom,
! [R2: list_fm > list_fm > $o,A2: list_fm,B2: list_fm,S: list_fm > list_fm > $o,C: list_fm] :
( ( R2 @ A2 @ B2 )
=> ( ( S @ B2 @ C )
=> ( relcom6651562906911586515ist_fm @ R2 @ S @ A2 @ C ) ) ) ).
% relcomppI
thf(fact_616_relcomppE,axiom,
! [R2: list_fm > list_fm > $o,S: list_fm > list_fm > $o,A2: list_fm,C: list_fm] :
( ( relcom6651562906911586515ist_fm @ R2 @ S @ A2 @ C )
=> ~ ! [B4: list_fm] :
( ( R2 @ A2 @ B4 )
=> ~ ( S @ B4 @ C ) ) ) ).
% relcomppE
thf(fact_617_eq__OO,axiom,
! [R3: list_fm > list_fm > $o] :
( ( relcom6651562906911586515ist_fm
@ ^ [Y: list_fm,Z: list_fm] : ( Y = Z )
@ R3 )
= R3 ) ).
% eq_OO
thf(fact_618_OO__eq,axiom,
! [R3: list_fm > list_fm > $o] :
( ( relcom6651562906911586515ist_fm @ R3
@ ^ [Y: list_fm,Z: list_fm] : ( Y = Z ) )
= R3 ) ).
% OO_eq
thf(fact_619_relcompp_Osimps,axiom,
( relcom6651562906911586515ist_fm
= ( ^ [R: list_fm > list_fm > $o,S3: list_fm > list_fm > $o,A1: list_fm,A22: list_fm] :
? [A3: list_fm,B5: list_fm,C4: list_fm] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( R @ A3 @ B5 )
& ( S3 @ B5 @ C4 ) ) ) ) ).
% relcompp.simps
thf(fact_620_relcompp_Ocases,axiom,
! [R2: list_fm > list_fm > $o,S: list_fm > list_fm > $o,A12: list_fm,A23: list_fm] :
( ( relcom6651562906911586515ist_fm @ R2 @ S @ A12 @ A23 )
=> ~ ! [B4: list_fm] :
( ( R2 @ A12 @ B4 )
=> ~ ( S @ B4 @ A23 ) ) ) ).
% relcompp.cases
thf(fact_621_pick__middlep,axiom,
! [P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o,A2: list_fm,C: list_fm] :
( ( relcom6651562906911586515ist_fm @ P2 @ Q2 @ A2 @ C )
=> ( ( P2 @ A2 @ ( bNF_pi3539088406881798022ist_fm @ P2 @ Q2 @ A2 @ C ) )
& ( Q2 @ ( bNF_pi3539088406881798022ist_fm @ P2 @ Q2 @ A2 @ C ) @ C ) ) ) ).
% pick_middlep
thf(fact_622_sndOp__in,axiom,
! [Ac2: product_prod_fm_fm,P2: fm > fm > $o,Q2: fm > fm > $o] :
( ( member8474499337054950954_fm_fm @ Ac2 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ ( relcompp_fm_fm_fm @ P2 @ Q2 ) ) ) )
=> ( member8474499337054950954_fm_fm @ ( bNF_sndOp_fm_fm_fm @ P2 @ Q2 @ Ac2 ) @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_623_sndOp__in,axiom,
! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom6651562906911586515ist_fm @ P2 @ Q2 ) ) ) )
=> ( member8102475879199740618ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q2 @ Ac2 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_624_sndOp__in,axiom,
! [Ac2: produc164195504107695125m_rule,P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] :
( ( member4220325220686508332m_rule @ Ac2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( relcom439222529760170038m_rule @ P2 @ Q2 ) ) ) )
=> ( member4220325220686508332m_rule @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q2 @ Ac2 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_625_fstOp__in,axiom,
! [Ac2: product_prod_fm_fm,P2: fm > fm > $o,Q2: fm > fm > $o] :
( ( member8474499337054950954_fm_fm @ Ac2 @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ ( relcompp_fm_fm_fm @ P2 @ Q2 ) ) ) )
=> ( member8474499337054950954_fm_fm @ ( bNF_fstOp_fm_fm_fm @ P2 @ Q2 @ Ac2 ) @ ( collec4425076577877259116_fm_fm @ ( produc2811014937302177098m_fm_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_626_fstOp__in,axiom,
! [Ac2: produc1996495991257130529ist_fm,P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
( ( member8102475879199740618ist_fm @ Ac2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( relcom6651562906911586515ist_fm @ P2 @ Q2 ) ) ) )
=> ( member8102475879199740618ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q2 @ Ac2 ) @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_627_fstOp__in,axiom,
! [Ac2: produc164195504107695125m_rule,P2: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o] :
( ( member4220325220686508332m_rule @ Ac2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( relcom6373949985761432764e_rule @ P2 @ Q2 ) ) ) )
=> ( member4220325220686508332m_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q2 @ Ac2 ) @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_628_csquare__def,axiom,
( bNF_cs3500245479755196598ist_fm
= ( ^ [A6: set_Pr7058068377845519745ist_fm,F1: produc1996495991257130529ist_fm > list_fm,F22: produc1996495991257130529ist_fm > list_fm,P1: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P22: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
! [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A6 )
=> ( ( F1 @ ( P1 @ X2 ) )
= ( F22 @ ( P22 @ X2 ) ) ) ) ) ) ).
% csquare_def
thf(fact_629_list__ex1__iff,axiom,
( list_e5200897398398297162_fm_fm
= ( ^ [P4: product_prod_fm_fm > $o,Xs2: list_P6934817935912052487_fm_fm] :
? [X2: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X2 @ ( set_Pr6624747810323803798_fm_fm @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: product_prod_fm_fm] :
( ( ( member8474499337054950954_fm_fm @ Y4 @ ( set_Pr6624747810323803798_fm_fm @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_630_list__ex1__iff,axiom,
( list_e1321013721834914026ist_fm
= ( ^ [P4: produc1996495991257130529ist_fm > $o,Xs2: list_P5616295576739893671ist_fm] :
? [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ ( set_Pr8767716839810916150ist_fm @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: produc1996495991257130529ist_fm] :
( ( ( member8102475879199740618ist_fm @ Y4 @ ( set_Pr8767716839810916150ist_fm @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_631_list__ex1__iff,axiom,
( list_ex1_set_fm
= ( ^ [P4: set_fm > $o,Xs2: list_set_fm] :
? [X2: set_fm] :
( ( member_set_fm2 @ X2 @ ( set_set_fm2 @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: set_fm] :
( ( ( member_set_fm2 @ Y4 @ ( set_set_fm2 @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_632_list__ex1__iff,axiom,
( list_ex1_set_set_fm
= ( ^ [P4: set_set_fm > $o,Xs2: list_set_set_fm] :
? [X2: set_set_fm] :
( ( member_set_set_fm2 @ X2 @ ( set_set_set_fm2 @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: set_set_fm] :
( ( ( member_set_set_fm2 @ Y4 @ ( set_set_set_fm2 @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_633_list__ex1__iff,axiom,
( list_ex1_list_fm
= ( ^ [P4: list_fm > $o,Xs2: list_list_fm] :
? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: list_fm] :
( ( ( member_list_fm2 @ Y4 @ ( set_list_fm2 @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_634_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P4: fm > $o,Xs2: list_fm] :
? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y4: fm] :
( ( ( member_fm2 @ Y4 @ ( set_fm2 @ Xs2 ) )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_635_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_636_verit__comp__simplify1_I2_J,axiom,
! [A2: set_list_fm] : ( ord_le3173885194822255862ist_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_637_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_fm] : ( ord_le1461404734466536732set_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_638_verit__comp__simplify1_I2_J,axiom,
! [A2: $o > set_fm] : ( ord_less_eq_o_set_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_639_verit__comp__simplify1_I2_J,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_640_verit__comp__simplify1_I2_J,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_641_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_642_can__select__set__list__ex1,axiom,
! [P2: list_fm > $o,A: list_list_fm] :
( ( can_select_list_fm @ P2 @ ( set_list_fm2 @ A ) )
= ( list_ex1_list_fm @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_643_can__select__set__list__ex1,axiom,
! [P2: fm > $o,A: list_fm] :
( ( can_select_fm @ P2 @ ( set_fm2 @ A ) )
= ( list_ex1_fm @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_644_Collect__split__mono__strong,axiom,
! [X5: set_fm,A: set_Pr8331943448356867980atural,Y7: set_Pr7564037697178849041atural,P2: fm > produc7822875418678951345atural > $o,Q2: fm > produc7822875418678951345atural > $o] :
( ( X5
= ( image_4328254823976585133ral_fm @ produc7697328821926353820atural @ A ) )
=> ( ( Y7
= ( image_4720309336624674452atural @ produc7769300925896546778atural @ A ) )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ X5 )
=> ! [Xa: produc7822875418678951345atural] :
( ( member8280416476884352090atural @ Xa @ Y7 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q2 @ X3 @ Xa ) ) ) )
=> ( ( ord_le3568868789420396524atural @ A @ ( collec7665630151490057899atural @ ( produc3617581063489722275ural_o @ P2 ) ) )
=> ( ord_le3568868789420396524atural @ A @ ( collec7665630151490057899atural @ ( produc3617581063489722275ural_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_645_Collect__split__mono__strong,axiom,
! [X5: set_rule,A: set_Pr2241365836748572487ist_fm,Y7: set_Pr7058068377845519745ist_fm,P2: rule > produc1996495991257130529ist_fm > $o,Q2: rule > produc1996495991257130529ist_fm > $o] :
( ( X5
= ( image_952663707700710365m_rule @ produc4228417144610596823ist_fm @ A ) )
=> ( ( Y7
= ( image_3681455213663431561ist_fm @ produc4514744107061996309ist_fm @ A ) )
=> ( ! [X3: rule] :
( ( member_rule2 @ X3 @ X5 )
=> ! [Xa: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ Xa @ Y7 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q2 @ X3 @ Xa ) ) ) )
=> ( ( ord_le6285472588696295335ist_fm @ A @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ P2 ) ) )
=> ( ord_le6285472588696295335ist_fm @ A @ ( collec6536177220190208742ist_fm @ ( produc1628407760719572648t_fm_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_646_Collect__split__mono__strong,axiom,
! [X5: set_Pr7058068377845519745ist_fm,A: set_Pr1008144964186165195m_rule,Y7: set_rule,P2: produc1996495991257130529ist_fm > rule > $o,Q2: produc1996495991257130529ist_fm > rule > $o] :
( ( X5
= ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ A ) )
=> ( ( Y7
= ( image_8178078329501008481e_rule @ produc7165828336582415457m_rule @ A ) )
=> ( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ X5 )
=> ! [Xa: rule] :
( ( member_rule2 @ Xa @ Y7 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q2 @ X3 @ Xa ) ) ) )
=> ( ( ord_le5052251716133888043m_rule @ A @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ P2 ) ) )
=> ( ord_le5052251716133888043m_rule @ A @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_647_Collect__split__mono__strong,axiom,
! [X5: set_list_fm,A: set_Pr7058068377845519745ist_fm,Y7: set_list_fm,P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
( ( X5
= ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ A ) )
=> ( ( Y7
= ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ A ) )
=> ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ X5 )
=> ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ Y7 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q2 @ X3 @ Xa ) ) ) )
=> ( ( ord_le2055072402069232929ist_fm @ A @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) ) )
=> ( ord_le2055072402069232929ist_fm @ A @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_648_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr7058068377845519745ist_fm,Y7: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ X5 @ Y7 )
=> ( ord_le2055072402069232929ist_fm @ X5 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( fun_in8167150049942790916ist_fm @ Y7 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_649_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr1008144964186165195m_rule,Y7: set_Pr1008144964186165195m_rule] :
( ( ord_le5052251716133888043m_rule @ X5 @ Y7 )
=> ( ord_le5052251716133888043m_rule @ X5 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( fun_in1891732760143430564m_rule @ Y7 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_650_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr7058068377845519745ist_fm,Y7: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ X5 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ ( fun_in8167150049942790916ist_fm @ Y7 ) ) ) )
=> ( ord_le2055072402069232929ist_fm @ X5 @ Y7 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_651_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr1008144964186165195m_rule,Y7: set_Pr1008144964186165195m_rule] :
( ( ord_le5052251716133888043m_rule @ X5 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ ( fun_in1891732760143430564m_rule @ Y7 ) ) ) )
=> ( ord_le5052251716133888043m_rule @ X5 @ Y7 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_652_eq__comp__r,axiom,
! [R3: list_fm > list_fm > $o] :
( ( relcom6651562906911586515ist_fm
@ ^ [Y: list_fm,Z: list_fm] : ( Y = Z )
@ ( relcom6651562906911586515ist_fm @ R3
@ ^ [Y: list_fm,Z: list_fm] : ( Y = Z ) ) )
= R3 ) ).
% eq_comp_r
thf(fact_653_select,axiom,
! [Xs: list_P6934817935912052487_fm_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_Pr991870136757000071_fm_fm )
=> ( member8474499337054950954_fm_fm @ ( produc4964226024804884979atural @ ( select875622366172221621_fm_fm @ Xs @ S ) ) @ ( set_Pr6624747810323803798_fm_fm @ Xs ) ) ) ).
% select
thf(fact_654_select,axiom,
! [Xs: list_set_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_set_fm )
=> ( member_set_fm2 @ ( produc2398674278646710758atural @ ( select_set_fm @ Xs @ S ) ) @ ( set_set_fm2 @ Xs ) ) ) ).
% select
thf(fact_655_select,axiom,
! [Xs: list_set_set_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_set_set_fm )
=> ( member_set_set_fm2 @ ( produc4578180781001494320atural @ ( select_set_set_fm @ Xs @ S ) ) @ ( set_set_set_fm2 @ Xs ) ) ) ).
% select
thf(fact_656_select,axiom,
! [Xs: list_P8801850819564216977m_rule,S: produc7822875418678951345atural] :
( ( Xs != nil_Pr3052741188495898641m_rule )
=> ( member2434505771962676916m_rule @ ( produc8992147475317826793atural @ ( select4229960543179501887m_rule @ Xs @ S ) ) @ ( set_Pr2379943022980910880m_rule @ Xs ) ) ) ).
% select
thf(fact_657_select,axiom,
! [Xs: list_rule,S: produc7822875418678951345atural] :
( ( Xs != nil_rule )
=> ( member_rule2 @ ( produc8728111071016758119atural @ ( select_rule @ Xs @ S ) ) @ ( set_rule2 @ Xs ) ) ) ).
% select
thf(fact_658_select,axiom,
! [Xs: list_P5616295576739893671ist_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_Pr6600767949821390631ist_fm )
=> ( member8102475879199740618ist_fm @ ( produc8353339014683232339atural @ ( select5195649533649406549ist_fm @ Xs @ S ) ) @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).
% select
thf(fact_659_select,axiom,
! [Xs: list_P8332459875677254181m_rule,S: produc7822875418678951345atural] :
( ( Xs != nil_Pr474642538098133391m_rule )
=> ( member4220325220686508332m_rule @ ( produc407180354247663767atural @ ( select8693855115942095585m_rule @ Xs @ S ) ) @ ( set_Pr8505323785428441536m_rule @ Xs ) ) ) ).
% select
thf(fact_660_select,axiom,
! [Xs: list_list_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_list_fm )
=> ( member_list_fm2 @ ( produc412862668981055756atural @ ( select_list_fm @ Xs @ S ) ) @ ( set_list_fm2 @ Xs ) ) ) ).
% select
thf(fact_661_select,axiom,
! [Xs: list_fm,S: produc7822875418678951345atural] :
( ( Xs != nil_fm )
=> ( member_fm2 @ ( produc7697328821926353820atural @ ( select_fm @ Xs @ S ) ) @ ( set_fm2 @ Xs ) ) ) ).
% select
thf(fact_662_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_fm @ N2 @ nil_fm )
= N2 ) ).
% gen_length_code(1)
thf(fact_663_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_le7484115818410561182m_rule @ N2 @ nil_Pr3052741188495898641m_rule )
= N2 ) ).
% gen_length_code(1)
thf(fact_664_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_rule @ N2 @ nil_rule )
= N2 ) ).
% gen_length_code(1)
thf(fact_665_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_le1831903222360256692ist_fm @ N2 @ nil_Pr6600767949821390631ist_fm )
= N2 ) ).
% gen_length_code(1)
thf(fact_666_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_le4128708808483664066m_rule @ N2 @ nil_Pr474642538098133391m_rule )
= N2 ) ).
% gen_length_code(1)
thf(fact_667_stream__all__def,axiom,
( stream919847744520025429m_rule
= ( ^ [P4: produc3793571666044074251m_rule > $o,S3: stream6494289010434245521m_rule] :
! [P3: nat] : ( P4 @ ( snth_P8950227884416379311m_rule @ S3 @ P3 ) ) ) ) ).
% stream_all_def
thf(fact_668_stream__all__def,axiom,
( stream_all_rule
= ( ^ [P4: rule > $o,S3: stream_rule] :
! [P3: nat] : ( P4 @ ( snth_rule @ S3 @ P3 ) ) ) ) ).
% stream_all_def
thf(fact_669_stream__all__def,axiom,
( stream720971363120138347ist_fm
= ( ^ [P4: produc1996495991257130529ist_fm > $o,S3: stream8299795917829157543ist_fm] :
! [P3: nat] : ( P4 @ ( snth_P2624749845746815685ist_fm @ S3 @ P3 ) ) ) ) ).
% stream_all_def
thf(fact_670_stream__all__def,axiom,
( stream456966007795584843m_rule
= ( ^ [P4: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
! [P3: nat] : ( P4 @ ( snth_P1712579846118164081m_rule @ S3 @ P3 ) ) ) ) ).
% stream_all_def
thf(fact_671_image__eqI,axiom,
! [B2: fm,F: fm > fm,X: fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member_fm2 @ B2 @ ( image_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_672_image__eqI,axiom,
! [B2: list_fm,F: fm > list_fm,X: fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member_list_fm2 @ B2 @ ( image_fm_list_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_673_image__eqI,axiom,
! [B2: set_fm,F: fm > set_fm,X: fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member_set_fm2 @ B2 @ ( image_fm_set_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_674_image__eqI,axiom,
! [B2: fm,F: list_fm > fm,X: list_fm,A: set_list_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_list_fm2 @ X @ A )
=> ( member_fm2 @ B2 @ ( image_list_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_675_image__eqI,axiom,
! [B2: fm,F: set_fm > fm,X: set_fm,A: set_set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_fm2 @ X @ A )
=> ( member_fm2 @ B2 @ ( image_set_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_676_image__eqI,axiom,
! [B2: product_prod_fm_fm,F: fm > product_prod_fm_fm,X: fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ B2 @ ( image_453549069922048106_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_677_image__eqI,axiom,
! [B2: set_set_fm,F: fm > set_set_fm,X: fm,A: set_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_fm2 @ X @ A )
=> ( member_set_set_fm2 @ B2 @ ( image_fm_set_set_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_678_image__eqI,axiom,
! [B2: fm,F: product_prod_fm_fm > fm,X: product_prod_fm_fm,A: set_Pr4463079037648049377_fm_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member8474499337054950954_fm_fm @ X @ A )
=> ( member_fm2 @ B2 @ ( image_5888559327631113100_fm_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_679_image__eqI,axiom,
! [B2: list_fm,F: list_fm > list_fm,X: list_fm,A: set_list_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_list_fm2 @ X @ A )
=> ( member_list_fm2 @ B2 @ ( image_2959368623641186503ist_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_680_image__eqI,axiom,
! [B2: set_fm,F: list_fm > set_fm,X: list_fm,A: set_list_fm] :
( ( B2
= ( F @ X ) )
=> ( ( member_list_fm2 @ X @ A )
=> ( member_set_fm2 @ B2 @ ( image_list_fm_set_fm @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_681_pair__in__swap__image,axiom,
! [Y3: produc164195504107695125m_rule,X: produc164195504107695125m_rule,A: set_Pr5090152306159784043m_rule] :
( ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ Y3 @ X ) @ ( image_3030693407660932293m_rule @ produc1276282870026899811m_rule @ A ) )
= ( member2434505771962676916m_rule @ ( produc5927390650430071747m_rule @ X @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_682_pair__in__swap__image,axiom,
! [Y3: fm,X: fm,A: set_Pr4463079037648049377_fm_fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y3 @ X ) @ ( image_7218310488025046853_fm_fm @ product_swap_fm_fm @ A ) )
= ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_683_pair__in__swap__image,axiom,
! [Y3: list_fm,X: list_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ X ) @ ( image_6982201141141708101ist_fm @ produc7936585717479832313ist_fm @ A ) )
= ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_684_pair__in__swap__image,axiom,
! [Y3: rule,X: produc1996495991257130529ist_fm,A: set_Pr1008144964186165195m_rule] :
( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ Y3 @ X ) @ ( image_6659396422799841965ist_fm @ produc2641608828089375343m_rule @ A ) )
= ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_685_pair__in__swap__image,axiom,
! [Y3: produc1996495991257130529ist_fm,X: rule,A: set_Pr2241365836748572487ist_fm] :
( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ Y3 @ X ) @ ( image_4130355632178425005m_rule @ produc9213896635423732003ist_fm @ A ) )
= ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_686_imageI,axiom,
! [X: fm,A: set_fm,F: fm > fm] :
( ( member_fm2 @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ ( image_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_687_imageI,axiom,
! [X: fm,A: set_fm,F: fm > list_fm] :
( ( member_fm2 @ X @ A )
=> ( member_list_fm2 @ ( F @ X ) @ ( image_fm_list_fm @ F @ A ) ) ) ).
% imageI
thf(fact_688_imageI,axiom,
! [X: fm,A: set_fm,F: fm > set_fm] :
( ( member_fm2 @ X @ A )
=> ( member_set_fm2 @ ( F @ X ) @ ( image_fm_set_fm @ F @ A ) ) ) ).
% imageI
thf(fact_689_imageI,axiom,
! [X: list_fm,A: set_list_fm,F: list_fm > fm] :
( ( member_list_fm2 @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ ( image_list_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_690_imageI,axiom,
! [X: set_fm,A: set_set_fm,F: set_fm > fm] :
( ( member_set_fm2 @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ ( image_set_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_691_imageI,axiom,
! [X: fm,A: set_fm,F: fm > product_prod_fm_fm] :
( ( member_fm2 @ X @ A )
=> ( member8474499337054950954_fm_fm @ ( F @ X ) @ ( image_453549069922048106_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_692_imageI,axiom,
! [X: fm,A: set_fm,F: fm > set_set_fm] :
( ( member_fm2 @ X @ A )
=> ( member_set_set_fm2 @ ( F @ X ) @ ( image_fm_set_set_fm @ F @ A ) ) ) ).
% imageI
thf(fact_693_imageI,axiom,
! [X: product_prod_fm_fm,A: set_Pr4463079037648049377_fm_fm,F: product_prod_fm_fm > fm] :
( ( member8474499337054950954_fm_fm @ X @ A )
=> ( member_fm2 @ ( F @ X ) @ ( image_5888559327631113100_fm_fm @ F @ A ) ) ) ).
% imageI
thf(fact_694_imageI,axiom,
! [X: list_fm,A: set_list_fm,F: list_fm > list_fm] :
( ( member_list_fm2 @ X @ A )
=> ( member_list_fm2 @ ( F @ X ) @ ( image_2959368623641186503ist_fm @ F @ A ) ) ) ).
% imageI
thf(fact_695_imageI,axiom,
! [X: list_fm,A: set_list_fm,F: list_fm > set_fm] :
( ( member_list_fm2 @ X @ A )
=> ( member_set_fm2 @ ( F @ X ) @ ( image_list_fm_set_fm @ F @ A ) ) ) ).
% imageI
thf(fact_696_image__iff,axiom,
! [Z2: rule,F: produc164195504107695125m_rule > rule,A: set_Pr1008144964186165195m_rule] :
( ( member_rule2 @ Z2 @ ( image_8178078329501008481e_rule @ F @ A ) )
= ( ? [X2: produc164195504107695125m_rule] :
( ( member4220325220686508332m_rule @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_697_image__iff,axiom,
! [Z2: fm,F: fm > fm,A: set_fm] :
( ( member_fm2 @ Z2 @ ( image_fm_fm @ F @ A ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_698_image__iff,axiom,
! [Z2: list_fm,F: produc1996495991257130529ist_fm > list_fm,A: set_Pr7058068377845519745ist_fm] :
( ( member_list_fm2 @ Z2 @ ( image_6369949096775035324ist_fm @ F @ A ) )
= ( ? [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_699_image__iff,axiom,
! [Z2: set_fm,F: list_fm > set_fm,A: set_list_fm] :
( ( member_set_fm2 @ Z2 @ ( image_list_fm_set_fm @ F @ A ) )
= ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_700_image__iff,axiom,
! [Z2: set_fm,F: set_fm > set_fm,A: set_set_fm] :
( ( member_set_fm2 @ Z2 @ ( image_set_fm_set_fm @ F @ A ) )
= ( ? [X2: set_fm] :
( ( member_set_fm2 @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_701_bex__imageD,axiom,
! [F: produc164195504107695125m_rule > rule,A: set_Pr1008144964186165195m_rule,P2: rule > $o] :
( ? [X6: rule] :
( ( member_rule2 @ X6 @ ( image_8178078329501008481e_rule @ F @ A ) )
& ( P2 @ X6 ) )
=> ? [X3: produc164195504107695125m_rule] :
( ( member4220325220686508332m_rule @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_702_bex__imageD,axiom,
! [F: produc1996495991257130529ist_fm > list_fm,A: set_Pr7058068377845519745ist_fm,P2: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm2 @ X6 @ ( image_6369949096775035324ist_fm @ F @ A ) )
& ( P2 @ X6 ) )
=> ? [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_703_bex__imageD,axiom,
! [F: fm > fm,A: set_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( image_fm_fm @ F @ A ) )
& ( P2 @ X6 ) )
=> ? [X3: fm] :
( ( member_fm2 @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_704_bex__imageD,axiom,
! [F: list_fm > set_fm,A: set_list_fm,P2: set_fm > $o] :
( ? [X6: set_fm] :
( ( member_set_fm2 @ X6 @ ( image_list_fm_set_fm @ F @ A ) )
& ( P2 @ X6 ) )
=> ? [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_705_bex__imageD,axiom,
! [F: set_fm > set_fm,A: set_set_fm,P2: set_fm > $o] :
( ? [X6: set_fm] :
( ( member_set_fm2 @ X6 @ ( image_set_fm_set_fm @ F @ A ) )
& ( P2 @ X6 ) )
=> ? [X3: set_fm] :
( ( member_set_fm2 @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_706_image__cong,axiom,
! [M: set_Pr1008144964186165195m_rule,N3: set_Pr1008144964186165195m_rule,F: produc164195504107695125m_rule > rule,G2: produc164195504107695125m_rule > rule] :
( ( M = N3 )
=> ( ! [X3: produc164195504107695125m_rule] :
( ( member4220325220686508332m_rule @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_8178078329501008481e_rule @ F @ M )
= ( image_8178078329501008481e_rule @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_707_image__cong,axiom,
! [M: set_fm,N3: set_fm,F: fm > fm,G2: fm > fm] :
( ( M = N3 )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_fm_fm @ F @ M )
= ( image_fm_fm @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_708_image__cong,axiom,
! [M: set_Pr7058068377845519745ist_fm,N3: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > list_fm,G2: produc1996495991257130529ist_fm > list_fm] :
( ( M = N3 )
=> ( ! [X3: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_6369949096775035324ist_fm @ F @ M )
= ( image_6369949096775035324ist_fm @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_709_image__cong,axiom,
! [M: set_list_fm,N3: set_list_fm,F: list_fm > set_fm,G2: list_fm > set_fm] :
( ( M = N3 )
=> ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_list_fm_set_fm @ F @ M )
= ( image_list_fm_set_fm @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_710_image__cong,axiom,
! [M: set_set_fm,N3: set_set_fm,F: set_fm > set_fm,G2: set_fm > set_fm] :
( ( M = N3 )
=> ( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_set_fm_set_fm @ F @ M )
= ( image_set_fm_set_fm @ G2 @ N3 ) ) ) ) ).
% image_cong
thf(fact_711_ball__imageD,axiom,
! [F: produc164195504107695125m_rule > rule,A: set_Pr1008144964186165195m_rule,P2: rule > $o] :
( ! [X3: rule] :
( ( member_rule2 @ X3 @ ( image_8178078329501008481e_rule @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X6: produc164195504107695125m_rule] :
( ( member4220325220686508332m_rule @ X6 @ A )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_712_ball__imageD,axiom,
! [F: produc1996495991257130529ist_fm > list_fm,A: set_Pr7058068377845519745ist_fm,P2: list_fm > $o] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( image_6369949096775035324ist_fm @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X6: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X6 @ A )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_713_ball__imageD,axiom,
! [F: fm > fm,A: set_fm,P2: fm > $o] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( image_fm_fm @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X6: fm] :
( ( member_fm2 @ X6 @ A )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_714_ball__imageD,axiom,
! [F: list_fm > set_fm,A: set_list_fm,P2: set_fm > $o] :
( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ ( image_list_fm_set_fm @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X6: list_fm] :
( ( member_list_fm2 @ X6 @ A )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_715_ball__imageD,axiom,
! [F: set_fm > set_fm,A: set_set_fm,P2: set_fm > $o] :
( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ ( image_set_fm_set_fm @ F @ A ) )
=> ( P2 @ X3 ) )
=> ! [X6: set_fm] :
( ( member_set_fm2 @ X6 @ A )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_716_rev__image__eqI,axiom,
! [X: fm,A: set_fm,B2: fm,F: fm > fm] :
( ( member_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_fm2 @ B2 @ ( image_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_717_rev__image__eqI,axiom,
! [X: fm,A: set_fm,B2: list_fm,F: fm > list_fm] :
( ( member_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_list_fm2 @ B2 @ ( image_fm_list_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_718_rev__image__eqI,axiom,
! [X: fm,A: set_fm,B2: set_fm,F: fm > set_fm] :
( ( member_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_fm2 @ B2 @ ( image_fm_set_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_719_rev__image__eqI,axiom,
! [X: list_fm,A: set_list_fm,B2: fm,F: list_fm > fm] :
( ( member_list_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_fm2 @ B2 @ ( image_list_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_720_rev__image__eqI,axiom,
! [X: set_fm,A: set_set_fm,B2: fm,F: set_fm > fm] :
( ( member_set_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_fm2 @ B2 @ ( image_set_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_721_rev__image__eqI,axiom,
! [X: fm,A: set_fm,B2: product_prod_fm_fm,F: fm > product_prod_fm_fm] :
( ( member_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member8474499337054950954_fm_fm @ B2 @ ( image_453549069922048106_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_722_rev__image__eqI,axiom,
! [X: fm,A: set_fm,B2: set_set_fm,F: fm > set_set_fm] :
( ( member_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_set_fm2 @ B2 @ ( image_fm_set_set_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_723_rev__image__eqI,axiom,
! [X: product_prod_fm_fm,A: set_Pr4463079037648049377_fm_fm,B2: fm,F: product_prod_fm_fm > fm] :
( ( member8474499337054950954_fm_fm @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_fm2 @ B2 @ ( image_5888559327631113100_fm_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_724_rev__image__eqI,axiom,
! [X: list_fm,A: set_list_fm,B2: list_fm,F: list_fm > list_fm] :
( ( member_list_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_list_fm2 @ B2 @ ( image_2959368623641186503ist_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_725_rev__image__eqI,axiom,
! [X: list_fm,A: set_list_fm,B2: set_fm,F: list_fm > set_fm] :
( ( member_list_fm2 @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_fm2 @ B2 @ ( image_list_fm_set_fm @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_726_subset__image__iff,axiom,
! [B: set_fm,F: fm > fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_fm_fm @ F @ A ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A )
& ( B
= ( image_fm_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_727_subset__image__iff,axiom,
! [B: set_fm,F: list_fm > fm,A: set_list_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_list_fm_fm @ F @ A ) )
= ( ? [AA: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ AA @ A )
& ( B
= ( image_list_fm_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_728_subset__image__iff,axiom,
! [B: set_fm,F: set_fm > fm,A: set_set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_set_fm_fm @ F @ A ) )
= ( ? [AA: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ AA @ A )
& ( B
= ( image_set_fm_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_729_subset__image__iff,axiom,
! [B: set_list_fm,F: fm > list_fm,A: set_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_fm_list_fm @ F @ A ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A )
& ( B
= ( image_fm_list_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_730_subset__image__iff,axiom,
! [B: set_set_fm,F: fm > set_fm,A: set_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_fm_set_fm @ F @ A ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A )
& ( B
= ( image_fm_set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_731_subset__image__iff,axiom,
! [B: set_list_fm,F: list_fm > list_fm,A: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_2959368623641186503ist_fm @ F @ A ) )
= ( ? [AA: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ AA @ A )
& ( B
= ( image_2959368623641186503ist_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_732_subset__image__iff,axiom,
! [B: set_list_fm,F: set_fm > list_fm,A: set_set_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_set_fm_list_fm @ F @ A ) )
= ( ? [AA: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ AA @ A )
& ( B
= ( image_set_fm_list_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_733_subset__image__iff,axiom,
! [B: set_set_fm,F: list_fm > set_fm,A: set_list_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_list_fm_set_fm @ F @ A ) )
= ( ? [AA: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ AA @ A )
& ( B
= ( image_list_fm_set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_734_subset__image__iff,axiom,
! [B: set_set_fm,F: set_fm > set_fm,A: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_set_fm_set_fm @ F @ A ) )
= ( ? [AA: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ AA @ A )
& ( B
= ( image_set_fm_set_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_735_subset__image__iff,axiom,
! [B: set_fm,F: produc1996495991257130529ist_fm > fm,A: set_Pr7058068377845519745ist_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_3983738456748166956_fm_fm @ F @ A ) )
= ( ? [AA: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ AA @ A )
& ( B
= ( image_3983738456748166956_fm_fm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_736_image__subset__iff,axiom,
! [F: produc164195504107695125m_rule > rule,A: set_Pr1008144964186165195m_rule,B: set_rule] :
( ( ord_less_eq_set_rule @ ( image_8178078329501008481e_rule @ F @ A ) @ B )
= ( ! [X2: produc164195504107695125m_rule] :
( ( member4220325220686508332m_rule @ X2 @ A )
=> ( member_rule2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_737_image__subset__iff,axiom,
! [F: fm > fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A ) @ B )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A )
=> ( member_fm2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_738_image__subset__iff,axiom,
! [F: produc1996495991257130529ist_fm > list_fm,A: set_Pr7058068377845519745ist_fm,B: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ ( image_6369949096775035324ist_fm @ F @ A ) @ B )
= ( ! [X2: produc1996495991257130529ist_fm] :
( ( member8102475879199740618ist_fm @ X2 @ A )
=> ( member_list_fm2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_739_image__subset__iff,axiom,
! [F: list_fm > set_fm,A: set_list_fm,B: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ ( image_list_fm_set_fm @ F @ A ) @ B )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A )
=> ( member_set_fm2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_740_image__subset__iff,axiom,
! [F: set_fm > set_fm,A: set_set_fm,B: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ ( image_set_fm_set_fm @ F @ A ) @ B )
= ( ! [X2: set_fm] :
( ( member_set_fm2 @ X2 @ A )
=> ( member_set_fm2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_741_subset__imageE,axiom,
! [B: set_fm,F: fm > fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_fm_fm @ F @ A ) )
=> ~ ! [C5: set_fm] :
( ( ord_less_eq_set_fm @ C5 @ A )
=> ( B
!= ( image_fm_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_742_subset__imageE,axiom,
! [B: set_fm,F: list_fm > fm,A: set_list_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_list_fm_fm @ F @ A ) )
=> ~ ! [C5: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ C5 @ A )
=> ( B
!= ( image_list_fm_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_743_subset__imageE,axiom,
! [B: set_fm,F: set_fm > fm,A: set_set_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_set_fm_fm @ F @ A ) )
=> ~ ! [C5: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ C5 @ A )
=> ( B
!= ( image_set_fm_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_744_subset__imageE,axiom,
! [B: set_list_fm,F: fm > list_fm,A: set_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_fm_list_fm @ F @ A ) )
=> ~ ! [C5: set_fm] :
( ( ord_less_eq_set_fm @ C5 @ A )
=> ( B
!= ( image_fm_list_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_745_subset__imageE,axiom,
! [B: set_set_fm,F: fm > set_fm,A: set_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_fm_set_fm @ F @ A ) )
=> ~ ! [C5: set_fm] :
( ( ord_less_eq_set_fm @ C5 @ A )
=> ( B
!= ( image_fm_set_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_746_subset__imageE,axiom,
! [B: set_list_fm,F: list_fm > list_fm,A: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_2959368623641186503ist_fm @ F @ A ) )
=> ~ ! [C5: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ C5 @ A )
=> ( B
!= ( image_2959368623641186503ist_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_747_subset__imageE,axiom,
! [B: set_list_fm,F: set_fm > list_fm,A: set_set_fm] :
( ( ord_le3173885194822255862ist_fm @ B @ ( image_set_fm_list_fm @ F @ A ) )
=> ~ ! [C5: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ C5 @ A )
=> ( B
!= ( image_set_fm_list_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_748_subset__imageE,axiom,
! [B: set_set_fm,F: list_fm > set_fm,A: set_list_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_list_fm_set_fm @ F @ A ) )
=> ~ ! [C5: set_list_fm] :
( ( ord_le3173885194822255862ist_fm @ C5 @ A )
=> ( B
!= ( image_list_fm_set_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_749_subset__imageE,axiom,
! [B: set_set_fm,F: set_fm > set_fm,A: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ B @ ( image_set_fm_set_fm @ F @ A ) )
=> ~ ! [C5: set_set_fm] :
( ( ord_le1461404734466536732set_fm @ C5 @ A )
=> ( B
!= ( image_set_fm_set_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_750_subset__imageE,axiom,
! [B: set_fm,F: produc1996495991257130529ist_fm > fm,A: set_Pr7058068377845519745ist_fm] :
( ( ord_less_eq_set_fm @ B @ ( image_3983738456748166956_fm_fm @ F @ A ) )
=> ~ ! [C5: set_Pr7058068377845519745ist_fm] :
( ( ord_le2055072402069232929ist_fm @ C5 @ A )
=> ( B
!= ( image_3983738456748166956_fm_fm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_751_image__subsetI,axiom,
! [A: set_fm,F: fm > fm,B: set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_752_image__subsetI,axiom,
! [A: set_list_fm,F: list_fm > fm,B: set_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A )
=> ( member_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_list_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_753_image__subsetI,axiom,
! [A: set_set_fm,F: set_fm > fm,B: set_fm] :
( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ A )
=> ( member_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_set_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_754_image__subsetI,axiom,
! [A: set_fm,F: fm > list_fm,B: set_list_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member_list_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_le3173885194822255862ist_fm @ ( image_fm_list_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_755_image__subsetI,axiom,
! [A: set_fm,F: fm > set_fm,B: set_set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member_set_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_le1461404734466536732set_fm @ ( image_fm_set_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_756_image__subsetI,axiom,
! [A: set_fm,F: fm > product_prod_fm_fm,B: set_Pr4463079037648049377_fm_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member8474499337054950954_fm_fm @ ( F @ X3 ) @ B ) )
=> ( ord_le7532831136298237057_fm_fm @ ( image_453549069922048106_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_757_image__subsetI,axiom,
! [A: set_fm,F: fm > set_set_fm,B: set_set_set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( member_set_set_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_le8219319623975992658set_fm @ ( image_fm_set_set_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_758_image__subsetI,axiom,
! [A: set_Pr4463079037648049377_fm_fm,F: product_prod_fm_fm > fm,B: set_fm] :
( ! [X3: product_prod_fm_fm] :
( ( member8474499337054950954_fm_fm @ X3 @ A )
=> ( member_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_5888559327631113100_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_759_image__subsetI,axiom,
! [A: set_set_set_fm,F: set_set_fm > fm,B: set_fm] :
( ! [X3: set_set_fm] :
( ( member_set_set_fm2 @ X3 @ A )
=> ( member_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_fm @ ( image_set_set_fm_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_760_image__subsetI,axiom,
! [A: set_list_fm,F: list_fm > list_fm,B: set_list_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A )
=> ( member_list_fm2 @ ( F @ X3 ) @ B ) )
=> ( ord_le3173885194822255862ist_fm @ ( image_2959368623641186503ist_fm @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_761_image__mono,axiom,
! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,F: produc1996495991257130529ist_fm > set_fm] :
( ( ord_le2055072402069232929ist_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( image_7490505654316676450set_fm @ F @ A ) @ ( image_7490505654316676450set_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_762_image__mono,axiom,
! [A: set_list_fm,B: set_list_fm,F: list_fm > fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ord_less_eq_set_fm @ ( image_list_fm_fm @ F @ A ) @ ( image_list_fm_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_763_image__mono,axiom,
! [A: set_list_fm,B: set_list_fm,F: list_fm > produc1996495991257130529ist_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ord_le2055072402069232929ist_fm @ ( image_1107023060279243642ist_fm @ F @ A ) @ ( image_1107023060279243642ist_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_764_image__mono,axiom,
! [A: set_list_fm,B: set_list_fm,F: list_fm > list_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ord_le3173885194822255862ist_fm @ ( image_2959368623641186503ist_fm @ F @ A ) @ ( image_2959368623641186503ist_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_765_image__mono,axiom,
! [A: set_list_fm,B: set_list_fm,F: list_fm > set_fm] :
( ( ord_le3173885194822255862ist_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( image_list_fm_set_fm @ F @ A ) @ ( image_list_fm_set_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_766_image__mono,axiom,
! [A: set_set_fm,B: set_set_fm,F: set_fm > fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ ( image_set_fm_fm @ F @ A ) @ ( image_set_fm_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_767_image__mono,axiom,
! [A: set_set_fm,B: set_set_fm,F: set_fm > produc1996495991257130529ist_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ord_le2055072402069232929ist_fm @ ( image_6232314896483315540ist_fm @ F @ A ) @ ( image_6232314896483315540ist_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_768_image__mono,axiom,
! [A: set_set_fm,B: set_set_fm,F: set_fm > list_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ord_le3173885194822255862ist_fm @ ( image_set_fm_list_fm @ F @ A ) @ ( image_set_fm_list_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_769_image__mono,axiom,
! [A: set_set_fm,B: set_set_fm,F: set_fm > set_fm] :
( ( ord_le1461404734466536732set_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( image_set_fm_set_fm @ F @ A ) @ ( image_set_fm_set_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_770_image__mono,axiom,
! [A: set_fm,B: set_fm,F: fm > fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A ) @ ( image_fm_fm @ F @ B ) ) ) ).
% image_mono
thf(fact_771_can__select__def,axiom,
( can_select_fm
= ( ^ [P4: fm > $o,A6: set_fm] :
? [X2: fm] :
( ( member_fm2 @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y4: fm] :
( ( ( member_fm2 @ Y4 @ A6 )
& ( P4 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_772_all__subset__image,axiom,
! [F: fm > fm,A: set_fm,P2: set_fm > $o] :
( ( ! [B6: set_fm] :
( ( ord_less_eq_set_fm @ B6 @ ( image_fm_fm @ F @ A ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_fm] :
( ( ord_less_eq_set_fm @ B6 @ A )
=> ( P2 @ ( image_fm_fm @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_773_in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_774_set__empty2,axiom,
! [Xs: list_fm] :
( ( bot_bot_set_fm
= ( set_fm2 @ Xs ) )
= ( Xs = nil_fm ) ) ).
% set_empty2
thf(fact_775_set__empty,axiom,
! [Xs: list_fm] :
( ( ( set_fm2 @ Xs )
= bot_bot_set_fm )
= ( Xs = nil_fm ) ) ).
% set_empty
thf(fact_776_all__not__in__conv,axiom,
! [A: set_fm] :
( ( ! [X2: fm] :
~ ( member_fm2 @ X2 @ A ) )
= ( A = bot_bot_set_fm ) ) ).
% all_not_in_conv
thf(fact_777_empty__iff,axiom,
! [C: fm] :
~ ( member_fm2 @ C @ bot_bot_set_fm ) ).
% empty_iff
thf(fact_778_subset__empty,axiom,
! [A: set_fm] :
( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
= ( A = bot_bot_set_fm ) ) ).
% subset_empty
thf(fact_779_empty__subsetI,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).
% empty_subsetI
thf(fact_780_ex__in__conv,axiom,
! [A: set_fm] :
( ( ? [X2: fm] : ( member_fm2 @ X2 @ A ) )
= ( A != bot_bot_set_fm ) ) ).
% ex_in_conv
thf(fact_781_equals0I,axiom,
! [A: set_fm] :
( ! [Y2: fm] :
~ ( member_fm2 @ Y2 @ A )
=> ( A = bot_bot_set_fm ) ) ).
% equals0I
thf(fact_782_equals0D,axiom,
! [A: set_fm,A2: fm] :
( ( A = bot_bot_set_fm )
=> ~ ( member_fm2 @ A2 @ A ) ) ).
% equals0D
thf(fact_783_emptyE,axiom,
! [A2: fm] :
~ ( member_fm2 @ A2 @ bot_bot_set_fm ) ).
% emptyE
thf(fact_784_bot_Oextremum,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).
% bot.extremum
thf(fact_785_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_786_bot_Oextremum__unique,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
= ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_unique
thf(fact_787_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_788_bot_Oextremum__uniqueI,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
=> ( A2 = bot_bot_set_fm ) ) ).
% bot.extremum_uniqueI
thf(fact_789_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_790_subset__emptyI,axiom,
! [A: set_fm] :
( ! [X3: fm] :
~ ( member_fm2 @ X3 @ A )
=> ( ord_less_eq_set_fm @ A @ bot_bot_set_fm ) ) ).
% subset_emptyI
thf(fact_791_empty__set,axiom,
( bot_bot_set_fm
= ( set_fm2 @ nil_fm ) ) ).
% empty_set
thf(fact_792_UNIV__I,axiom,
! [X: fm] : ( member_fm2 @ X @ top_top_set_fm ) ).
% UNIV_I
thf(fact_793_member__remove,axiom,
! [X: fm,Y3: fm,A: set_fm] :
( ( member_fm2 @ X @ ( remove_fm @ Y3 @ A ) )
= ( ( member_fm2 @ X @ A )
& ( X != Y3 ) ) ) ).
% member_remove
thf(fact_794_range__fst,axiom,
( ( image_6369949096775035324ist_fm @ produc1501393135466168645ist_fm @ top_to728902761071950545ist_fm )
= top_top_set_list_fm ) ).
% range_fst
thf(fact_795_range__fst,axiom,
( ( image_3223098899990291461ist_fm @ produc6879501374131015971m_rule @ top_to1603951219661559419m_rule )
= top_to728902761071950545ist_fm ) ).
% range_fst
thf(fact_796_range__snd,axiom,
( ( image_6369949096775035324ist_fm @ produc4588648349897876871ist_fm @ top_to728902761071950545ist_fm )
= top_top_set_list_fm ) ).
% range_snd
thf(fact_797_UNIV__eq__I,axiom,
! [A: set_fm] :
( ! [X3: fm] : ( member_fm2 @ X3 @ A )
=> ( top_top_set_fm = A ) ) ).
% UNIV_eq_I
thf(fact_798_UNIV__witness,axiom,
? [X3: fm] : ( member_fm2 @ X3 @ top_top_set_fm ) ).
% UNIV_witness
thf(fact_799_bot__empty__eq,axiom,
( bot_bot_fm_o
= ( ^ [X2: fm] : ( member_fm2 @ X2 @ bot_bot_set_fm ) ) ) ).
% bot_empty_eq
thf(fact_800_top_Oextremum__uniqueI,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ top_top_set_fm @ A2 )
=> ( A2 = top_top_set_fm ) ) ).
% top.extremum_uniqueI
thf(fact_801_top_Oextremum__unique,axiom,
! [A2: set_fm] :
( ( ord_less_eq_set_fm @ top_top_set_fm @ A2 )
= ( A2 = top_top_set_fm ) ) ).
% top.extremum_unique
thf(fact_802_top__greatest,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ top_top_set_fm ) ).
% top_greatest
thf(fact_803_subset__UNIV,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).
% subset_UNIV
thf(fact_804_Fpow__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( finite_Fpow_fm @ A ) @ ( finite_Fpow_fm @ B ) ) ) ).
% Fpow_mono
thf(fact_805_is__empty__set,axiom,
! [Xs: list_fm] :
( ( is_empty_fm @ ( set_fm2 @ Xs ) )
= ( null_fm @ Xs ) ) ).
% is_empty_set
thf(fact_806_the__elem__set,axiom,
! [X: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X @ nil_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_807_remove__code_I1_J,axiom,
! [X: fm,Xs: list_fm] :
( ( remove_fm @ X @ ( set_fm2 @ Xs ) )
= ( set_fm2 @ ( removeAll_fm @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_808_PowI,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( member_set_fm2 @ A @ ( pow_fm @ B ) ) ) ).
% PowI
thf(fact_809_Pow__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( member_set_fm2 @ A @ ( pow_fm @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ).
% Pow_iff
thf(fact_810_fst__map__prod,axiom,
! [F: list_fm > list_fm,G2: list_fm > list_fm,X: produc1996495991257130529ist_fm] :
( ( produc1501393135466168645ist_fm @ ( produc3421166768339006330ist_fm @ F @ G2 @ X ) )
= ( F @ ( produc1501393135466168645ist_fm @ X ) ) ) ).
% fst_map_prod
thf(fact_811_fst__map__prod,axiom,
! [F: produc1996495991257130529ist_fm > list_fm,G2: rule > list_fm,X: produc164195504107695125m_rule] :
( ( produc1501393135466168645ist_fm @ ( produc1965614570530558770ist_fm @ F @ G2 @ X ) )
= ( F @ ( produc6879501374131015971m_rule @ X ) ) ) ).
% fst_map_prod
thf(fact_812_fst__map__prod,axiom,
! [F: list_fm > produc1996495991257130529ist_fm,G2: list_fm > rule,X: produc1996495991257130529ist_fm] :
( ( produc6879501374131015971m_rule @ ( produc8124276167727720944m_rule @ F @ G2 @ X ) )
= ( F @ ( produc1501393135466168645ist_fm @ X ) ) ) ).
% fst_map_prod
thf(fact_813_fst__map__prod,axiom,
! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G2: rule > rule,X: produc164195504107695125m_rule] :
( ( produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F @ G2 @ X ) )
= ( F @ ( produc6879501374131015971m_rule @ X ) ) ) ).
% fst_map_prod
thf(fact_814_snd__map__prod,axiom,
! [F: list_fm > list_fm,G2: list_fm > list_fm,X: produc1996495991257130529ist_fm] :
( ( produc4588648349897876871ist_fm @ ( produc3421166768339006330ist_fm @ F @ G2 @ X ) )
= ( G2 @ ( produc4588648349897876871ist_fm @ X ) ) ) ).
% snd_map_prod
thf(fact_815_removeAll__id,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( removeAll_fm @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_816_asym__onI,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm] :
( ! [X3: fm,Y2: fm] :
( ( member_fm2 @ X3 @ A )
=> ( ( member_fm2 @ Y2 @ A )
=> ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y2 ) @ R2 )
=> ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y2 @ X3 ) @ R2 ) ) ) )
=> ( asym_on_fm @ A @ R2 ) ) ).
% asym_onI
thf(fact_817_Id__onI,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A2 @ A2 ) @ ( id_on_fm @ A ) ) ) ).
% Id_onI
thf(fact_818_not__in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= ( cons_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_819_top__empty__eq,axiom,
( top_top_fm_o
= ( ^ [X2: fm] : ( member_fm2 @ X2 @ top_top_set_fm ) ) ) ).
% top_empty_eq
thf(fact_820_set__ConsD,axiom,
! [Y3: fm,X: fm,Xs: list_fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_fm2 @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_821_list_Oset__cases,axiom,
! [E: fm,A2: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A2 ) )
=> ( ! [Z22: list_fm] :
( A2
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_822_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_823_list_Oset__intros_I2_J,axiom,
! [Y3: fm,X22: list_fm,X21: fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ X22 ) )
=> ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_824_PowD,axiom,
! [A: set_fm,B: set_fm] :
( ( member_set_fm2 @ A @ ( pow_fm @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% PowD
thf(fact_825_asym__onD,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm,X: fm,Y3: fm] :
( ( asym_on_fm @ A @ R2 )
=> ( ( member_fm2 @ X @ A )
=> ( ( member_fm2 @ Y3 @ A )
=> ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R2 )
=> ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y3 @ X ) @ R2 ) ) ) ) ) ).
% asym_onD
thf(fact_826_asym__on__subset,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm,B: set_fm] :
( ( asym_on_fm @ A @ R2 )
=> ( ( ord_less_eq_set_fm @ B @ A )
=> ( asym_on_fm @ B @ R2 ) ) ) ).
% asym_on_subset
thf(fact_827_Id__onE,axiom,
! [C: product_prod_fm_fm,A: set_fm] :
( ( member8474499337054950954_fm_fm @ C @ ( id_on_fm @ A ) )
=> ~ ! [X3: fm] :
( ( member_fm2 @ X3 @ A )
=> ( C
!= ( product_Pair_fm_fm @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_828_Id__on__eqI,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( A2 = B2 )
=> ( ( member_fm2 @ A2 @ A )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A2 @ B2 ) @ ( id_on_fm @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_829_Id__on__iff,axiom,
! [X: fm,Y3: fm,A: set_fm] :
( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ ( id_on_fm @ A ) )
= ( ( X = Y3 )
& ( member_fm2 @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_830_member__rec_I1_J,axiom,
! [X: fm,Xs: list_fm,Y3: fm] :
( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y3 )
= ( ( X = Y3 )
| ( member_fm @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_831_set__subset__Cons,axiom,
! [Xs: list_fm,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_832_Pow__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_le1461404734466536732set_fm @ ( pow_fm @ A ) @ ( pow_fm @ B ) ) ) ).
% Pow_mono
thf(fact_833_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm2 @ X2 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_834_insort__is__Cons,axiom,
! [Xs: list_fm,F: fm > nat,A2: fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ X3 ) ) )
=> ( ( linord8899536026354856477fm_nat @ F @ A2 @ Xs )
= ( cons_fm @ A2 @ Xs ) ) ) ).
% insort_is_Cons
thf(fact_835_Pow__set_I1_J,axiom,
( ( pow_fm @ ( set_fm2 @ nil_fm ) )
= ( insert_set_fm @ bot_bot_set_fm @ bot_bot_set_set_fm ) ) ).
% Pow_set(1)
thf(fact_836_insertCI,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( ~ ( member_fm2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_fm2 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_837_insert__iff,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ ( insert_fm2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_fm2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_838_singletonI,axiom,
! [A2: fm] : ( member_fm2 @ A2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ).
% singletonI
thf(fact_839_insert__subset,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( ( member_fm2 @ X @ B )
& ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% insert_subset
thf(fact_840_singleton__insert__inj__eq_H,axiom,
! [A2: fm,A: set_fm,B2: fm] :
( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B2 @ bot_bot_set_fm ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_841_singleton__insert__inj__eq,axiom,
! [B2: fm,A2: fm,A: set_fm] :
( ( ( insert_fm2 @ B2 @ bot_bot_set_fm )
= ( insert_fm2 @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_842_list_Osimps_I15_J,axiom,
! [X21: fm,X22: list_fm] :
( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
= ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_843_List_Oset__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( set_fm2 @ ( insert_fm @ X @ Xs ) )
= ( insert_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_844_insertE,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ ( insert_fm2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_fm2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_845_insertI1,axiom,
! [A2: fm,B: set_fm] : ( member_fm2 @ A2 @ ( insert_fm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_846_insertI2,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( member_fm2 @ A2 @ B )
=> ( member_fm2 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_847_Set_Oset__insert,axiom,
! [X: fm,A: set_fm] :
( ( member_fm2 @ X @ A )
=> ~ ! [B7: set_fm] :
( ( A
= ( insert_fm2 @ X @ B7 ) )
=> ( member_fm2 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_848_insert__ident,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X @ A )
=> ( ~ ( member_fm2 @ X @ B )
=> ( ( ( insert_fm2 @ X @ A )
= ( insert_fm2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_849_insert__absorb,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_850_insert__eq__iff,axiom,
! [A2: fm,A: set_fm,B2: fm,B: set_fm] :
( ~ ( member_fm2 @ A2 @ A )
=> ( ~ ( member_fm2 @ B2 @ B )
=> ( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C6: set_fm] :
( ( A
= ( insert_fm2 @ B2 @ C6 ) )
& ~ ( member_fm2 @ B2 @ C6 )
& ( B
= ( insert_fm2 @ A2 @ C6 ) )
& ~ ( member_fm2 @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_851_mk__disjoint__insert,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ? [B7: set_fm] :
( ( A
= ( insert_fm2 @ A2 @ B7 ) )
& ~ ( member_fm2 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_852_singletonD,axiom,
! [B2: fm,A2: fm] :
( ( member_fm2 @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_853_singleton__iff,axiom,
! [B2: fm,A2: fm] :
( ( member_fm2 @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_854_insert__subsetI,axiom,
! [X: fm,A: set_fm,X5: set_fm] :
( ( member_fm2 @ X @ A )
=> ( ( ord_less_eq_set_fm @ X5 @ A )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_855_subset__insertI2,axiom,
! [A: set_fm,B: set_fm,B2: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_856_subset__insertI,axiom,
! [B: set_fm,A2: fm] : ( ord_less_eq_set_fm @ B @ ( insert_fm2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_857_subset__insert,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X @ A )
=> ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% subset_insert
thf(fact_858_insert__mono,axiom,
! [C3: set_fm,D: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ C3 @ D )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ A2 @ C3 ) @ ( insert_fm2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_859_subset__singleton__iff,axiom,
! [X5: set_fm,A2: fm] :
( ( ord_less_eq_set_fm @ X5 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
= ( ( X5 = bot_bot_set_fm )
| ( X5
= ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ) ).
% subset_singleton_iff
thf(fact_860_subset__singletonD,axiom,
! [A: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
=> ( ( A = bot_bot_set_fm )
| ( A
= ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ) ).
% subset_singletonD
thf(fact_861_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_862_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_863_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_864_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_865_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P2: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_866_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_867_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_868_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_869_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_870_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_871_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_872_order__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_873_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_874_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_875_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_876_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_877_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_878_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_879_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_880_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_881_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_882_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_883_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_884_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_885_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_886_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_887_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_888_less__induct,axiom,
! [P2: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A2 ) ) ).
% less_induct
thf(fact_889_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_890_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_891_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_892_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_893_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_894_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_895_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_896_leD,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ~ ( ord_less_set_fm @ X @ Y3 ) ) ).
% leD
thf(fact_897_leD,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_nat @ X @ Y3 ) ) ).
% leD
thf(fact_898_leI,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% leI
thf(fact_899_nless__le,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ~ ( ord_less_set_fm @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_fm @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_900_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_901_antisym__conv1,axiom,
! [X: set_fm,Y3: set_fm] :
( ~ ( ord_less_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_902_antisym__conv1,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_903_antisym__conv2,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ~ ( ord_less_set_fm @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_904_antisym__conv2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_905_less__le__not__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ~ ( ord_less_eq_set_fm @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_906_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_907_not__le__imp__less,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X )
=> ( ord_less_nat @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_908_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
( ( ord_less_set_fm @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_909_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_nat @ A3 @ B5 )
| ( A3 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_910_order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_911_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( A3 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_912_order_Ostrict__trans1,axiom,
! [A2: set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_set_fm @ B2 @ C )
=> ( ord_less_set_fm @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_913_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_914_order_Ostrict__trans2,axiom,
! [A2: set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ord_less_set_fm @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_915_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_916_order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A3 @ B5 )
& ~ ( ord_less_eq_set_fm @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_917_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_918_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A3: set_fm] :
( ( ord_less_set_fm @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_919_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_nat @ B5 @ A3 )
| ( A3 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_920_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [B5: set_fm,A3: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_921_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( A3 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_922_dual__order_Ostrict__trans1,axiom,
! [B2: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ( ord_less_set_fm @ C @ B2 )
=> ( ord_less_set_fm @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_923_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_924_dual__order_Ostrict__trans2,axiom,
! [B2: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_set_fm @ C @ B2 )
=> ( ord_less_set_fm @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_925_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_926_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [B5: set_fm,A3: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A3 )
& ~ ( ord_less_eq_set_fm @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_927_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_928_order_Ostrict__implies__order,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ord_less_set_fm @ A2 @ B2 )
=> ( ord_less_eq_set_fm @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_929_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_930_dual__order_Ostrict__implies__order,axiom,
! [B2: set_fm,A2: set_fm] :
( ( ord_less_set_fm @ B2 @ A2 )
=> ( ord_less_eq_set_fm @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_931_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_932_order__le__less,axiom,
( ord_less_eq_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_set_fm @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_933_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_934_order__less__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_935_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_936_linorder__not__le,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
= ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_937_linorder__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_938_order__less__imp__le,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_set_fm @ X @ Y3 )
=> ( ord_less_eq_set_fm @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_939_order__less__imp__le,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_940_order__le__neq__trans,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_fm @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_941_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_942_order__neq__le__trans,axiom,
! [A2: set_fm,B2: set_fm] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ord_less_set_fm @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_943_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_944_order__le__less__trans,axiom,
! [X: set_fm,Y3: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_set_fm @ Y3 @ Z2 )
=> ( ord_less_set_fm @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_945_order__le__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_946_order__less__le__trans,axiom,
! [X: set_fm,Y3: set_fm,Z2: set_fm] :
( ( ord_less_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ Z2 )
=> ( ord_less_set_fm @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_947_order__less__le__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_948_order__le__less__subst1,axiom,
! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_949_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_950_order__le__less__subst2,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_951_order__le__less__subst2,axiom,
! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_952_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_953_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_954_order__less__le__subst1,axiom,
! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
( ( ord_less_set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_955_order__less__le__subst1,axiom,
! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_956_order__less__le__subst1,axiom,
! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
( ( ord_less_set_fm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_957_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_958_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_959_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_960_linorder__le__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_961_order__le__imp__less__or__eq,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_set_fm @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_962_order__le__imp__less__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_963_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_964_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_965_nat__descend__induct,axiom,
! [N2: nat,P2: nat > $o,M3: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K @ I2 )
=> ( P2 @ I2 ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_966_is__singletonI_H,axiom,
! [A: set_fm] :
( ( A != bot_bot_set_fm )
=> ( ! [X3: fm,Y2: fm] :
( ( member_fm2 @ X3 @ A )
=> ( ( member_fm2 @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_fm @ A ) ) ) ).
% is_singletonI'
thf(fact_967_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_968_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_969_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_970_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_971_psubsetI,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_fm @ A @ B ) ) ) ).
% psubsetI
thf(fact_972_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
( ( ord_less_set_fm @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_973_subset__psubset__trans,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_set_fm @ B @ C3 )
=> ( ord_less_set_fm @ A @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_974_subset__not__subset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B6 )
& ~ ( ord_less_eq_set_fm @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_975_psubset__subset__trans,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C3 )
=> ( ord_less_set_fm @ A @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_976_psubset__imp__subset,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_977_psubset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_978_psubsetE,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ~ ( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ B @ A ) ) ) ).
% psubsetE
thf(fact_979_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_980_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_981_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_982_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_983_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_984_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_985_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_986_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_987_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_988_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_989_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_990_complete__interval,axiom,
! [A2: nat,B2: nat,P2: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P2 @ X6 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_991_pinf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% pinf(6)
thf(fact_992_pinf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% pinf(8)
thf(fact_993_minf_I6_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T2 ) ) ).
% minf(6)
thf(fact_994_psubsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_995_minf_I8_J,axiom,
! [T2: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T2 @ X6 ) ) ).
% minf(8)
thf(fact_996_set__removeAll,axiom,
! [X: fm,Xs: list_fm] :
( ( set_fm2 @ ( removeAll_fm @ X @ Xs ) )
= ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) ) ).
% set_removeAll
thf(fact_997_DiffI,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ A )
=> ( ~ ( member_fm2 @ C @ B )
=> ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) ) ) ) ).
% DiffI
thf(fact_998_Diff__iff,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) )
= ( ( member_fm2 @ C @ A )
& ~ ( member_fm2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_999_insert__Diff1,axiom,
! [X: fm,B: set_fm,A: set_fm] :
( ( member_fm2 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1000_Diff__insert0,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm2 @ X @ A )
=> ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( minus_minus_set_fm @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1001_Diff__eq__empty__iff,axiom,
! [A: set_fm,B: set_fm] :
( ( ( minus_minus_set_fm @ A @ B )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1002_psubset__imp__ex__mem,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ? [B4: fm] : ( member_fm2 @ B4 @ ( minus_minus_set_fm @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1003_prod_Omap__ident__strong,axiom,
! [T2: product_prod_fm_fm,F12: fm > fm,F23: fm > fm] :
( ! [Z1: fm] :
( ( member_fm2 @ Z1 @ ( basic_fsts_fm_fm @ T2 ) )
=> ( ( F12 @ Z1 )
= Z1 ) )
=> ( ! [Z22: fm] :
( ( member_fm2 @ Z22 @ ( basic_snds_fm_fm @ T2 ) )
=> ( ( F23 @ Z22 )
= Z22 ) )
=> ( ( produc8016985811614767674_fm_fm @ F12 @ F23 @ T2 )
= T2 ) ) ) ).
% prod.map_ident_strong
thf(fact_1004_DiffE,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ~ ( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% DiffE
thf(fact_1005_DiffD1,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ( member_fm2 @ C @ A ) ) ).
% DiffD1
thf(fact_1006_DiffD2,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( minus_minus_set_fm @ A @ B ) )
=> ~ ( member_fm2 @ C @ B ) ) ).
% DiffD2
thf(fact_1007_insert__Diff__if,axiom,
! [X: fm,B: set_fm,A: set_fm] :
( ( ( member_fm2 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( minus_minus_set_fm @ A @ B ) ) )
& ( ~ ( member_fm2 @ X @ B )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( insert_fm2 @ X @ ( minus_minus_set_fm @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1008_Diff__mono,axiom,
! [A: set_fm,C3: set_fm,D: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ C3 )
=> ( ( ord_less_eq_set_fm @ D @ B )
=> ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ ( minus_minus_set_fm @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1009_Diff__subset,axiom,
! [A: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1010_double__diff,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C3 )
=> ( ( minus_minus_set_fm @ B @ ( minus_minus_set_fm @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1011_fsts_Ocases,axiom,
! [A2: list_fm,P: produc1996495991257130529ist_fm] :
( ( member_list_fm2 @ A2 @ ( basic_2778179344159109705ist_fm @ P ) )
=> ( A2
= ( produc1501393135466168645ist_fm @ P ) ) ) ).
% fsts.cases
thf(fact_1012_fsts_Ocases,axiom,
! [A2: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
( ( member8102475879199740618ist_fm @ A2 @ ( basic_4306050201691295519m_rule @ P ) )
=> ( A2
= ( produc6879501374131015971m_rule @ P ) ) ) ).
% fsts.cases
thf(fact_1013_fsts_Osimps,axiom,
! [A2: list_fm,P: produc1996495991257130529ist_fm] :
( ( member_list_fm2 @ A2 @ ( basic_2778179344159109705ist_fm @ P ) )
= ( A2
= ( produc1501393135466168645ist_fm @ P ) ) ) ).
% fsts.simps
thf(fact_1014_fsts_Osimps,axiom,
! [A2: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
( ( member8102475879199740618ist_fm @ A2 @ ( basic_4306050201691295519m_rule @ P ) )
= ( A2
= ( produc6879501374131015971m_rule @ P ) ) ) ).
% fsts.simps
thf(fact_1015_fsts_Ointros,axiom,
! [P: produc1996495991257130529ist_fm] : ( member_list_fm2 @ ( produc1501393135466168645ist_fm @ P ) @ ( basic_2778179344159109705ist_fm @ P ) ) ).
% fsts.intros
thf(fact_1016_fsts_Ointros,axiom,
! [P: produc164195504107695125m_rule] : ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ P ) @ ( basic_4306050201691295519m_rule @ P ) ) ).
% fsts.intros
thf(fact_1017_snds_Ointros,axiom,
! [P: produc1996495991257130529ist_fm] : ( member_list_fm2 @ ( produc4588648349897876871ist_fm @ P ) @ ( basic_6053177629474610823ist_fm @ P ) ) ).
% snds.intros
thf(fact_1018_snds_Osimps,axiom,
! [A2: list_fm,P: produc1996495991257130529ist_fm] :
( ( member_list_fm2 @ A2 @ ( basic_6053177629474610823ist_fm @ P ) )
= ( A2
= ( produc4588648349897876871ist_fm @ P ) ) ) ).
% snds.simps
thf(fact_1019_snds_Ocases,axiom,
! [A2: list_fm,P: produc1996495991257130529ist_fm] :
( ( member_list_fm2 @ A2 @ ( basic_6053177629474610823ist_fm @ P ) )
=> ( A2
= ( produc4588648349897876871ist_fm @ P ) ) ) ).
% snds.cases
thf(fact_1020_Diff__insert__absorb,axiom,
! [X: fm,A: set_fm] :
( ~ ( member_fm2 @ X @ A )
=> ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1021_insert__Diff,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm2 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1022_subset__Diff__insert,axiom,
! [A: set_fm,B: set_fm,X: fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ ( insert_fm2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ C3 ) )
& ~ ( member_fm2 @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1023_subset__insert__iff,axiom,
! [A: set_fm,X: fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ( ( member_fm2 @ X @ A )
=> ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B ) )
& ( ~ ( member_fm2 @ X @ A )
=> ( ord_less_eq_set_fm @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1024_Diff__single__insert,axiom,
! [A: set_fm,X: fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B )
=> ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1025_subset__subseqs,axiom,
! [X5: set_fm,Xs: list_fm] :
( ( ord_less_eq_set_fm @ X5 @ ( set_fm2 @ Xs ) )
=> ( member_set_fm2 @ X5 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1026_in__image__insert__iff,axiom,
! [B: set_set_fm,X: fm,A: set_fm] :
( ! [C5: set_fm] :
( ( member_set_fm2 @ C5 @ B )
=> ~ ( member_fm2 @ X @ C5 ) )
=> ( ( member_set_fm2 @ A @ ( image_set_fm_set_fm @ ( insert_fm2 @ X ) @ B ) )
= ( ( member_fm2 @ X @ A )
& ( member_set_fm2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1027_subseqs__powset,axiom,
! [Xs: list_fm] :
( ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
= ( pow_fm @ ( set_fm2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_1028_prod__set__defs_I1_J,axiom,
( basic_2778179344159109705ist_fm
= ( ^ [P3: produc1996495991257130529ist_fm] : ( insert_list_fm @ ( produc1501393135466168645ist_fm @ P3 ) @ bot_bot_set_list_fm ) ) ) ).
% prod_set_defs(1)
thf(fact_1029_prod__set__defs_I1_J,axiom,
( basic_4306050201691295519m_rule
= ( ^ [P3: produc164195504107695125m_rule] : ( insert589278167076416497ist_fm @ ( produc6879501374131015971m_rule @ P3 ) @ bot_bo4530011010606209517ist_fm ) ) ) ).
% prod_set_defs(1)
thf(fact_1030_prod__set__defs_I2_J,axiom,
( basic_6053177629474610823ist_fm
= ( ^ [P3: produc1996495991257130529ist_fm] : ( insert_list_fm @ ( produc4588648349897876871ist_fm @ P3 ) @ bot_bot_set_list_fm ) ) ) ).
% prod_set_defs(2)
thf(fact_1031_psubset__insert__iff,axiom,
! [A: set_fm,X: fm,B: set_fm] :
( ( ord_less_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ( ( member_fm2 @ X @ B )
=> ( ord_less_set_fm @ A @ B ) )
& ( ~ ( member_fm2 @ X @ B )
=> ( ( ( member_fm2 @ X @ A )
=> ( ord_less_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B ) )
& ( ~ ( member_fm2 @ X @ A )
=> ( ord_less_eq_set_fm @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1032_diff__shunt__var,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ( minus_minus_set_fm @ X @ Y3 )
= bot_bot_set_fm )
= ( ord_less_eq_set_fm @ X @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1033_diff__diff__cancel,axiom,
! [I3: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1034_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1035_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1036_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_1037_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1038_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1039_le__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1040_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1041_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1042_less__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1043_top_Oordering__top__axioms,axiom,
ordering_top_set_fm @ ord_less_eq_set_fm @ ord_less_set_fm @ top_top_set_fm ).
% top.ordering_top_axioms
thf(fact_1044_lexord__partial__trans,axiom,
! [Xs: list_fm,R2: set_Pr4463079037648049377_fm_fm,Ys: list_fm,Zs: list_fm] :
( ! [X3: fm,Y2: fm,Z3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y2 ) @ R2 )
=> ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y2 @ Z3 ) @ R2 )
=> ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Z3 ) @ R2 ) ) ) )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys ) @ ( lexord_fm @ R2 ) )
=> ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R2 ) )
=> ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Zs ) @ ( lexord_fm @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1045_length__removeAll__less,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_fm @ ( removeAll_fm @ X @ Xs ) ) @ ( size_size_list_fm @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1046_shift__snth__ge,axiom,
! [Xs: list_P8332459875677254181m_rule,P: nat,S: stream727092118206550309m_rule] :
( ( ord_less_eq_nat @ ( size_s9206449984754302649m_rule @ Xs ) @ P )
=> ( ( snth_P1712579846118164081m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) @ P )
= ( snth_P1712579846118164081m_rule @ S @ ( minus_minus_nat @ P @ ( size_s9206449984754302649m_rule @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_1047_Un__iff,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( sup_sup_set_fm @ A @ B ) )
= ( ( member_fm2 @ C @ A )
| ( member_fm2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_1048_UnCI,axiom,
! [C: fm,B: set_fm,A: set_fm] :
( ( ~ ( member_fm2 @ C @ B )
=> ( member_fm2 @ C @ A ) )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnCI
thf(fact_1049_Un__subset__iff,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B ) @ C3 )
= ( ( ord_less_eq_set_fm @ A @ C3 )
& ( ord_less_eq_set_fm @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_1050_UnI2,axiom,
! [C: fm,B: set_fm,A: set_fm] :
( ( member_fm2 @ C @ B )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnI2
thf(fact_1051_UnI1,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A @ B ) ) ) ).
% UnI1
thf(fact_1052_UnE,axiom,
! [C: fm,A: set_fm,B: set_fm] :
( ( member_fm2 @ C @ ( sup_sup_set_fm @ A @ B ) )
=> ( ~ ( member_fm2 @ C @ A )
=> ( member_fm2 @ C @ B ) ) ) ).
% UnE
thf(fact_1053_Un__mono,axiom,
! [A: set_fm,C3: set_fm,B: set_fm,D: set_fm] :
( ( ord_less_eq_set_fm @ A @ C3 )
=> ( ( ord_less_eq_set_fm @ B @ D )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B ) @ ( sup_sup_set_fm @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_1054_Un__least,axiom,
! [A: set_fm,C3: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ C3 )
=> ( ( ord_less_eq_set_fm @ B @ C3 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_1055_Un__upper1,axiom,
! [A: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ A @ ( sup_sup_set_fm @ A @ B ) ) ).
% Un_upper1
thf(fact_1056_Un__upper2,axiom,
! [B: set_fm,A: set_fm] : ( ord_less_eq_set_fm @ B @ ( sup_sup_set_fm @ A @ B ) ) ).
% Un_upper2
thf(fact_1057_Un__absorb1,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( sup_sup_set_fm @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1058_Un__absorb2,axiom,
! [B: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ A )
=> ( ( sup_sup_set_fm @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1059_subset__UnE,axiom,
! [C3: set_fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ C3 @ ( sup_sup_set_fm @ A @ B ) )
=> ~ ! [A7: set_fm] :
( ( ord_less_eq_set_fm @ A7 @ A )
=> ! [B8: set_fm] :
( ( ord_less_eq_set_fm @ B8 @ B )
=> ( C3
!= ( sup_sup_set_fm @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1060_subset__Un__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B6: set_fm] :
( ( sup_sup_set_fm @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_1061_Diff__partition,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( sup_sup_set_fm @ A @ ( minus_minus_set_fm @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1062_Diff__subset__conv,axiom,
! [A: set_fm,B: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ C3 )
= ( ord_less_eq_set_fm @ A @ ( sup_sup_set_fm @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1063_set__shuffles,axiom,
! [Zs: list_fm,Xs: list_fm,Ys: list_fm] :
( ( member_list_fm2 @ Zs @ ( shuffles_fm @ Xs @ Ys ) )
=> ( ( set_fm2 @ Zs )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1064_le__sup__iff,axiom,
! [X: set_fm,Y3: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ X @ Y3 ) @ Z2 )
= ( ( ord_less_eq_set_fm @ X @ Z2 )
& ( ord_less_eq_set_fm @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1065_le__sup__iff,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y3 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1066_sup_Obounded__iff,axiom,
! [B2: set_fm,C: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_fm @ B2 @ A2 )
& ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1067_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1068_set__union,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( union_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_1069_sup_OcoboundedI2,axiom,
! [C: set_fm,B2: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ C @ B2 )
=> ( ord_less_eq_set_fm @ C @ ( sup_sup_set_fm @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1070_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1071_sup_OcoboundedI1,axiom,
! [C: set_fm,A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ C @ A2 )
=> ( ord_less_eq_set_fm @ C @ ( sup_sup_set_fm @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1072_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1073_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B5: set_fm] :
( ( sup_sup_set_fm @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1074_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1075_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A3: set_fm] :
( ( sup_sup_set_fm @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1076_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1077_sup_Ocobounded2,axiom,
! [B2: set_fm,A2: set_fm] : ( ord_less_eq_set_fm @ B2 @ ( sup_sup_set_fm @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1078_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_1079_sup_Ocobounded1,axiom,
! [A2: set_fm,B2: set_fm] : ( ord_less_eq_set_fm @ A2 @ ( sup_sup_set_fm @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1080_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_1081_sup_Oorder__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A3: set_fm] :
( A3
= ( sup_sup_set_fm @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1082_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1083_sup_OboundedI,axiom,
! [B2: set_fm,A2: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ( ord_less_eq_set_fm @ C @ A2 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1084_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1085_sup_OboundedE,axiom,
! [B2: set_fm,C: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ~ ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1086_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1087_sup__absorb2,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( sup_sup_set_fm @ X @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_1088_sup__absorb2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( sup_sup_nat @ X @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_1089_sup__absorb1,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( ( sup_sup_set_fm @ X @ Y3 )
= X ) ) ).
% sup_absorb1
thf(fact_1090_sup__absorb1,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( sup_sup_nat @ X @ Y3 )
= X ) ) ).
% sup_absorb1
thf(fact_1091_sup_Oabsorb2,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( sup_sup_set_fm @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1092_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1093_sup_Oabsorb1,axiom,
! [B2: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( ( sup_sup_set_fm @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1094_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1095_sup__unique,axiom,
! [F: set_fm > set_fm > set_fm,X: set_fm,Y3: set_fm] :
( ! [X3: set_fm,Y2: set_fm] : ( ord_less_eq_set_fm @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_fm,Y2: set_fm] : ( ord_less_eq_set_fm @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_fm,Y2: set_fm,Z3: set_fm] :
( ( ord_less_eq_set_fm @ Y2 @ X3 )
=> ( ( ord_less_eq_set_fm @ Z3 @ X3 )
=> ( ord_less_eq_set_fm @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_fm @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_1096_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y3: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_1097_sup_OorderI,axiom,
! [A2: set_fm,B2: set_fm] :
( ( A2
= ( sup_sup_set_fm @ A2 @ B2 ) )
=> ( ord_less_eq_set_fm @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_1098_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_1099_sup_OorderE,axiom,
! [B2: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_fm @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1100_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_1101_le__iff__sup,axiom,
( ord_less_eq_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( sup_sup_set_fm @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1102_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1103_sup__least,axiom,
! [Y3: set_fm,X: set_fm,Z2: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( ( ord_less_eq_set_fm @ Z2 @ X )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ Y3 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1104_sup__least,axiom,
! [Y3: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1105_sup__mono,axiom,
! [A2: set_fm,C: set_fm,B2: set_fm,D3: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ C )
=> ( ( ord_less_eq_set_fm @ B2 @ D3 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B2 ) @ ( sup_sup_set_fm @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_1106_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D3 ) ) ) ) ).
% sup_mono
thf(fact_1107_sup_Omono,axiom,
! [C: set_fm,A2: set_fm,D3: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ C @ A2 )
=> ( ( ord_less_eq_set_fm @ D3 @ B2 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ C @ D3 ) @ ( sup_sup_set_fm @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1108_sup_Omono,axiom,
! [C: nat,A2: nat,D3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D3 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D3 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1109_le__supI2,axiom,
! [X: set_fm,B2: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ X @ B2 )
=> ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_1110_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_1111_le__supI1,axiom,
! [X: set_fm,A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ X @ A2 )
=> ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_1112_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_1113_sup__ge2,axiom,
! [Y3: set_fm,X: set_fm] : ( ord_less_eq_set_fm @ Y3 @ ( sup_sup_set_fm @ X @ Y3 ) ) ).
% sup_ge2
thf(fact_1114_sup__ge2,axiom,
! [Y3: nat,X: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X @ Y3 ) ) ).
% sup_ge2
thf(fact_1115_sup__ge1,axiom,
! [X: set_fm,Y3: set_fm] : ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ X @ Y3 ) ) ).
% sup_ge1
thf(fact_1116_sup__ge1,axiom,
! [X: nat,Y3: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y3 ) ) ).
% sup_ge1
thf(fact_1117_le__supI,axiom,
! [A2: set_fm,X: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ X )
=> ( ( ord_less_eq_set_fm @ B2 @ X )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1118_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1119_le__supE,axiom,
! [A2: set_fm,B2: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_fm @ A2 @ X )
=> ~ ( ord_less_eq_set_fm @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1120_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1121_inf__sup__ord_I3_J,axiom,
! [X: set_fm,Y3: set_fm] : ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ X @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_1122_inf__sup__ord_I3_J,axiom,
! [X: nat,Y3: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_1123_inf__sup__ord_I4_J,axiom,
! [Y3: set_fm,X: set_fm] : ( ord_less_eq_set_fm @ Y3 @ ( sup_sup_set_fm @ X @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_1124_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_1125_chains__extend,axiom,
! [C: set_set_fm,S6: set_set_fm,Z2: set_fm] :
( ( member_set_set_fm2 @ C @ ( chains_fm @ S6 ) )
=> ( ( member_set_fm2 @ Z2 @ S6 )
=> ( ! [X3: set_fm] :
( ( member_set_fm2 @ X3 @ C )
=> ( ord_less_eq_set_fm @ X3 @ Z2 ) )
=> ( member_set_set_fm2 @ ( sup_sup_set_set_fm @ ( insert_set_fm @ Z2 @ bot_bot_set_set_fm ) @ C ) @ ( chains_fm @ S6 ) ) ) ) ) ).
% chains_extend
thf(fact_1126_sdrop__cycle__eq,axiom,
! [U: list_P8332459875677254181m_rule] :
( ( U != nil_Pr474642538098133391m_rule )
=> ( ( sdrop_7224736112439592940m_rule @ ( size_s9206449984754302649m_rule @ U ) @ ( cycle_1017677951696034864m_rule @ U ) )
= ( cycle_1017677951696034864m_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_1127_sdrop__cycle__eq,axiom,
! [U: list_P5616295576739893671ist_fm] :
( ( U != nil_Pr6600767949821390631ist_fm )
=> ( ( sdrop_4442373711808556042ist_fm @ ( size_s2722402132374190611ist_fm @ U ) @ ( cycle_271291504775723206ist_fm @ U ) )
= ( cycle_271291504775723206ist_fm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_1128_sdrop__cycle__eq,axiom,
! [U: list_rule] :
( ( U != nil_rule )
=> ( ( sdrop_rule @ ( size_size_list_rule @ U ) @ ( cycle_rule @ U ) )
= ( cycle_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_1129_sdrop__cycle__eq,axiom,
! [U: list_P8801850819564216977m_rule] :
( ( U != nil_Pr3052741188495898641m_rule )
=> ( ( sdrop_7373388980841566196m_rule @ ( size_s8174888084367447805m_rule @ U ) @ ( cycle_6612116590585936048m_rule @ U ) )
= ( cycle_6612116590585936048m_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_1130_Cons__in__lists__iff,axiom,
! [X: fm,Xs: list_fm,A: set_fm] :
( ( member_list_fm2 @ ( cons_fm @ X @ Xs ) @ ( lists_fm @ A ) )
= ( ( member_fm2 @ X @ A )
& ( member_list_fm2 @ Xs @ ( lists_fm @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1131_in__listsI,axiom,
! [Xs: list_fm,A: set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X3 @ A ) )
=> ( member_list_fm2 @ Xs @ ( lists_fm @ A ) ) ) ).
% in_listsI
thf(fact_1132_lists_OCons,axiom,
! [A2: fm,A: set_fm,L: list_fm] :
( ( member_fm2 @ A2 @ A )
=> ( ( member_list_fm2 @ L @ ( lists_fm @ A ) )
=> ( member_list_fm2 @ ( cons_fm @ A2 @ L ) @ ( lists_fm @ A ) ) ) ) ).
% lists.Cons
thf(fact_1133_listsE,axiom,
! [X: fm,L: list_fm,A: set_fm] :
( ( member_list_fm2 @ ( cons_fm @ X @ L ) @ ( lists_fm @ A ) )
=> ~ ( ( member_fm2 @ X @ A )
=> ~ ( member_list_fm2 @ L @ ( lists_fm @ A ) ) ) ) ).
% listsE
thf(fact_1134_in__lists__conv__set,axiom,
! [Xs: list_fm,A: set_fm] :
( ( member_list_fm2 @ Xs @ ( lists_fm @ A ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X2 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1135_in__listsD,axiom,
! [Xs: list_fm,A: set_fm] :
( ( member_list_fm2 @ Xs @ ( lists_fm @ A ) )
=> ! [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X6 @ A ) ) ) ).
% in_listsD
thf(fact_1136_Zorn__Lemma2,axiom,
! [A: set_set_fm] :
( ! [X3: set_set_fm] :
( ( member_set_set_fm2 @ X3 @ ( chains_fm @ A ) )
=> ? [Xa2: set_fm] :
( ( member_set_fm2 @ Xa2 @ A )
& ! [Xb: set_fm] :
( ( member_set_fm2 @ Xb @ X3 )
=> ( ord_less_eq_set_fm @ Xb @ Xa2 ) ) ) )
=> ? [X3: set_fm] :
( ( member_set_fm2 @ X3 @ A )
& ! [Xa2: set_fm] :
( ( member_set_fm2 @ Xa2 @ A )
=> ( ( ord_less_eq_set_fm @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1137_chainsD,axiom,
! [C: set_set_fm,S6: set_set_fm,X: set_fm,Y3: set_fm] :
( ( member_set_set_fm2 @ C @ ( chains_fm @ S6 ) )
=> ( ( member_set_fm2 @ X @ C )
=> ( ( member_set_fm2 @ Y3 @ C )
=> ( ( ord_less_eq_set_fm @ X @ Y3 )
| ( ord_less_eq_set_fm @ Y3 @ X ) ) ) ) ) ).
% chainsD
thf(fact_1138_lists_Osimps,axiom,
! [A2: list_fm,A: set_fm] :
( ( member_list_fm2 @ A2 @ ( lists_fm @ A ) )
= ( ( A2 = nil_fm )
| ? [A3: fm,L2: list_fm] :
( ( A2
= ( cons_fm @ A3 @ L2 ) )
& ( member_fm2 @ A3 @ A )
& ( member_list_fm2 @ L2 @ ( lists_fm @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1139_lists_Ocases,axiom,
! [A2: list_fm,A: set_fm] :
( ( member_list_fm2 @ A2 @ ( lists_fm @ A ) )
=> ( ( A2 != nil_fm )
=> ~ ! [A5: fm,L3: list_fm] :
( ( A2
= ( cons_fm @ A5 @ L3 ) )
=> ( ( member_fm2 @ A5 @ A )
=> ~ ( member_list_fm2 @ L3 @ ( lists_fm @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1140_lists__mono,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_le3173885194822255862ist_fm @ ( lists_fm @ A ) @ ( lists_fm @ B ) ) ) ).
% lists_mono
thf(fact_1141_partition__set,axiom,
! [P2: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P2 @ Xs )
= ( produc381145313068854617ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_fm @ ( set_fm2 @ Yes ) @ ( set_fm2 @ No ) )
= ( set_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_1142_shift__snth,axiom,
! [N2: nat,Xs: list_P8332459875677254181m_rule,S: stream727092118206550309m_rule] :
( ( ( ord_less_nat @ N2 @ ( size_s9206449984754302649m_rule @ Xs ) )
=> ( ( snth_P1712579846118164081m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) @ N2 )
= ( nth_Pr8225284241310773582m_rule @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s9206449984754302649m_rule @ Xs ) )
=> ( ( snth_P1712579846118164081m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) @ N2 )
= ( snth_P1712579846118164081m_rule @ S @ ( minus_minus_nat @ N2 @ ( size_s9206449984754302649m_rule @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_1143_shift__snth__less,axiom,
! [P: nat,Xs: list_P8332459875677254181m_rule,S: stream727092118206550309m_rule] :
( ( ord_less_nat @ P @ ( size_s9206449984754302649m_rule @ Xs ) )
=> ( ( snth_P1712579846118164081m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) @ P )
= ( nth_Pr8225284241310773582m_rule @ Xs @ P ) ) ) ).
% shift_snth_less
thf(fact_1144_nth__mem,axiom,
! [N2: nat,Xs: list_fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( member_fm2 @ ( nth_fm @ Xs @ N2 ) @ ( set_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1145_list__ball__nth,axiom,
! [N2: nat,Xs: list_fm,P2: fm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1146_in__set__conv__nth,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1147_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P2: fm > $o,X: fm] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
=> ( P2 @ ( nth_fm @ Xs @ I4 ) ) )
=> ( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1148_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
=> ( P2 @ ( nth_fm @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1149_partition__P,axiom,
! [P2: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P2 @ Xs )
= ( produc381145313068854617ist_fm @ Yes @ No ) )
=> ( ! [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Yes ) )
=> ( P2 @ X6 ) )
& ! [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ No ) )
=> ~ ( P2 @ X6 ) ) ) ) ).
% partition_P
thf(fact_1150_nth__equal__first__eq,axiom,
! [X: fm,Xs: list_fm,N2: nat] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ( ( nth_fm @ ( cons_fm @ X @ Xs ) @ N2 )
= X )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1151_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1152_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1153_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1154_diff__is__0__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1155_diff__is__0__eq_H,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( minus_minus_nat @ M3 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1156_sdrop_Osimps_I1_J,axiom,
! [S: stream727092118206550309m_rule] :
( ( sdrop_7224736112439592940m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_1157_sdrop_Osimps_I1_J,axiom,
! [S: stream8299795917829157543ist_fm] :
( ( sdrop_4442373711808556042ist_fm @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_1158_sdrop_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( sdrop_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_1159_sdrop_Osimps_I1_J,axiom,
! [S: stream6494289010434245521m_rule] :
( ( sdrop_7373388980841566196m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_1160_in__set__takeD,axiom,
! [X: fm,N2: nat,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) )
=> ( member_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1161_set__take__subset,axiom,
! [N2: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1162_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1163_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_1164_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_1165_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1166_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1167_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1168_snth_Osimps_I1_J,axiom,
! [S: stream727092118206550309m_rule] :
( ( snth_P1712579846118164081m_rule @ S @ zero_zero_nat )
= ( shd_Pr7235097944458474089m_rule @ S ) ) ).
% snth.simps(1)
thf(fact_1169_set__take__subset__set__take,axiom,
! [M3: nat,N2: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ M3 @ Xs ) ) @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1170_length__pos__if__in__set,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1171_min_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ).
% min.absorb1
thf(fact_1172_min_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ).
% min.absorb2
thf(fact_1173_min_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.bounded_iff
thf(fact_1174_stake__nth,axiom,
! [M3: nat,N2: nat,S: stream727092118206550309m_rule] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( nth_Pr8225284241310773582m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S ) @ M3 )
= ( snth_P1712579846118164081m_rule @ S @ M3 ) ) ) ).
% stake_nth
thf(fact_1175_min_Omono,axiom,
! [A2: nat,C: nat,B2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D3 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C @ D3 ) ) ) ) ).
% min.mono
thf(fact_1176_min_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( ord_min_nat @ A2 @ B2 ) ) ) ).
% min.orderE
thf(fact_1177_min_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% min.orderI
thf(fact_1178_min_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.boundedE
thf(fact_1179_min_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ) ).
% min.boundedI
thf(fact_1180_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( A3
= ( ord_min_nat @ A3 @ B5 ) ) ) ) ).
% min.order_iff
thf(fact_1181_min_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).
% min.cobounded1
thf(fact_1182_min_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).
% min.cobounded2
thf(fact_1183_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( ord_min_nat @ A3 @ B5 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_1184_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( ord_min_nat @ A3 @ B5 )
= B5 ) ) ) ).
% min.absorb_iff2
thf(fact_1185_min_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI1
thf(fact_1186_min_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI2
thf(fact_1187_min__le__iff__disj,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y3 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
| ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_1188_min__absorb2,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( ( ord_min_set_fm @ X @ Y3 )
= Y3 ) ) ).
% min_absorb2
thf(fact_1189_min__absorb2,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_min_nat @ X @ Y3 )
= Y3 ) ) ).
% min_absorb2
thf(fact_1190_min__absorb1,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_min_set_fm @ X @ Y3 )
= X ) ) ).
% min_absorb1
thf(fact_1191_min__absorb1,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_min_nat @ X @ Y3 )
= X ) ) ).
% min_absorb1
thf(fact_1192_min__def,axiom,
( ord_min_set_fm
= ( ^ [A3: set_fm,B5: set_fm] : ( if_set_fm @ ( ord_less_eq_set_fm @ A3 @ B5 ) @ A3 @ B5 ) ) ) ).
% min_def
thf(fact_1193_min__def,axiom,
( ord_min_nat
= ( ^ [A3: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B5 ) @ A3 @ B5 ) ) ) ).
% min_def
thf(fact_1194_stake__sdrop,axiom,
! [N2: nat,S: stream727092118206550309m_rule] :
( ( shift_5008521991580687660m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_1195_stake__sdrop,axiom,
! [N2: nat,S: stream8299795917829157543ist_fm] :
( ( shift_5848029981951141578ist_fm @ ( stake_2930890243849202354ist_fm @ N2 @ S ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_1196_stake__sdrop,axiom,
! [N2: nat,S: stream_rule] :
( ( shift_rule @ ( stake_rule @ N2 @ S ) @ ( sdrop_rule @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_1197_stake__sdrop,axiom,
! [N2: nat,S: stream6494289010434245521m_rule] :
( ( shift_3194717208791370420m_rule @ ( stake_8908532221384226460m_rule @ N2 @ S ) @ ( sdrop_7373388980841566196m_rule @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_1198_set__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( append_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_append
thf(fact_1199_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: fm] :
( ( member_fm2 @ Y4 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1200_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y4: fm] :
( ( member_fm2 @ Y4 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1201_in__set__conv__decomp__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1202_in__set__conv__decomp__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1203_split__list__first__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1204_split__list__last__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_fm,X3: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1205_split__list__first__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1206_split__list__last__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_fm,X3: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1207_in__set__conv__decomp,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1208_append__Cons__eq__iff,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Xs3: list_fm,Ys4: list_fm] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm2 @ X @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) )
= ( append_fm @ Xs3 @ ( cons_fm @ X @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1209_split__list__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_1210_split__list__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1211_split__list__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X6: fm] :
( ( member_fm2 @ X6 @ ( set_fm2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_1212_split__list__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1213_split__list,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1214_lexord__same__pref__iff,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lexord_fm @ R2 ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X2 @ X2 ) @ R2 ) )
| ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1215_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1216_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1217_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1218_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1219_lift__Suc__antimono__le,axiom,
! [F: nat > set_fm,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_set_fm @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_fm @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1220_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1221_lift__Suc__mono__le,axiom,
! [F: nat > set_fm,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_set_fm @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_fm @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1222_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1223_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1224_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1225_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1226_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1227_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1228_inc__induct,axiom,
! [I3: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J2 )
=> ( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1229_dec__induct,axiom,
! [I3: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P2 @ I3 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J2 )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1230_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1231_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1232_Suc__diff__le,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ N2 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1233_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N: nat] : ( R3 @ N @ ( suc @ N ) )
=> ( R3 @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1234_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P2 @ M3 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1235_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( P2 @ M4 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_1236_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1237_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1238_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1239_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_1240_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1241_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1242_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1243_ex__least__nat__less,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1244_stake__Suc,axiom,
! [N2: nat,S: stream727092118206550309m_rule] :
( ( stake_1447931197033250628m_rule @ ( suc @ N2 ) @ S )
= ( append1085632490384598586m_rule @ ( stake_1447931197033250628m_rule @ N2 @ S ) @ ( cons_P768676612401224159m_rule @ ( snth_P1712579846118164081m_rule @ S @ N2 ) @ nil_Pr474642538098133391m_rule ) ) ) ).
% stake_Suc
thf(fact_1245_SuccD,axiom,
! [K2: fm,Kl: set_list_fm,Kl2: list_fm] :
( ( member_fm2 @ K2 @ ( bNF_Greatest_Succ_fm @ Kl @ Kl2 ) )
=> ( member_list_fm2 @ ( append_fm @ Kl2 @ ( cons_fm @ K2 @ nil_fm ) ) @ Kl ) ) ).
% SuccD
thf(fact_1246_SuccI,axiom,
! [Kl2: list_fm,K2: fm,Kl: set_list_fm] :
( ( member_list_fm2 @ ( append_fm @ Kl2 @ ( cons_fm @ K2 @ nil_fm ) ) @ Kl )
=> ( member_fm2 @ K2 @ ( bNF_Greatest_Succ_fm @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1247_empty__Shift,axiom,
! [Kl: set_list_fm,K2: fm] :
( ( member_list_fm2 @ nil_fm @ Kl )
=> ( ( member_fm2 @ K2 @ ( bNF_Greatest_Succ_fm @ Kl @ nil_fm ) )
=> ( member_list_fm2 @ nil_fm @ ( bNF_Gr3717417623296551025ift_fm @ Kl @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_1248_set__drop__subset,axiom,
! [N2: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ N2 @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1249_irrefl__onI,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm] :
( ! [A5: fm] :
( ( member_fm2 @ A5 @ A )
=> ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A5 @ A5 ) @ R2 ) )
=> ( irrefl_on_fm @ A @ R2 ) ) ).
% irrefl_onI
thf(fact_1250_irrefl__onD,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm,A2: fm] :
( ( irrefl_on_fm @ A @ R2 )
=> ( ( member_fm2 @ A2 @ A )
=> ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A2 @ A2 ) @ R2 ) ) ) ).
% irrefl_onD
thf(fact_1251_in__set__dropD,axiom,
! [X: fm,N2: nat,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ ( drop_fm @ N2 @ Xs ) ) )
=> ( member_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1252_irrefl__on__subset,axiom,
! [A: set_fm,R2: set_Pr4463079037648049377_fm_fm,B: set_fm] :
( ( irrefl_on_fm @ A @ R2 )
=> ( ( ord_less_eq_set_fm @ B @ A )
=> ( irrefl_on_fm @ B @ R2 ) ) ) ).
% irrefl_on_subset
thf(fact_1253_set__drop__subset__set__drop,axiom,
! [N2: nat,M3: nat,Xs: list_fm] :
( ( ord_less_eq_nat @ N2 @ M3 )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( drop_fm @ M3 @ Xs ) ) @ ( set_fm2 @ ( drop_fm @ N2 @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1254_drop__stake,axiom,
! [N2: nat,M3: nat,S: stream727092118206550309m_rule] :
( ( drop_P7527609239214822479m_rule @ N2 @ ( stake_1447931197033250628m_rule @ M3 @ S ) )
= ( stake_1447931197033250628m_rule @ ( minus_minus_nat @ M3 @ N2 ) @ ( sdrop_7224736112439592940m_rule @ N2 @ S ) ) ) ).
% drop_stake
thf(fact_1255_drop__stake,axiom,
! [N2: nat,M3: nat,S: stream8299795917829157543ist_fm] :
( ( drop_P1437753125557294183ist_fm @ N2 @ ( stake_2930890243849202354ist_fm @ M3 @ S ) )
= ( stake_2930890243849202354ist_fm @ ( minus_minus_nat @ M3 @ N2 ) @ ( sdrop_4442373711808556042ist_fm @ N2 @ S ) ) ) ).
% drop_stake
thf(fact_1256_drop__stake,axiom,
! [N2: nat,M3: nat,S: stream_rule] :
( ( drop_rule @ N2 @ ( stake_rule @ M3 @ S ) )
= ( stake_rule @ ( minus_minus_nat @ M3 @ N2 ) @ ( sdrop_rule @ N2 @ S ) ) ) ).
% drop_stake
thf(fact_1257_drop__stake,axiom,
! [N2: nat,M3: nat,S: stream6494289010434245521m_rule] :
( ( drop_P7779951988903599441m_rule @ N2 @ ( stake_8908532221384226460m_rule @ M3 @ S ) )
= ( stake_8908532221384226460m_rule @ ( minus_minus_nat @ M3 @ N2 ) @ ( sdrop_7373388980841566196m_rule @ N2 @ S ) ) ) ).
% drop_stake
thf(fact_1258_sdrop__shift,axiom,
! [I3: nat,W: list_P8332459875677254181m_rule,S: stream727092118206550309m_rule] :
( ( sdrop_7224736112439592940m_rule @ I3 @ ( shift_5008521991580687660m_rule @ W @ S ) )
= ( shift_5008521991580687660m_rule @ ( drop_P7527609239214822479m_rule @ I3 @ W ) @ ( sdrop_7224736112439592940m_rule @ ( minus_minus_nat @ I3 @ ( size_s9206449984754302649m_rule @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1259_sdrop__shift,axiom,
! [I3: nat,W: list_P5616295576739893671ist_fm,S: stream8299795917829157543ist_fm] :
( ( sdrop_4442373711808556042ist_fm @ I3 @ ( shift_5848029981951141578ist_fm @ W @ S ) )
= ( shift_5848029981951141578ist_fm @ ( drop_P1437753125557294183ist_fm @ I3 @ W ) @ ( sdrop_4442373711808556042ist_fm @ ( minus_minus_nat @ I3 @ ( size_s2722402132374190611ist_fm @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1260_sdrop__shift,axiom,
! [I3: nat,W: list_rule,S: stream_rule] :
( ( sdrop_rule @ I3 @ ( shift_rule @ W @ S ) )
= ( shift_rule @ ( drop_rule @ I3 @ W ) @ ( sdrop_rule @ ( minus_minus_nat @ I3 @ ( size_size_list_rule @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1261_sdrop__shift,axiom,
! [I3: nat,W: list_P8801850819564216977m_rule,S: stream6494289010434245521m_rule] :
( ( sdrop_7373388980841566196m_rule @ I3 @ ( shift_3194717208791370420m_rule @ W @ S ) )
= ( shift_3194717208791370420m_rule @ ( drop_P7779951988903599441m_rule @ I3 @ W ) @ ( sdrop_7373388980841566196m_rule @ ( minus_minus_nat @ I3 @ ( size_s8174888084367447805m_rule @ W ) ) @ S ) ) ) ).
% sdrop_shift
thf(fact_1262_shift__simps_I1_J,axiom,
! [Xs: list_P8332459875677254181m_rule,S: stream727092118206550309m_rule] :
( ( ( Xs = nil_Pr474642538098133391m_rule )
=> ( ( shd_Pr7235097944458474089m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) )
= ( shd_Pr7235097944458474089m_rule @ S ) ) )
& ( ( Xs != nil_Pr474642538098133391m_rule )
=> ( ( shd_Pr7235097944458474089m_rule @ ( shift_5008521991580687660m_rule @ Xs @ S ) )
= ( hd_Pro3588770950400194538m_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1263_set__swap,axiom,
! [I3: nat,Xs: list_fm,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ ( list_update_fm @ Xs @ I3 @ ( nth_fm @ Xs @ J2 ) ) @ J2 @ ( nth_fm @ Xs @ I3 ) ) )
= ( set_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1264_set__update__subset__insert,axiom,
! [Xs: list_fm,I3: nat,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( list_update_fm @ Xs @ I3 @ X ) ) @ ( insert_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1265_set__update__memI,axiom,
! [N2: nat,Xs: list_fm,X: fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( member_fm2 @ X @ ( set_fm2 @ ( list_update_fm @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1266_set__update__subsetI,axiom,
! [Xs: list_fm,A: set_fm,X: fm,I3: nat] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ A )
=> ( ( member_fm2 @ X @ A )
=> ( ord_less_eq_set_fm @ ( set_fm2 @ ( list_update_fm @ Xs @ I3 @ X ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1267_hd__in__set,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( member_fm2 @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1268_list_Oset__sel_I1_J,axiom,
! [A2: list_fm] :
( ( A2 != nil_fm )
=> ( member_fm2 @ ( hd_fm @ A2 ) @ ( set_fm2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1269_cycle_Osimps_I1_J,axiom,
! [Xs: list_P8332459875677254181m_rule] :
( ( shd_Pr7235097944458474089m_rule @ ( cycle_1017677951696034864m_rule @ Xs ) )
= ( hd_Pro3588770950400194538m_rule @ Xs ) ) ).
% cycle.simps(1)
thf(fact_1270_set__update__distinct,axiom,
! [Xs: list_fm,N2: nat,X: fm] :
( ( distinct_fm @ Xs )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ Xs @ N2 @ X ) )
= ( insert_fm2 @ X @ ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ ( nth_fm @ Xs @ N2 ) @ bot_bot_set_fm ) ) ) ) ) ) ).
% set_update_distinct
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Syntax__Ofm_J_T,axiom,
! [X: set_fm,Y3: set_fm] :
( ( if_set_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Syntax__Ofm_J_T,axiom,
! [X: set_fm,Y3: set_fm] :
( ( if_set_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [X: list_fm,Y3: list_fm] :
( ( if_list_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
! [X: list_fm,Y3: list_fm] :
( ( if_list_fm @ $true @ X @ Y3 )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [M4: nat] :
( ( member_fm2 @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M4 @ steps ) ) ) ) ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------