TPTP Problem File: SLH0214^1.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 : Separation_Logic_Unbounded/0003_FixedPoint/prob_00970_029240__6948054_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1510 ( 475 unt; 232 typ; 0 def)
% Number of atoms : 4488 (1189 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 11512 ( 419 ~; 51 |; 255 &;8684 @)
% ( 0 <=>;2103 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 3952 (3952 >; 0 *; 0 +; 0 <<)
% Number of symbols : 216 ( 215 usr; 20 con; 0-4 aty)
% Number of variables : 3656 ( 243 ^;3262 !; 151 ?;3656 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:42.556
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J_J,type,
set_se1522906970093639477_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J_J,type,
set_se3584202636623819855_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
set_option_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
set_set_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_J,type,
set_c_d_set_a_o: $tType ).
thf(ty_n_t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
option_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
set_set_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__d,type,
d: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (215)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
comple5290581719055393889et_a_o: set_c_d_set_a_o > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple3834726295627996700_set_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple6131501996466690428_set_a: set_set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
comple6619285321642111682_set_a: set_se1522906970093639477_set_a > set_option_c_d_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
comple4629436989800923665tion_a: set_set_option_a > set_option_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
comple2723893648999256284_set_a: set_se3584202636623819855_set_a > set_set_c_d_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple1957918121334358780_set_a: ( set_c_d_set_a > ( c > d ) > set_a ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple3108528645285135042_set_a: ( set_option_c_d_set_a > option_c_d_set_a ) > ( option_c_d_set_a > option_c_d_set_a > $o ) > ( option_c_d_set_a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__Option__Ooption_Itf__a_J,type,
comple2995885364255664145tion_a: ( set_option_a > option_a ) > ( option_a > option_a > $o ) > ( option_a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple2100625987831660124_set_a: ( set_set_c_d_set_a > set_c_d_set_a ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__Set__Oset_Itf__a_J,type,
comple8887300225568239275_set_a: ( set_set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001tf__a,type,
comple72871723935627595ible_a: ( set_a > a ) > ( a > a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple2361085228800170300_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp_001t__Set__Oset_Itf__a_J,type,
comple6813827801316615403_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
comple8961357239255020671et_a_o: ( ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a_o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple4855714899335171198_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001_062_Itf__a_M_Eo_J,type,
comple4809514086394661360es_a_o: ( ( a > $o ) > a > $o ) > set_a_o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple3976681931473715166_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_set_c_d_set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple1698038292816285837_set_a: ( set_set_a > set_set_a ) > set_set_set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates_001t__Set__Oset_Itf__a_J,type,
comple4964449497533277997_set_a: ( set_a > set_a ) > set_set_a ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
comple235802229605493583et_a_o: ( ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple8462753965213938094_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001_062_Itf__a_M_Eo_J,type,
comple1243119448443558080sp_a_o: ( ( a > $o ) > a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple8844751925773657358_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_c_d_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple4561909299121069501_set_a: ( set_set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001t__Set__Oset_Itf__a_J,type,
comple8134540176031052893_set_a: ( set_a > set_a ) > set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
comple4742651485759770334et_a_o: ( ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ) > set_c_d_set_a_o > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple7455786223818501471_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001_062_Itf__a_M_Eo_J,type,
comple1735739171376666831in_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > set_a_o > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple2313665252861385893_set_a: ( option_c_d_set_a > option_c_d_set_a > $o ) > set_option_c_d_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Option__Ooption_Itf__a_J,type,
comple6692356590997824628tion_a: ( option_a > option_a > $o ) > set_option_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple2185443536470187199_set_a: ( set_c_d_set_a > set_c_d_set_a > $o ) > set_set_c_d_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple2480829807455835374_set_a: ( set_set_a > set_set_a > $o ) > set_set_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Set__Oset_Itf__a_J,type,
comple4316259127148425102_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001tf__a,type,
comple1697357536187991598hain_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit6926915774301931483_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit9007271454129256903_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
condit6526574527931036330et_a_o: set_c_d_set_a_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit7392869265169887891_set_a: set_c_d_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_062_Itf__a_M_Eo_J,type,
condit5969422546283407003ve_a_o: set_a_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
condit2151140496889778675_set_a: set_set_c_d_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
condit5548584133349953570_set_a: set_set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_Itf__a_J,type,
condit3373647341569784514_set_a: set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit5292637031048566470_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_Itf__a_J,type,
condit6315317455391067509_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__a,type,
condit4103000493307248661_bdd_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
condit8154225043310684324_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Set__Oset_Itf__a_J,type,
condit4774827555938943059_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001tf__a,type,
condit6541519642617408243_bdd_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
finite3010068450757450645_set_a: set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
finite3330819693523053784_set_a: set_c_d_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
finite1674126218327898605tion_a: set_option_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_FixedPoint_Ologic_OInf_001tf__c_001tf__d_001tf__a,type,
inf_c_d_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_OSup_001tf__c_001tf__d_001tf__a,type,
sup_c_d_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oempty__interp_001_062_Itf__c_Mtf__d_J_001tf__a,type,
empty_interp_c_d_a: ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Ofull__interp_001tf__c_001tf__d_001tf__a,type,
full_interp_c_d_a: ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oinf_001tf__c_001tf__d_001tf__a,type,
inf_c_d_a2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oless_001tf__c_001tf__d_001tf__a,type,
less_c_d_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Oset__closure__property_001tf__a_001tf__c_001tf__d,type,
set_cl2807270042661212426_a_c_d: ( a > a > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Osmaller__interp_001tf__c_001tf__d_001tf__a,type,
smaller_interp_c_d_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Osupp_001tf__a_001tf__c_001tf__d,type,
supp_a_c_d: ( a > a > option_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
monoto2937423850181994535_set_a: set_c_d_set_a > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
monoto6316088450447394390_set_a: set_c_d_set_a > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( set_a > set_a > $o ) > ( ( ( c > d ) > set_a ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
monoto2748056057003999288_set_a: set_set_a > ( set_a > set_a > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( set_a > ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
monoto2502030104860647832_set_a: set_a > ( a > a > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( a > ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
monotone_on_a_set_a: set_a > ( a > a > $o ) > ( set_a > set_a > $o ) > ( a > set_a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
minus_6165026464846083862_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
minus_1665977719694084726_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
inf_inf_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
inf_in754637537901350525_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
semila6957839794703059165_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
semila1630236661048524575_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Set__Oset_Itf__a_J,type,
semila4706084620769370446_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic1898000229760699588_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic1995125144389820681_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Option_Ooption_ONone_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
none_c_d_set_a: option_c_d_set_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_OSome_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
some_c_d_set_a: ( ( c > d ) > set_a ) > option_c_d_set_a ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Othese_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
these_c_d_set_a: set_option_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Option_Othese_001tf__a,type,
these_a: set_option_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
bot_bot_c_d_set_a_o: ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
bot_bot_c_d_set_a: ( c > d ) > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_M_Eo_J,type,
bot_bo2369851049062976348et_a_o: option_c_d_set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
bot_bot_option_a_o: option_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_M_Eo_J,type,
bot_bo3591254198091563330et_a_o: set_c_d_set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_J,type,
bot_bo848287272940216920et_a_o: set_c_d_set_a_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
bot_bo738396921950161403_set_a: set_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
bot_bot_set_a_o2: set_a_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
bot_bo6666349697208826049_set_a: set_option_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
bot_bot_set_option_a: set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
bot_bo58555506362910043_set_a: set_set_c_d_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le3685282097655362107_set_a: set_c_d_set_a > set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
ord_le961293222253252206et_a_o: ( ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le8464990428230162895_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_J,type,
ord_le8757755980619729956et_a_o: set_c_d_set_a_o > set_c_d_set_a_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le5982164083705284911_set_a: set_c_d_set_a > set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_set_a_o: set_a_o > set_a_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le20137369925261813_set_a: set_option_c_d_set_a > set_option_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
ord_le1955136853071979460tion_a: set_option_a > set_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le7272806397018272911_set_a: set_set_c_d_set_a > set_set_c_d_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
ordering_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Set__Oset_Itf__a_J,type,
ordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
orderi5785346111247480928_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
partia701112543150332005_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_Itf__a_J,type,
partia6602192050731689876_set_a: ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
preord7021486942077351306_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Set__Oset_Itf__a_J,type,
preordering_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
top_top_c_d_set_a_o: ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
top_top_c_d_set_a: ( c > d ) > set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
top_to4267977599310771935_set_a: set_c_d_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
collect_c_d_set_a: ( ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
collec4582986676398431712_set_a: ( option_c_d_set_a > $o ) > set_option_c_d_set_a ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_Itf__a_J,type,
collect_option_a: ( option_a > $o ) > set_option_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
collec3354561713582630522_set_a: ( set_c_d_set_a > $o ) > set_set_c_d_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_8552787320881293370_set_a: ( ( ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a ) > set_c_d_set_a_o > set_set_c_d_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_5710119992958135237_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_5050625251388476148_set_a: ( ( ( c > d ) > set_a ) > set_a ) > set_c_d_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001tf__a,type,
image_c_d_set_a_a: ( ( ( c > d ) > set_a ) > a ) > set_c_d_set_a > set_a ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_6768572723705120255_set_a: ( option_c_d_set_a > ( c > d ) > set_a ) > set_option_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_928718206969431365_set_a: ( option_c_d_set_a > option_c_d_set_a ) > set_option_c_d_set_a > set_option_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Option__Ooption_Itf__a_J,type,
image_9195261480396307732tion_a: ( option_c_d_set_a > option_a ) > set_option_c_d_set_a > set_option_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_317793290637937008_set_a: ( option_a > ( c > d ) > set_a ) > set_option_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4141439534252489855_set_a: ( option_a > set_set_a ) > set_option_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_option_a_set_a: ( option_a > set_a ) > set_option_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
image_option_a_a: ( option_a > a ) > set_option_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_212549500329102437_set_a: ( set_c_d_set_a > ( c > d ) > set_a ) > set_set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
image_5418612861375423429_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_1482592857945081046_set_a: ( set_a > ( c > d ) > set_a ) > set_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
image_set_a_option_a: ( set_a > option_a ) > set_set_a > set_option_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
image_a_c_d_set_a: ( a > ( c > d ) > set_a ) > set_a > set_c_d_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_M_Eo_J,type,
image_a_a_o: ( a > a > $o ) > set_a > set_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
insert_c_d_set_a: ( ( c > d ) > set_a ) > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
insert1935891768494221125_set_a: option_c_d_set_a > set_option_c_d_set_a > set_option_c_d_set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
insert_set_c_d_set_a: set_c_d_set_a > set_set_c_d_set_a > set_set_c_d_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
is_empty_c_d_set_a: set_c_d_set_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_Itf__a_J,type,
is_empty_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
is_sin6979784932356128547_set_a: set_c_d_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
the_elem_c_d_set_a: set_c_d_set_a > ( c > d ) > set_a ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_at2224545791267470424_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_at2139306834251651636_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OatLeast_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_at4358065015900363374_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OatMost_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_atMost_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_gr4053032598485390707_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_gr2245648953767368143_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_gr5532796609634356233_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_Set__Interval_Oord_OlessThan_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
set_le5418582716766741598_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_UnboundedLogic_Opre__logic_Ocompatible_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
pre_co1390589184961732201_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > option_c_d_set_a ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Ocompatible_001tf__a,type,
pre_compatible_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
pre_larger_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > option_c_d_set_a ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001tf__a,type,
pre_larger_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J,type,
member_c_d_set_a_o: ( ( ( c > d ) > set_a ) > $o ) > set_c_d_set_a_o > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
member_c_d_set_a: ( ( c > d ) > set_a ) > set_c_d_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001t__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
member4306893881663408030_set_a: option_c_d_set_a > set_option_c_d_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
member_set_c_d_set_a: set_c_d_set_a > set_set_c_d_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
member8260580452227636350_set_a: set_option_c_d_set_a > set_se1522906970093639477_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
member_set_option_a: set_option_a > set_set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
member6574826897039512728_set_a: set_set_c_d_set_a > set_se3584202636623819855_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A____,type,
a2: set_c_d_set_a ).
thf(sy_v__092_060Delta_062a____,type,
delta_a: ( c > d ) > set_a ).
thf(sy_v__092_060Delta_062b____,type,
delta_b: ( c > d ) > set_a ).
thf(sy_v_a____,type,
a3: a ).
thf(sy_v_b____,type,
b: a ).
thf(sy_v_plus,type,
plus: a > a > option_a ).
thf(sy_v_s____,type,
s: c > d ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1277)
thf(fact_0__092_060open_062x_A_092_060in_062_A_092_060Delta_062a_As_092_060close_062,axiom,
member_a @ x @ ( delta_a @ s ) ).
% \<open>x \<in> \<Delta>a s\<close>
thf(fact_1__092_060open_062_092_060Delta_062a_A_092_060in_062_AA_092_060close_062,axiom,
member_c_d_set_a @ delta_a @ a2 ).
% \<open>\<Delta>a \<in> A\<close>
thf(fact_2_commutative,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% commutative
thf(fact_3__092_060open_062b_A_092_060succeq_062_Ax_092_060close_062,axiom,
pre_larger_a @ plus @ b @ x ).
% \<open>b \<succeq> x\<close>
thf(fact_4__092_060open_062a_A_092_060succeq_062_Ax_092_060close_062,axiom,
pre_larger_a @ plus @ a3 @ x ).
% \<open>a \<succeq> x\<close>
thf(fact_5__092_060open_062_092_060Delta_062b_A_092_060in_062_AA_092_060close_062,axiom,
member_c_d_set_a @ delta_b @ a2 ).
% \<open>\<Delta>b \<in> A\<close>
thf(fact_6__092_060open_062a_A_092_060in_062_A_092_060Delta_062a_As_092_060close_062,axiom,
member_a @ a3 @ ( delta_a @ s ) ).
% \<open>a \<in> \<Delta>a s\<close>
thf(fact_7_asm0_I3_J,axiom,
! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ a2 )
=> ( supp_a_c_d @ plus @ X ) ) ).
% asm0(3)
thf(fact_8__092_060open_062b_A_092_060in_062_A_092_060Delta_062b_As_092_060close_062,axiom,
member_a @ b @ ( delta_b @ s ) ).
% \<open>b \<in> \<Delta>b s\<close>
thf(fact_9_asm,axiom,
( ( member_a @ a3 @ ( comple3834726295627996700_set_a @ a2 @ s ) )
& ( member_a @ b @ ( comple3834726295627996700_set_a @ a2 @ s ) ) ) ).
% asm
thf(fact_10__092_060open_062b_A_092_060in_062_A_092_060Delta_062a_As_092_060close_062,axiom,
member_a @ b @ ( delta_a @ s ) ).
% \<open>b \<in> \<Delta>a s\<close>
thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060Delta_062a_A_092_060Delta_062b_O_A_092_060lbrakk_062_092_060Delta_062a_A_092_060in_062_AA_059_A_092_060Delta_062b_A_092_060in_062_AA_059_Aa_A_092_060in_062_A_092_060Delta_062a_As_059_Ab_A_092_060in_062_A_092_060Delta_062b_As_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Delta_a: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Delta_a @ a2 )
=> ! [Delta_b: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Delta_b @ a2 )
=> ( ( member_a @ a3 @ ( Delta_a @ s ) )
=> ~ ( member_a @ b @ ( Delta_b @ s ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>\<Delta>a \<Delta>b. \<lbrakk>\<Delta>a \<in> A; \<Delta>b \<in> A; a \<in> \<Delta>a s; b \<in> \<Delta>b s\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_12_asm0_I2_J,axiom,
a2 != bot_bo738396921950161403_set_a ).
% asm0(2)
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062a_A_092_060succeq_062_Ax_059_Ab_A_092_060succeq_062_Ax_059_Ax_A_092_060in_062_A_092_060Delta_062a_As_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X2: a] :
( ( pre_larger_a @ plus @ a3 @ X2 )
=> ( ( pre_larger_a @ plus @ b @ X2 )
=> ~ ( member_a @ X2 @ ( delta_a @ s ) ) ) ) ).
% \<open>\<And>thesis. (\<And>x. \<lbrakk>a \<succeq> x; b \<succeq> x; x \<in> \<Delta>a s\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_suppI,axiom,
! [Delta: ( c > d ) > set_a] :
( ! [A2: a,B2: a,S: c > d] :
( ( ( member_a @ A2 @ ( Delta @ S ) )
& ( member_a @ B2 @ ( Delta @ S ) ) )
=> ? [X: a] :
( ( pre_larger_a @ plus @ A2 @ X )
& ( pre_larger_a @ plus @ B2 @ X )
& ( member_a @ X @ ( Delta @ S ) ) ) )
=> ( supp_a_c_d @ plus @ Delta ) ) ).
% suppI
thf(fact_15_supp__def,axiom,
! [Delta: ( c > d ) > set_a] :
( ( supp_a_c_d @ plus @ Delta )
= ( ! [A3: a,B3: a,S2: c > d] :
( ( ( member_a @ A3 @ ( Delta @ S2 ) )
& ( member_a @ B3 @ ( Delta @ S2 ) ) )
=> ? [X3: a] :
( ( pre_larger_a @ plus @ A3 @ X3 )
& ( pre_larger_a @ plus @ B3 @ X3 )
& ( member_a @ X3 @ ( Delta @ S2 ) ) ) ) ) ) ).
% supp_def
thf(fact_16_compatible__smaller,axiom,
! [A: a,B: a,X4: a] :
( ( pre_larger_a @ plus @ A @ B )
=> ( ( pre_compatible_a @ plus @ X4 @ A )
=> ( pre_compatible_a @ plus @ X4 @ B ) ) ) ).
% compatible_smaller
thf(fact_17_larger__implies__compatible,axiom,
! [X4: a,Y: a] :
( ( pre_larger_a @ plus @ X4 @ Y )
=> ( pre_compatible_a @ plus @ X4 @ Y ) ) ).
% larger_implies_compatible
thf(fact_18_larger__def,axiom,
! [A: a,B: a] :
( ( pre_larger_a @ plus @ A @ B )
= ( ? [C: a] :
( ( some_a @ A )
= ( plus @ B @ C ) ) ) ) ).
% larger_def
thf(fact_19_larger__first__sum,axiom,
! [Y: a,A: a,B: a,X4: a] :
( ( ( some_a @ Y )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ X4 @ Y )
=> ? [A4: a] :
( ( ( some_a @ X4 )
= ( plus @ A4 @ B ) )
& ( pre_larger_a @ plus @ A4 @ A ) ) ) ) ).
% larger_first_sum
thf(fact_20_sum__both__larger,axiom,
! [X5: a,A5: a,B4: a,X4: a,A: a,B: a] :
( ( ( some_a @ X5 )
= ( plus @ A5 @ B4 ) )
=> ( ( ( some_a @ X4 )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ A5 @ A )
=> ( ( pre_larger_a @ plus @ B4 @ B )
=> ( pre_larger_a @ plus @ X5 @ X4 ) ) ) ) ) ).
% sum_both_larger
thf(fact_21_set__closure__property__instantiate,axiom,
! [S3: a > a > set_a,Delta: ( c > d ) > set_a,A: a,S4: c > d,B: a,X4: a] :
( ( set_cl2807270042661212426_a_c_d @ S3 @ Delta )
=> ( ( member_a @ A @ ( Delta @ S4 ) )
=> ( ( member_a @ B @ ( Delta @ S4 ) )
=> ( ( member_a @ X4 @ ( S3 @ A @ B ) )
=> ( member_a @ X4 @ ( Delta @ S4 ) ) ) ) ) ) ).
% set_closure_property_instantiate
thf(fact_22_False,axiom,
~ ( ord_less_eq_set_a @ ( delta_a @ s ) @ ( delta_b @ s ) ) ).
% False
thf(fact_23_asso1,axiom,
! [A: a,B: a,Ab: a,C2: a,Bc: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ( ( plus @ B @ C2 )
= ( some_a @ Bc ) ) )
=> ( ( plus @ Ab @ C2 )
= ( plus @ A @ Bc ) ) ) ).
% asso1
thf(fact_24_move__sum,axiom,
! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X4: a,X1: a,X22: a] :
( ( ( some_a @ A )
= ( plus @ A1 @ A22 ) )
=> ( ( ( some_a @ B )
= ( plus @ B1 @ B22 ) )
=> ( ( ( some_a @ X4 )
= ( plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X22 )
= ( plus @ A22 @ B22 ) )
=> ( ( some_a @ X4 )
= ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).
% move_sum
thf(fact_25_pre__logic_Olarger_Ocong,axiom,
pre_larger_a = pre_larger_a ).
% pre_logic.larger.cong
thf(fact_26_empty__interp__def,axiom,
( empty_interp_c_d_a
= ( ^ [S2: c > d] : bot_bot_set_a ) ) ).
% empty_interp_def
thf(fact_27_set__closure__propertyI,axiom,
! [Delta: ( c > d ) > set_a,S3: a > a > set_a] :
( ! [A2: a,B2: a,S: c > d] :
( ( ( member_a @ A2 @ ( Delta @ S ) )
& ( member_a @ B2 @ ( Delta @ S ) ) )
=> ( ord_less_eq_set_a @ ( S3 @ A2 @ B2 ) @ ( Delta @ S ) ) )
=> ( set_cl2807270042661212426_a_c_d @ S3 @ Delta ) ) ).
% set_closure_propertyI
thf(fact_28_set__closure__property__def,axiom,
( set_cl2807270042661212426_a_c_d
= ( ^ [S5: a > a > set_a,Delta2: ( c > d ) > set_a] :
! [A3: a,B3: a,S2: c > d] :
( ( ( member_a @ A3 @ ( Delta2 @ S2 ) )
& ( member_a @ B3 @ ( Delta2 @ S2 ) ) )
=> ( ord_less_eq_set_a @ ( S5 @ A3 @ B3 ) @ ( Delta2 @ S2 ) ) ) ) ) ).
% set_closure_property_def
thf(fact_29_asso3,axiom,
! [A: a,B: a,C2: a,Bc: a] :
( ~ ( pre_compatible_a @ plus @ A @ B )
=> ( ( ( plus @ B @ C2 )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).
% asso3
thf(fact_30_asso2,axiom,
! [A: a,B: a,Ab: a,C2: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ plus @ B @ C2 ) )
=> ~ ( pre_compatible_a @ plus @ Ab @ C2 ) ) ).
% asso2
thf(fact_31_pre__logic_Ocompatible_Ocong,axiom,
pre_compatible_a = pre_compatible_a ).
% pre_logic.compatible.cong
thf(fact_32_pre__logic_Olarger__def,axiom,
( pre_larger_c_d_set_a
= ( ^ [Plus: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > option_c_d_set_a,A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
? [C: ( c > d ) > set_a] :
( ( some_c_d_set_a @ A3 )
= ( Plus @ B3 @ C ) ) ) ) ).
% pre_logic.larger_def
thf(fact_33_pre__logic_Olarger__def,axiom,
( pre_larger_a
= ( ^ [Plus: a > a > option_a,A3: a,B3: a] :
? [C: a] :
( ( some_a @ A3 )
= ( Plus @ B3 @ C ) ) ) ) ).
% pre_logic.larger_def
thf(fact_34_logic_Osupp_Ocong,axiom,
supp_a_c_d = supp_a_c_d ).
% logic.supp.cong
thf(fact_35_complete__lattice__class_OSup__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% complete_lattice_class.Sup_empty
thf(fact_36_complete__lattice__class_OSup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% complete_lattice_class.Sup_empty
thf(fact_37_complete__lattice__class_OSup__empty,axiom,
( ( comple5290581719055393889et_a_o @ bot_bo848287272940216920et_a_o )
= bot_bot_c_d_set_a_o ) ).
% complete_lattice_class.Sup_empty
thf(fact_38_complete__lattice__class_OSup__empty,axiom,
( ( complete_Sup_Sup_a_o @ bot_bot_set_a_o2 )
= bot_bot_a_o ) ).
% complete_lattice_class.Sup_empty
thf(fact_39_complete__lattice__class_OSup__empty,axiom,
( ( comple6131501996466690428_set_a @ bot_bo58555506362910043_set_a )
= bot_bo738396921950161403_set_a ) ).
% complete_lattice_class.Sup_empty
thf(fact_40_complete__lattice__class_OSup__empty,axiom,
( ( comple3834726295627996700_set_a @ bot_bo738396921950161403_set_a )
= bot_bot_c_d_set_a ) ).
% complete_lattice_class.Sup_empty
thf(fact_41_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A6 )
= bot_bot_set_set_a )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_42_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A6 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_43_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_c_d_set_a_o] :
( ( ( comple5290581719055393889et_a_o @ A6 )
= bot_bot_c_d_set_a_o )
= ( ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ A6 )
=> ( X3 = bot_bot_c_d_set_a_o ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_44_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_a_o] :
( ( ( complete_Sup_Sup_a_o @ A6 )
= bot_bot_a_o )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
=> ( X3 = bot_bot_a_o ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_45_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_set_c_d_set_a] :
( ( ( comple6131501996466690428_set_a @ A6 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bo738396921950161403_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_46_complete__lattice__class_OSup__bot__conv_I1_J,axiom,
! [A6: set_c_d_set_a] :
( ( ( comple3834726295627996700_set_a @ A6 )
= bot_bot_c_d_set_a )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_c_d_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(1)
thf(fact_47_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A6 ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_48_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A6 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_49_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_c_d_set_a_o] :
( ( bot_bot_c_d_set_a_o
= ( comple5290581719055393889et_a_o @ A6 ) )
= ( ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ A6 )
=> ( X3 = bot_bot_c_d_set_a_o ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_50_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_a_o] :
( ( bot_bot_a_o
= ( complete_Sup_Sup_a_o @ A6 ) )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
=> ( X3 = bot_bot_a_o ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_51_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( comple6131501996466690428_set_a @ A6 ) )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bo738396921950161403_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_52_complete__lattice__class_OSup__bot__conv_I2_J,axiom,
! [A6: set_c_d_set_a] :
( ( bot_bot_c_d_set_a
= ( comple3834726295627996700_set_a @ A6 ) )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_c_d_set_a ) ) ) ) ).
% complete_lattice_class.Sup_bot_conv(2)
thf(fact_53_subset__empty,axiom,
! [A6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ bot_bot_set_set_a )
= ( A6 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_54_subset__empty,axiom,
! [A6: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ bot_bo738396921950161403_set_a )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% subset_empty
thf(fact_55_subset__empty,axiom,
! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ bot_bot_set_a )
= ( A6 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_56_empty__subsetI,axiom,
! [A6: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A6 ) ).
% empty_subsetI
thf(fact_57_empty__subsetI,axiom,
! [A6: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A6 ) ).
% empty_subsetI
thf(fact_58_empty__subsetI,axiom,
! [A6: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A6 ) ).
% empty_subsetI
thf(fact_59_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_set_set_a,B5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A6 @ B5 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ ( comple3958522678809307947_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_60_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A6 @ B5 )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ ( comple6131501996466690428_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_61_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_c_d_set_a_o,B5: set_c_d_set_a_o] :
( ( ord_le8757755980619729956et_a_o @ A6 @ B5 )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ ( comple5290581719055393889et_a_o @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_62_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_a_o,B5: set_a_o] :
( ( ord_less_eq_set_a_o @ A6 @ B5 )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ ( complete_Sup_Sup_a_o @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_63_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_64_complete__lattice__class_OSup__subset__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ ( comple3834726295627996700_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_subset_mono
thf(fact_65_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( member_set_c_d_set_a @ A @ ( collec3354561713582630522_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: option_a,P: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: option_c_d_set_a,P: option_c_d_set_a > $o] :
( ( member4306893881663408030_set_a @ A @ ( collec4582986676398431712_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: ( c > d ) > set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a @ A @ ( collect_c_d_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A6: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A6: set_set_c_d_set_a] :
( ( collec3354561713582630522_set_a
@ ^ [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A6: set_option_a] :
( ( collect_option_a
@ ^ [X3: option_a] : ( member_option_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A6: set_option_c_d_set_a] :
( ( collec4582986676398431712_set_a
@ ^ [X3: option_c_d_set_a] : ( member4306893881663408030_set_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A6: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A6: set_c_d_set_a] :
( ( collect_c_d_set_a
@ ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_77_Collect__cong,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ! [X2: ( c > d ) > set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_c_d_set_a @ P )
= ( collect_c_d_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_78_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_79_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_set_set_a,U: set_set_a] :
( ! [V: set_set_a] :
( ( member_set_set_a @ V @ A6 )
=> ( ord_le3724670747650509150_set_a @ U @ V ) )
=> ( ( A6 != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ U @ ( comple3958522678809307947_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_80_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_set_c_d_set_a,U: set_c_d_set_a] :
( ! [V: set_c_d_set_a] :
( ( member_set_c_d_set_a @ V @ A6 )
=> ( ord_le5982164083705284911_set_a @ U @ V ) )
=> ( ( A6 != bot_bo58555506362910043_set_a )
=> ( ord_le5982164083705284911_set_a @ U @ ( comple6131501996466690428_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_81_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_c_d_set_a_o,U: ( ( c > d ) > set_a ) > $o] :
( ! [V: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ V @ A6 )
=> ( ord_le961293222253252206et_a_o @ U @ V ) )
=> ( ( A6 != bot_bo848287272940216920et_a_o )
=> ( ord_le961293222253252206et_a_o @ U @ ( comple5290581719055393889et_a_o @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_82_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_a_o,U: a > $o] :
( ! [V: a > $o] :
( ( member_a_o @ V @ A6 )
=> ( ord_less_eq_a_o @ U @ V ) )
=> ( ( A6 != bot_bot_set_a_o2 )
=> ( ord_less_eq_a_o @ U @ ( complete_Sup_Sup_a_o @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_83_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_set_a,U: set_a] :
( ! [V: set_a] :
( ( member_set_a @ V @ A6 )
=> ( ord_less_eq_set_a @ U @ V ) )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_84_complete__lattice__class_Oless__eq__Sup,axiom,
! [A6: set_c_d_set_a,U: ( c > d ) > set_a] :
( ! [V: ( c > d ) > set_a] :
( ( member_c_d_set_a @ V @ A6 )
=> ( ord_le8464990428230162895_set_a @ U @ V ) )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ord_le8464990428230162895_set_a @ U @ ( comple3834726295627996700_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.less_eq_Sup
thf(fact_85_cSup__least,axiom,
! [X6: set_set_set_a,Z: set_set_a] :
( ( X6 != bot_bo3380559777022489994_set_a )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_86_cSup__least,axiom,
! [X6: set_set_c_d_set_a,Z: set_c_d_set_a] :
( ( X6 != bot_bo58555506362910043_set_a )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ X6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_87_cSup__least,axiom,
! [X6: set_c_d_set_a_o,Z: ( ( c > d ) > set_a ) > $o] :
( ( X6 != bot_bo848287272940216920et_a_o )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ X6 )
=> ( ord_le961293222253252206et_a_o @ X2 @ Z ) )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_88_cSup__least,axiom,
! [X6: set_a_o,Z: a > $o] :
( ( X6 != bot_bot_set_a_o2 )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ X6 )
=> ( ord_less_eq_a_o @ X2 @ Z ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_89_cSup__least,axiom,
! [X6: set_set_a,Z: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_90_cSup__least,axiom,
! [X6: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ( X6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ X6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_91_cSup__eq__non__empty,axiom,
! [X6: set_set_set_a,A: set_set_a] :
( ( X6 != bot_bo3380559777022489994_set_a )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ A ) )
=> ( ! [Y2: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ X6 )
=> ( ord_le3724670747650509150_set_a @ X @ Y2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ Y2 ) )
=> ( ( comple3958522678809307947_set_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_92_cSup__eq__non__empty,axiom,
! [X6: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( X6 != bot_bo58555506362910043_set_a )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ X6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ A ) )
=> ( ! [Y2: set_c_d_set_a] :
( ! [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ X6 )
=> ( ord_le5982164083705284911_set_a @ X @ Y2 ) )
=> ( ord_le5982164083705284911_set_a @ A @ Y2 ) )
=> ( ( comple6131501996466690428_set_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_93_cSup__eq__non__empty,axiom,
! [X6: set_c_d_set_a_o,A: ( ( c > d ) > set_a ) > $o] :
( ( X6 != bot_bo848287272940216920et_a_o )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ X6 )
=> ( ord_le961293222253252206et_a_o @ X2 @ A ) )
=> ( ! [Y2: ( ( c > d ) > set_a ) > $o] :
( ! [X: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X @ X6 )
=> ( ord_le961293222253252206et_a_o @ X @ Y2 ) )
=> ( ord_le961293222253252206et_a_o @ A @ Y2 ) )
=> ( ( comple5290581719055393889et_a_o @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_94_cSup__eq__non__empty,axiom,
! [X6: set_a_o,A: a > $o] :
( ( X6 != bot_bot_set_a_o2 )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ X6 )
=> ( ord_less_eq_a_o @ X2 @ A ) )
=> ( ! [Y2: a > $o] :
( ! [X: a > $o] :
( ( member_a_o @ X @ X6 )
=> ( ord_less_eq_a_o @ X @ Y2 ) )
=> ( ord_less_eq_a_o @ A @ Y2 ) )
=> ( ( complete_Sup_Sup_a_o @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_95_cSup__eq__non__empty,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( ! [Y2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ X6 )
=> ( ord_less_eq_set_a @ X @ Y2 ) )
=> ( ord_less_eq_set_a @ A @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_96_cSup__eq__non__empty,axiom,
! [X6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( X6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ X6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ A ) )
=> ( ! [Y2: ( c > d ) > set_a] :
( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ X6 )
=> ( ord_le8464990428230162895_set_a @ X @ Y2 ) )
=> ( ord_le8464990428230162895_set_a @ A @ Y2 ) )
=> ( ( comple3834726295627996700_set_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_97_option_Oinject,axiom,
! [X22: ( c > d ) > set_a,Y22: ( c > d ) > set_a] :
( ( ( some_c_d_set_a @ X22 )
= ( some_c_d_set_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_98_option_Oinject,axiom,
! [X22: a,Y22: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_99_subsetI,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ( member_set_c_d_set_a @ X2 @ B5 ) )
=> ( ord_le7272806397018272911_set_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_100_subsetI,axiom,
! [A6: set_option_a,B5: set_option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A6 )
=> ( member_option_a @ X2 @ B5 ) )
=> ( ord_le1955136853071979460tion_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_101_subsetI,axiom,
! [A6: set_option_c_d_set_a,B5: set_option_c_d_set_a] :
( ! [X2: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ X2 @ A6 )
=> ( member4306893881663408030_set_a @ X2 @ B5 ) )
=> ( ord_le20137369925261813_set_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_102_subsetI,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( member_set_a @ X2 @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_103_subsetI,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( member_c_d_set_a @ X2 @ B5 ) )
=> ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_104_subsetI,axiom,
! [A6: set_a,B5: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ X2 @ B5 ) )
=> ( ord_less_eq_set_a @ A6 @ B5 ) ) ).
% subsetI
thf(fact_105_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_106_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_107_empty__Collect__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( bot_bo738396921950161403_set_a
= ( collect_c_d_set_a @ P ) )
= ( ! [X3: ( c > d ) > set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_108_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_109_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_110_Collect__empty__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( ( collect_c_d_set_a @ P )
= bot_bo738396921950161403_set_a )
= ( ! [X3: ( c > d ) > set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_111_all__not__in__conv,axiom,
! [A6: set_set_c_d_set_a] :
( ( ! [X3: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ X3 @ A6 ) )
= ( A6 = bot_bo58555506362910043_set_a ) ) ).
% all_not_in_conv
thf(fact_112_all__not__in__conv,axiom,
! [A6: set_option_a] :
( ( ! [X3: option_a] :
~ ( member_option_a @ X3 @ A6 ) )
= ( A6 = bot_bot_set_option_a ) ) ).
% all_not_in_conv
thf(fact_113_all__not__in__conv,axiom,
! [A6: set_option_c_d_set_a] :
( ( ! [X3: option_c_d_set_a] :
~ ( member4306893881663408030_set_a @ X3 @ A6 ) )
= ( A6 = bot_bo6666349697208826049_set_a ) ) ).
% all_not_in_conv
thf(fact_114_all__not__in__conv,axiom,
! [A6: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A6 ) )
= ( A6 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_115_all__not__in__conv,axiom,
! [A6: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A6 ) )
= ( A6 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_116_all__not__in__conv,axiom,
! [A6: set_c_d_set_a] :
( ( ! [X3: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ X3 @ A6 ) )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% all_not_in_conv
thf(fact_117_empty__iff,axiom,
! [C2: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ C2 @ bot_bo58555506362910043_set_a ) ).
% empty_iff
thf(fact_118_empty__iff,axiom,
! [C2: option_a] :
~ ( member_option_a @ C2 @ bot_bot_set_option_a ) ).
% empty_iff
thf(fact_119_empty__iff,axiom,
! [C2: option_c_d_set_a] :
~ ( member4306893881663408030_set_a @ C2 @ bot_bo6666349697208826049_set_a ) ).
% empty_iff
thf(fact_120_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_121_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_122_empty__iff,axiom,
! [C2: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ C2 @ bot_bo738396921950161403_set_a ) ).
% empty_iff
thf(fact_123_subset__antisym,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ A6 )
=> ( A6 = B5 ) ) ) ).
% subset_antisym
thf(fact_124_subset__antisym,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ A6 )
=> ( A6 = B5 ) ) ) ).
% subset_antisym
thf(fact_125_subset__antisym,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A6 )
=> ( A6 = B5 ) ) ) ).
% subset_antisym
thf(fact_126_asm0_I1_J,axiom,
comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ a2 ).
% asm0(1)
thf(fact_127_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_128_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_129_bot__set__def,axiom,
( bot_bo738396921950161403_set_a
= ( collect_c_d_set_a @ bot_bot_c_d_set_a_o ) ) ).
% bot_set_def
thf(fact_130_empty__Union__conv,axiom,
! [A6: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A6 ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_131_empty__Union__conv,axiom,
! [A6: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A6 ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_132_empty__Union__conv,axiom,
! [A6: set_set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( comple6131501996466690428_set_a @ A6 ) )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bo738396921950161403_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_133_Union__empty__conv,axiom,
! [A6: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A6 )
= bot_bot_set_set_a )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_134_Union__empty__conv,axiom,
! [A6: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A6 )
= bot_bot_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( X3 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_135_Union__empty__conv,axiom,
! [A6: set_set_c_d_set_a] :
( ( ( comple6131501996466690428_set_a @ A6 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ( X3 = bot_bo738396921950161403_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_136_Union__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% Union_empty
thf(fact_137_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_138_Union__empty,axiom,
( ( comple6131501996466690428_set_a @ bot_bo58555506362910043_set_a )
= bot_bo738396921950161403_set_a ) ).
% Union_empty
thf(fact_139_Union__subsetI,axiom,
! [A6: set_set_set_a,B5: set_set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A6 )
=> ? [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ B5 )
& ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ ( comple3958522678809307947_set_a @ B5 ) ) ) ).
% Union_subsetI
thf(fact_140_Union__subsetI,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ? [Y3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Y3 @ B5 )
& ( ord_le5982164083705284911_set_a @ X2 @ Y3 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ ( comple6131501996466690428_set_a @ B5 ) ) ) ).
% Union_subsetI
thf(fact_141_Union__subsetI,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ? [Y3: set_a] :
( ( member_set_a @ Y3 @ B5 )
& ( ord_less_eq_set_a @ X2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% Union_subsetI
thf(fact_142_Union__upper,axiom,
! [B5: set_set_a,A6: set_set_set_a] :
( ( member_set_set_a @ B5 @ A6 )
=> ( ord_le3724670747650509150_set_a @ B5 @ ( comple3958522678809307947_set_a @ A6 ) ) ) ).
% Union_upper
thf(fact_143_Union__upper,axiom,
! [B5: set_c_d_set_a,A6: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ B5 @ A6 )
=> ( ord_le5982164083705284911_set_a @ B5 @ ( comple6131501996466690428_set_a @ A6 ) ) ) ).
% Union_upper
thf(fact_144_Union__upper,axiom,
! [B5: set_a,A6: set_set_a] :
( ( member_set_a @ B5 @ A6 )
=> ( ord_less_eq_set_a @ B5 @ ( comple2307003609928055243_set_a @ A6 ) ) ) ).
% Union_upper
thf(fact_145_Union__least,axiom,
! [A6: set_set_set_a,C3: set_set_a] :
( ! [X7: set_set_a] :
( ( member_set_set_a @ X7 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X7 @ C3 ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ C3 ) ) ).
% Union_least
thf(fact_146_Union__least,axiom,
! [A6: set_set_c_d_set_a,C3: set_c_d_set_a] :
( ! [X7: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X7 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X7 @ C3 ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ C3 ) ) ).
% Union_least
thf(fact_147_Union__least,axiom,
! [A6: set_set_a,C3: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A6 )
=> ( ord_less_eq_set_a @ X7 @ C3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ C3 ) ) ).
% Union_least
thf(fact_148_Union__mono,axiom,
! [A6: set_set_set_a,B5: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A6 @ B5 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ ( comple3958522678809307947_set_a @ B5 ) ) ) ).
% Union_mono
thf(fact_149_Union__mono,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A6 @ B5 )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ ( comple6131501996466690428_set_a @ B5 ) ) ) ).
% Union_mono
thf(fact_150_Union__mono,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% Union_mono
thf(fact_151_ex__in__conv,axiom,
! [A6: set_set_c_d_set_a] :
( ( ? [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ A6 ) )
= ( A6 != bot_bo58555506362910043_set_a ) ) ).
% ex_in_conv
thf(fact_152_ex__in__conv,axiom,
! [A6: set_option_a] :
( ( ? [X3: option_a] : ( member_option_a @ X3 @ A6 ) )
= ( A6 != bot_bot_set_option_a ) ) ).
% ex_in_conv
thf(fact_153_ex__in__conv,axiom,
! [A6: set_option_c_d_set_a] :
( ( ? [X3: option_c_d_set_a] : ( member4306893881663408030_set_a @ X3 @ A6 ) )
= ( A6 != bot_bo6666349697208826049_set_a ) ) ).
% ex_in_conv
thf(fact_154_ex__in__conv,axiom,
! [A6: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A6 ) )
= ( A6 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_155_ex__in__conv,axiom,
! [A6: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A6 ) )
= ( A6 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_156_ex__in__conv,axiom,
! [A6: set_c_d_set_a] :
( ( ? [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ A6 ) )
= ( A6 != bot_bo738396921950161403_set_a ) ) ).
% ex_in_conv
thf(fact_157_equals0I,axiom,
! [A6: set_set_c_d_set_a] :
( ! [Y2: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ Y2 @ A6 )
=> ( A6 = bot_bo58555506362910043_set_a ) ) ).
% equals0I
thf(fact_158_equals0I,axiom,
! [A6: set_option_a] :
( ! [Y2: option_a] :
~ ( member_option_a @ Y2 @ A6 )
=> ( A6 = bot_bot_set_option_a ) ) ).
% equals0I
thf(fact_159_equals0I,axiom,
! [A6: set_option_c_d_set_a] :
( ! [Y2: option_c_d_set_a] :
~ ( member4306893881663408030_set_a @ Y2 @ A6 )
=> ( A6 = bot_bo6666349697208826049_set_a ) ) ).
% equals0I
thf(fact_160_equals0I,axiom,
! [A6: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A6 )
=> ( A6 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_161_equals0I,axiom,
! [A6: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A6 )
=> ( A6 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_162_equals0I,axiom,
! [A6: set_c_d_set_a] :
( ! [Y2: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ Y2 @ A6 )
=> ( A6 = bot_bo738396921950161403_set_a ) ) ).
% equals0I
thf(fact_163_equals0D,axiom,
! [A6: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( A6 = bot_bo58555506362910043_set_a )
=> ~ ( member_set_c_d_set_a @ A @ A6 ) ) ).
% equals0D
thf(fact_164_equals0D,axiom,
! [A6: set_option_a,A: option_a] :
( ( A6 = bot_bot_set_option_a )
=> ~ ( member_option_a @ A @ A6 ) ) ).
% equals0D
thf(fact_165_equals0D,axiom,
! [A6: set_option_c_d_set_a,A: option_c_d_set_a] :
( ( A6 = bot_bo6666349697208826049_set_a )
=> ~ ( member4306893881663408030_set_a @ A @ A6 ) ) ).
% equals0D
thf(fact_166_equals0D,axiom,
! [A6: set_set_a,A: set_a] :
( ( A6 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A6 ) ) ).
% equals0D
thf(fact_167_equals0D,axiom,
! [A6: set_a,A: a] :
( ( A6 = bot_bot_set_a )
=> ~ ( member_a @ A @ A6 ) ) ).
% equals0D
thf(fact_168_equals0D,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( A6 = bot_bo738396921950161403_set_a )
=> ~ ( member_c_d_set_a @ A @ A6 ) ) ).
% equals0D
thf(fact_169_emptyE,axiom,
! [A: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ A @ bot_bo58555506362910043_set_a ) ).
% emptyE
thf(fact_170_emptyE,axiom,
! [A: option_a] :
~ ( member_option_a @ A @ bot_bot_set_option_a ) ).
% emptyE
thf(fact_171_emptyE,axiom,
! [A: option_c_d_set_a] :
~ ( member4306893881663408030_set_a @ A @ bot_bo6666349697208826049_set_a ) ).
% emptyE
thf(fact_172_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_173_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_174_emptyE,axiom,
! [A: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% emptyE
thf(fact_175_Collect__mono__iff,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ( ord_le5982164083705284911_set_a @ ( collect_c_d_set_a @ P ) @ ( collect_c_d_set_a @ Q ) )
= ( ! [X3: ( c > d ) > set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_176_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_177_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_178_set__eq__subset,axiom,
( ( ^ [Y4: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_c_d_set_a,B6: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A7 @ B6 )
& ( ord_le5982164083705284911_set_a @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_179_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_180_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_181_subset__trans,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,C3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ C3 )
=> ( ord_le5982164083705284911_set_a @ A6 @ C3 ) ) ) ).
% subset_trans
thf(fact_182_subset__trans,axiom,
! [A6: set_set_a,B5: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ C3 )
=> ( ord_le3724670747650509150_set_a @ A6 @ C3 ) ) ) ).
% subset_trans
thf(fact_183_subset__trans,axiom,
! [A6: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ord_less_eq_set_a @ A6 @ C3 ) ) ) ).
% subset_trans
thf(fact_184_Collect__mono,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ! [X2: ( c > d ) > set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5982164083705284911_set_a @ ( collect_c_d_set_a @ P ) @ ( collect_c_d_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_185_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_186_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_187_subset__refl,axiom,
! [A6: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A6 @ A6 ) ).
% subset_refl
thf(fact_188_subset__refl,axiom,
! [A6: set_set_a] : ( ord_le3724670747650509150_set_a @ A6 @ A6 ) ).
% subset_refl
thf(fact_189_subset__refl,axiom,
! [A6: set_a] : ( ord_less_eq_set_a @ A6 @ A6 ) ).
% subset_refl
thf(fact_190_subset__iff,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A7: set_set_c_d_set_a,B6: set_set_c_d_set_a] :
! [T: set_c_d_set_a] :
( ( member_set_c_d_set_a @ T @ A7 )
=> ( member_set_c_d_set_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A7: set_option_a,B6: set_option_a] :
! [T: option_a] :
( ( member_option_a @ T @ A7 )
=> ( member_option_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__iff,axiom,
( ord_le20137369925261813_set_a
= ( ^ [A7: set_option_c_d_set_a,B6: set_option_c_d_set_a] :
! [T: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ T @ A7 )
=> ( member4306893881663408030_set_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_193_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B6: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A7 )
=> ( member_set_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A7: set_c_d_set_a,B6: set_c_d_set_a] :
! [T: ( c > d ) > set_a] :
( ( member_c_d_set_a @ T @ A7 )
=> ( member_c_d_set_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_195_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A7 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_196_equalityD2,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( A6 = B5 )
=> ( ord_le5982164083705284911_set_a @ B5 @ A6 ) ) ).
% equalityD2
thf(fact_197_equalityD2,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( A6 = B5 )
=> ( ord_le3724670747650509150_set_a @ B5 @ A6 ) ) ).
% equalityD2
thf(fact_198_equalityD2,axiom,
! [A6: set_a,B5: set_a] :
( ( A6 = B5 )
=> ( ord_less_eq_set_a @ B5 @ A6 ) ) ).
% equalityD2
thf(fact_199_equalityD1,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( A6 = B5 )
=> ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ).
% equalityD1
thf(fact_200_equalityD1,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( A6 = B5 )
=> ( ord_le3724670747650509150_set_a @ A6 @ B5 ) ) ).
% equalityD1
thf(fact_201_equalityD1,axiom,
! [A6: set_a,B5: set_a] :
( ( A6 = B5 )
=> ( ord_less_eq_set_a @ A6 @ B5 ) ) ).
% equalityD1
thf(fact_202_subset__eq,axiom,
( ord_le7272806397018272911_set_a
= ( ^ [A7: set_set_c_d_set_a,B6: set_set_c_d_set_a] :
! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A7 )
=> ( member_set_c_d_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_203_subset__eq,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A7: set_option_a,B6: set_option_a] :
! [X3: option_a] :
( ( member_option_a @ X3 @ A7 )
=> ( member_option_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_204_subset__eq,axiom,
( ord_le20137369925261813_set_a
= ( ^ [A7: set_option_c_d_set_a,B6: set_option_c_d_set_a] :
! [X3: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ X3 @ A7 )
=> ( member4306893881663408030_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_205_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( member_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_206_subset__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A7: set_c_d_set_a,B6: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A7 )
=> ( member_c_d_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_207_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_208_equalityE,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( A6 = B5 )
=> ~ ( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ~ ( ord_le5982164083705284911_set_a @ B5 @ A6 ) ) ) ).
% equalityE
thf(fact_209_equalityE,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( A6 = B5 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ~ ( ord_le3724670747650509150_set_a @ B5 @ A6 ) ) ) ).
% equalityE
thf(fact_210_equalityE,axiom,
! [A6: set_a,B5: set_a] :
( ( A6 = B5 )
=> ~ ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ~ ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ).
% equalityE
thf(fact_211_subsetD,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A6 @ B5 )
=> ( ( member_set_c_d_set_a @ C2 @ A6 )
=> ( member_set_c_d_set_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_212_subsetD,axiom,
! [A6: set_option_a,B5: set_option_a,C2: option_a] :
( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
=> ( ( member_option_a @ C2 @ A6 )
=> ( member_option_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_213_subsetD,axiom,
! [A6: set_option_c_d_set_a,B5: set_option_c_d_set_a,C2: option_c_d_set_a] :
( ( ord_le20137369925261813_set_a @ A6 @ B5 )
=> ( ( member4306893881663408030_set_a @ C2 @ A6 )
=> ( member4306893881663408030_set_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_214_subsetD,axiom,
! [A6: set_set_a,B5: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ( member_set_a @ C2 @ A6 )
=> ( member_set_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_215_subsetD,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,C2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ( member_c_d_set_a @ C2 @ A6 )
=> ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_216_subsetD,axiom,
! [A6: set_a,B5: set_a,C2: a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( member_a @ C2 @ A6 )
=> ( member_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_217_in__mono,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a,X4: set_c_d_set_a] :
( ( ord_le7272806397018272911_set_a @ A6 @ B5 )
=> ( ( member_set_c_d_set_a @ X4 @ A6 )
=> ( member_set_c_d_set_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_218_in__mono,axiom,
! [A6: set_option_a,B5: set_option_a,X4: option_a] :
( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
=> ( ( member_option_a @ X4 @ A6 )
=> ( member_option_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_219_in__mono,axiom,
! [A6: set_option_c_d_set_a,B5: set_option_c_d_set_a,X4: option_c_d_set_a] :
( ( ord_le20137369925261813_set_a @ A6 @ B5 )
=> ( ( member4306893881663408030_set_a @ X4 @ A6 )
=> ( member4306893881663408030_set_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_220_in__mono,axiom,
! [A6: set_set_a,B5: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ( member_set_a @ X4 @ A6 )
=> ( member_set_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_221_in__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( member_c_d_set_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_222_in__mono,axiom,
! [A6: set_a,B5: set_a,X4: a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_223_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_set_set_a,X4: set_set_a] :
( ! [Y2: set_set_a] :
( ( member_set_set_a @ Y2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ Y2 @ X4 ) )
=> ( ! [Y2: set_set_a] :
( ! [Z3: set_set_a] :
( ( member_set_set_a @ Z3 @ A6 )
=> ( ord_le3724670747650509150_set_a @ Z3 @ Y2 ) )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y2 ) )
=> ( ( comple3958522678809307947_set_a @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_224_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_set_c_d_set_a,X4: set_c_d_set_a] :
( ! [Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Y2 @ A6 )
=> ( ord_le5982164083705284911_set_a @ Y2 @ X4 ) )
=> ( ! [Y2: set_c_d_set_a] :
( ! [Z3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Z3 @ A6 )
=> ( ord_le5982164083705284911_set_a @ Z3 @ Y2 ) )
=> ( ord_le5982164083705284911_set_a @ X4 @ Y2 ) )
=> ( ( comple6131501996466690428_set_a @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_225_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_c_d_set_a_o,X4: ( ( c > d ) > set_a ) > $o] :
( ! [Y2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ Y2 @ A6 )
=> ( ord_le961293222253252206et_a_o @ Y2 @ X4 ) )
=> ( ! [Y2: ( ( c > d ) > set_a ) > $o] :
( ! [Z3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ Z3 @ A6 )
=> ( ord_le961293222253252206et_a_o @ Z3 @ Y2 ) )
=> ( ord_le961293222253252206et_a_o @ X4 @ Y2 ) )
=> ( ( comple5290581719055393889et_a_o @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_226_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_a_o,X4: a > $o] :
( ! [Y2: a > $o] :
( ( member_a_o @ Y2 @ A6 )
=> ( ord_less_eq_a_o @ Y2 @ X4 ) )
=> ( ! [Y2: a > $o] :
( ! [Z3: a > $o] :
( ( member_a_o @ Z3 @ A6 )
=> ( ord_less_eq_a_o @ Z3 @ Y2 ) )
=> ( ord_less_eq_a_o @ X4 @ Y2 ) )
=> ( ( complete_Sup_Sup_a_o @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_227_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_set_a,X4: set_a] :
( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A6 )
=> ( ord_less_eq_set_a @ Y2 @ X4 ) )
=> ( ! [Y2: set_a] :
( ! [Z3: set_a] :
( ( member_set_a @ Z3 @ A6 )
=> ( ord_less_eq_set_a @ Z3 @ Y2 ) )
=> ( ord_less_eq_set_a @ X4 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_228_complete__lattice__class_OSup__eqI,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ! [Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ Y2 @ X4 ) )
=> ( ! [Y2: ( c > d ) > set_a] :
( ! [Z3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Z3 @ A6 )
=> ( ord_le8464990428230162895_set_a @ Z3 @ Y2 ) )
=> ( ord_le8464990428230162895_set_a @ X4 @ Y2 ) )
=> ( ( comple3834726295627996700_set_a @ A6 )
= X4 ) ) ) ).
% complete_lattice_class.Sup_eqI
thf(fact_229_complete__lattice__class_OSup__mono,axiom,
! [A6: set_set_set_a,B5: set_set_set_a] :
( ! [A2: set_set_a] :
( ( member_set_set_a @ A2 @ A6 )
=> ? [X: set_set_a] :
( ( member_set_set_a @ X @ B5 )
& ( ord_le3724670747650509150_set_a @ A2 @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ ( comple3958522678809307947_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_230_complete__lattice__class_OSup__mono,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ! [A2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ A2 @ A6 )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ B5 )
& ( ord_le5982164083705284911_set_a @ A2 @ X ) ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ ( comple6131501996466690428_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_231_complete__lattice__class_OSup__mono,axiom,
! [A6: set_c_d_set_a_o,B5: set_c_d_set_a_o] :
( ! [A2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ A2 @ A6 )
=> ? [X: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X @ B5 )
& ( ord_le961293222253252206et_a_o @ A2 @ X ) ) )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ ( comple5290581719055393889et_a_o @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_232_complete__lattice__class_OSup__mono,axiom,
! [A6: set_a_o,B5: set_a_o] :
( ! [A2: a > $o] :
( ( member_a_o @ A2 @ A6 )
=> ? [X: a > $o] :
( ( member_a_o @ X @ B5 )
& ( ord_less_eq_a_o @ A2 @ X ) ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ ( complete_Sup_Sup_a_o @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_233_complete__lattice__class_OSup__mono,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ! [A2: set_a] :
( ( member_set_a @ A2 @ A6 )
=> ? [X: set_a] :
( ( member_set_a @ X @ B5 )
& ( ord_less_eq_set_a @ A2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_234_complete__lattice__class_OSup__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ! [A2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( ord_le8464990428230162895_set_a @ A2 @ X ) ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ ( comple3834726295627996700_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_235_complete__lattice__class_OSup__least,axiom,
! [A6: set_set_set_a,Z: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_236_complete__lattice__class_OSup__least,axiom,
! [A6: set_set_c_d_set_a,Z: set_c_d_set_a] :
( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_237_complete__lattice__class_OSup__least,axiom,
! [A6: set_c_d_set_a_o,Z: ( ( c > d ) > set_a ) > $o] :
( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ A6 )
=> ( ord_le961293222253252206et_a_o @ X2 @ Z ) )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_238_complete__lattice__class_OSup__least,axiom,
! [A6: set_a_o,Z: a > $o] :
( ! [X2: a > $o] :
( ( member_a_o @ X2 @ A6 )
=> ( ord_less_eq_a_o @ X2 @ Z ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_239_complete__lattice__class_OSup__least,axiom,
! [A6: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_240_complete__lattice__class_OSup__least,axiom,
! [A6: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ Z ) ) ).
% complete_lattice_class.Sup_least
thf(fact_241_complete__lattice__class_OSup__upper,axiom,
! [X4: set_set_a,A6: set_set_set_a] :
( ( member_set_set_a @ X4 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( comple3958522678809307947_set_a @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_242_complete__lattice__class_OSup__upper,axiom,
! [X4: set_c_d_set_a,A6: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X4 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X4 @ ( comple6131501996466690428_set_a @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_243_complete__lattice__class_OSup__upper,axiom,
! [X4: ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a_o] :
( ( member_c_d_set_a_o @ X4 @ A6 )
=> ( ord_le961293222253252206et_a_o @ X4 @ ( comple5290581719055393889et_a_o @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_244_complete__lattice__class_OSup__upper,axiom,
! [X4: a > $o,A6: set_a_o] :
( ( member_a_o @ X4 @ A6 )
=> ( ord_less_eq_a_o @ X4 @ ( complete_Sup_Sup_a_o @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_245_complete__lattice__class_OSup__upper,axiom,
! [X4: set_a,A6: set_set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_246_complete__lattice__class_OSup__upper,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( comple3834726295627996700_set_a @ A6 ) ) ) ).
% complete_lattice_class.Sup_upper
thf(fact_247_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ B )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_248_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ B )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_249_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_c_d_set_a_o,B: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ B )
= ( ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ A6 )
=> ( ord_le961293222253252206et_a_o @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_250_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_a_o,B: a > $o] :
( ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ B )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
=> ( ord_less_eq_a_o @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_251_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ B )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ( ord_less_eq_set_a @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_252_complete__lattice__class_OSup__le__iff,axiom,
! [A6: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ B )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X3 @ B ) ) ) ) ).
% complete_lattice_class.Sup_le_iff
thf(fact_253_complete__lattice__class_OSup__upper2,axiom,
! [U: set_set_a,A6: set_set_set_a,V2: set_set_a] :
( ( member_set_set_a @ U @ A6 )
=> ( ( ord_le3724670747650509150_set_a @ V2 @ U )
=> ( ord_le3724670747650509150_set_a @ V2 @ ( comple3958522678809307947_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_254_complete__lattice__class_OSup__upper2,axiom,
! [U: set_c_d_set_a,A6: set_set_c_d_set_a,V2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ U @ A6 )
=> ( ( ord_le5982164083705284911_set_a @ V2 @ U )
=> ( ord_le5982164083705284911_set_a @ V2 @ ( comple6131501996466690428_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_255_complete__lattice__class_OSup__upper2,axiom,
! [U: ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a_o,V2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ U @ A6 )
=> ( ( ord_le961293222253252206et_a_o @ V2 @ U )
=> ( ord_le961293222253252206et_a_o @ V2 @ ( comple5290581719055393889et_a_o @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_256_complete__lattice__class_OSup__upper2,axiom,
! [U: a > $o,A6: set_a_o,V2: a > $o] :
( ( member_a_o @ U @ A6 )
=> ( ( ord_less_eq_a_o @ V2 @ U )
=> ( ord_less_eq_a_o @ V2 @ ( complete_Sup_Sup_a_o @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_257_complete__lattice__class_OSup__upper2,axiom,
! [U: set_a,A6: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A6 )
=> ( ( ord_less_eq_set_a @ V2 @ U )
=> ( ord_less_eq_set_a @ V2 @ ( comple2307003609928055243_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_258_complete__lattice__class_OSup__upper2,axiom,
! [U: ( c > d ) > set_a,A6: set_c_d_set_a,V2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ U @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ V2 @ U )
=> ( ord_le8464990428230162895_set_a @ V2 @ ( comple3834726295627996700_set_a @ A6 ) ) ) ) ).
% complete_lattice_class.Sup_upper2
thf(fact_259_cSup__eq__maximum,axiom,
! [Z: set_set_a,X6: set_set_set_a] :
( ( member_set_set_a @ Z @ X6 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) )
=> ( ( comple3958522678809307947_set_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_260_cSup__eq__maximum,axiom,
! [Z: set_c_d_set_a,X6: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ Z @ X6 )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ X6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z ) )
=> ( ( comple6131501996466690428_set_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_261_cSup__eq__maximum,axiom,
! [Z: ( ( c > d ) > set_a ) > $o,X6: set_c_d_set_a_o] :
( ( member_c_d_set_a_o @ Z @ X6 )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ X6 )
=> ( ord_le961293222253252206et_a_o @ X2 @ Z ) )
=> ( ( comple5290581719055393889et_a_o @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_262_cSup__eq__maximum,axiom,
! [Z: a > $o,X6: set_a_o] :
( ( member_a_o @ Z @ X6 )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ X6 )
=> ( ord_less_eq_a_o @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_a_o @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_263_cSup__eq__maximum,axiom,
! [Z: set_a,X6: set_set_a] :
( ( member_set_a @ Z @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_264_cSup__eq__maximum,axiom,
! [Z: ( c > d ) > set_a,X6: set_c_d_set_a] :
( ( member_c_d_set_a @ Z @ X6 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ X6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z ) )
=> ( ( comple3834726295627996700_set_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_265_set__closure__property__admissible,axiom,
! [S3: a > a > set_a] : ( comple1957918121334358780_set_a @ comple3834726295627996700_set_a @ ord_le8464990428230162895_set_a @ ( set_cl2807270042661212426_a_c_d @ S3 ) ) ).
% set_closure_property_admissible
thf(fact_266_bot__apply,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_267_bot__apply,axiom,
( bot_bot_c_d_set_a
= ( ^ [X3: c > d] : bot_bot_set_a ) ) ).
% bot_apply
thf(fact_268_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_269_preorder__class_Odual__order_Orefl,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_270_preorder__class_Odual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_271_preorder__class_Odual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_272_preorder__class_Odual__order_Orefl,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_273_preorder__class_Oorder__refl,axiom,
! [X4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X4 @ X4 ) ).
% preorder_class.order_refl
thf(fact_274_preorder__class_Oorder__refl,axiom,
! [X4: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ X4 ) ).
% preorder_class.order_refl
thf(fact_275_preorder__class_Oorder__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% preorder_class.order_refl
thf(fact_276_preorder__class_Oorder__refl,axiom,
! [X4: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X4 @ X4 ) ).
% preorder_class.order_refl
thf(fact_277_compatible__def,axiom,
! [A: a,B: a] :
( ( pre_compatible_a @ plus @ A @ B )
= ( ( plus @ A @ B )
!= none_a ) ) ).
% compatible_def
thf(fact_278_subset__emptyI,axiom,
! [A6: set_set_c_d_set_a] :
( ! [X2: set_c_d_set_a] :
~ ( member_set_c_d_set_a @ X2 @ A6 )
=> ( ord_le7272806397018272911_set_a @ A6 @ bot_bo58555506362910043_set_a ) ) ).
% subset_emptyI
thf(fact_279_subset__emptyI,axiom,
! [A6: set_option_a] :
( ! [X2: option_a] :
~ ( member_option_a @ X2 @ A6 )
=> ( ord_le1955136853071979460tion_a @ A6 @ bot_bot_set_option_a ) ) ).
% subset_emptyI
thf(fact_280_subset__emptyI,axiom,
! [A6: set_option_c_d_set_a] :
( ! [X2: option_c_d_set_a] :
~ ( member4306893881663408030_set_a @ X2 @ A6 )
=> ( ord_le20137369925261813_set_a @ A6 @ bot_bo6666349697208826049_set_a ) ) ).
% subset_emptyI
thf(fact_281_subset__emptyI,axiom,
! [A6: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ A6 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_282_subset__emptyI,axiom,
! [A6: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le5982164083705284911_set_a @ A6 @ bot_bo738396921950161403_set_a ) ) ).
% subset_emptyI
thf(fact_283_subset__emptyI,axiom,
! [A6: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ A6 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_284_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ A @ bot_bot_c_d_set_a_o )
=> ( A = bot_bot_c_d_set_a_o ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_285_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_286_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_287_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ bot_bo738396921950161403_set_a )
=> ( A = bot_bo738396921950161403_set_a ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_288_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_289_order__bot__class_Obot_Oextremum__uniqueI,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ bot_bot_c_d_set_a )
=> ( A = bot_bot_c_d_set_a ) ) ).
% order_bot_class.bot.extremum_uniqueI
thf(fact_290_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: ( ( c > d ) > set_a ) > $o] :
( ( ord_le961293222253252206et_a_o @ A @ bot_bot_c_d_set_a_o )
= ( A = bot_bot_c_d_set_a_o ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_291_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_292_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_293_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ bot_bo738396921950161403_set_a )
= ( A = bot_bo738396921950161403_set_a ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_294_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_295_order__bot__class_Obot_Oextremum__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ bot_bot_c_d_set_a )
= ( A = bot_bot_c_d_set_a ) ) ).
% order_bot_class.bot.extremum_unique
thf(fact_296_order__bot__class_Obot_Oextremum,axiom,
! [A: ( ( c > d ) > set_a ) > $o] : ( ord_le961293222253252206et_a_o @ bot_bot_c_d_set_a_o @ A ) ).
% order_bot_class.bot.extremum
thf(fact_297_order__bot__class_Obot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% order_bot_class.bot.extremum
thf(fact_298_order__bot__class_Obot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% order_bot_class.bot.extremum
thf(fact_299_order__bot__class_Obot_Oextremum,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A ) ).
% order_bot_class.bot.extremum
thf(fact_300_order__bot__class_Obot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% order_bot_class.bot.extremum
thf(fact_301_order__bot__class_Obot_Oextremum,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ bot_bot_c_d_set_a @ A ) ).
% order_bot_class.bot.extremum
thf(fact_302_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A7: set_set_a] : ( A7 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_303_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A7: set_a] : ( A7 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_304_Set_Ois__empty__def,axiom,
( is_empty_c_d_set_a
= ( ^ [A7: set_c_d_set_a] : ( A7 = bot_bo738396921950161403_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_305_cSup__subset__mono,axiom,
! [A6: set_set_set_a,B5: set_set_set_a] :
( ( A6 != bot_bo3380559777022489994_set_a )
=> ( ( condit5548584133349953570_set_a @ B5 )
=> ( ( ord_le5722252365846178494_set_a @ A6 @ B5 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ ( comple3958522678809307947_set_a @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_306_cSup__subset__mono,axiom,
! [A6: set_set_c_d_set_a,B5: set_set_c_d_set_a] :
( ( A6 != bot_bo58555506362910043_set_a )
=> ( ( condit2151140496889778675_set_a @ B5 )
=> ( ( ord_le7272806397018272911_set_a @ A6 @ B5 )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ ( comple6131501996466690428_set_a @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_307_cSup__subset__mono,axiom,
! [A6: set_c_d_set_a_o,B5: set_c_d_set_a_o] :
( ( A6 != bot_bo848287272940216920et_a_o )
=> ( ( condit6526574527931036330et_a_o @ B5 )
=> ( ( ord_le8757755980619729956et_a_o @ A6 @ B5 )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ ( comple5290581719055393889et_a_o @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_308_cSup__subset__mono,axiom,
! [A6: set_a_o,B5: set_a_o] :
( ( A6 != bot_bot_set_a_o2 )
=> ( ( condit5969422546283407003ve_a_o @ B5 )
=> ( ( ord_less_eq_set_a_o @ A6 @ B5 )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ ( complete_Sup_Sup_a_o @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_309_cSup__subset__mono,axiom,
! [A6: set_set_a,B5: set_set_a] :
( ( A6 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_310_cSup__subset__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( condit7392869265169887891_set_a @ B5 )
=> ( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ ( comple3834726295627996700_set_a @ B5 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_311_Union__iff,axiom,
! [A6: set_a,C3: set_set_set_a] :
( ( member_set_a @ A6 @ ( comple3958522678809307947_set_a @ C3 ) )
= ( ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ C3 )
& ( member_set_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_312_Union__iff,axiom,
! [A6: set_c_d_set_a,C3: set_se3584202636623819855_set_a] :
( ( member_set_c_d_set_a @ A6 @ ( comple2723893648999256284_set_a @ C3 ) )
= ( ? [X3: set_set_c_d_set_a] :
( ( member6574826897039512728_set_a @ X3 @ C3 )
& ( member_set_c_d_set_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_313_Union__iff,axiom,
! [A6: option_a,C3: set_set_option_a] :
( ( member_option_a @ A6 @ ( comple4629436989800923665tion_a @ C3 ) )
= ( ? [X3: set_option_a] :
( ( member_set_option_a @ X3 @ C3 )
& ( member_option_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_314_Union__iff,axiom,
! [A6: option_c_d_set_a,C3: set_se1522906970093639477_set_a] :
( ( member4306893881663408030_set_a @ A6 @ ( comple6619285321642111682_set_a @ C3 ) )
= ( ? [X3: set_option_c_d_set_a] :
( ( member8260580452227636350_set_a @ X3 @ C3 )
& ( member4306893881663408030_set_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_315_Union__iff,axiom,
! [A6: a,C3: set_set_a] :
( ( member_a @ A6 @ ( comple2307003609928055243_set_a @ C3 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ C3 )
& ( member_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_316_Union__iff,axiom,
! [A6: ( c > d ) > set_a,C3: set_set_c_d_set_a] :
( ( member_c_d_set_a @ A6 @ ( comple6131501996466690428_set_a @ C3 ) )
= ( ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ C3 )
& ( member_c_d_set_a @ A6 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_317_UnionI,axiom,
! [X6: set_set_a,C3: set_set_set_a,A6: set_a] :
( ( member_set_set_a @ X6 @ C3 )
=> ( ( member_set_a @ A6 @ X6 )
=> ( member_set_a @ A6 @ ( comple3958522678809307947_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_318_UnionI,axiom,
! [X6: set_set_c_d_set_a,C3: set_se3584202636623819855_set_a,A6: set_c_d_set_a] :
( ( member6574826897039512728_set_a @ X6 @ C3 )
=> ( ( member_set_c_d_set_a @ A6 @ X6 )
=> ( member_set_c_d_set_a @ A6 @ ( comple2723893648999256284_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_319_UnionI,axiom,
! [X6: set_option_a,C3: set_set_option_a,A6: option_a] :
( ( member_set_option_a @ X6 @ C3 )
=> ( ( member_option_a @ A6 @ X6 )
=> ( member_option_a @ A6 @ ( comple4629436989800923665tion_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_320_UnionI,axiom,
! [X6: set_option_c_d_set_a,C3: set_se1522906970093639477_set_a,A6: option_c_d_set_a] :
( ( member8260580452227636350_set_a @ X6 @ C3 )
=> ( ( member4306893881663408030_set_a @ A6 @ X6 )
=> ( member4306893881663408030_set_a @ A6 @ ( comple6619285321642111682_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_321_UnionI,axiom,
! [X6: set_a,C3: set_set_a,A6: a] :
( ( member_set_a @ X6 @ C3 )
=> ( ( member_a @ A6 @ X6 )
=> ( member_a @ A6 @ ( comple2307003609928055243_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_322_UnionI,axiom,
! [X6: set_c_d_set_a,C3: set_set_c_d_set_a,A6: ( c > d ) > set_a] :
( ( member_set_c_d_set_a @ X6 @ C3 )
=> ( ( member_c_d_set_a @ A6 @ X6 )
=> ( member_c_d_set_a @ A6 @ ( comple6131501996466690428_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_323_UN__ball__bex__simps_I1_J,axiom,
! [A6: set_set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ ( comple6131501996466690428_set_a @ A6 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
=> ! [Y5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y5 @ X3 )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_324_UN__ball__bex__simps_I1_J,axiom,
! [A6: set_set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ A6 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ X3 )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_325_UN__ball__bex__simps_I3_J,axiom,
! [A6: set_set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ ( comple6131501996466690428_set_a @ A6 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A6 )
& ? [Y5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y5 @ X3 )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_326_UN__ball__bex__simps_I3_J,axiom,
! [A6: set_set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ A6 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
& ? [Y5: a] :
( ( member_a @ Y5 @ X3 )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_327_order__top__class_Obdd__above__top,axiom,
! [A6: set_c_d_set_a] : ( condit7392869265169887891_set_a @ A6 ) ).
% order_top_class.bdd_above_top
thf(fact_328_order__top__class_Obdd__above__top,axiom,
! [A6: set_set_a] : ( condit3373647341569784514_set_a @ A6 ) ).
% order_top_class.bdd_above_top
thf(fact_329_preorder__class_Obdd__above_OI,axiom,
! [A6: set_set_c_d_set_a,M: set_c_d_set_a] :
( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ M ) )
=> ( condit2151140496889778675_set_a @ A6 ) ) ).
% preorder_class.bdd_above.I
thf(fact_330_preorder__class_Obdd__above_OI,axiom,
! [A6: set_set_set_a,M: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ M ) )
=> ( condit5548584133349953570_set_a @ A6 ) ) ).
% preorder_class.bdd_above.I
thf(fact_331_preorder__class_Obdd__above_OI,axiom,
! [A6: set_set_a,M: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ X2 @ M ) )
=> ( condit3373647341569784514_set_a @ A6 ) ) ).
% preorder_class.bdd_above.I
thf(fact_332_preorder__class_Obdd__above_OI,axiom,
! [A6: set_c_d_set_a,M: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ M ) )
=> ( condit7392869265169887891_set_a @ A6 ) ) ).
% preorder_class.bdd_above.I
thf(fact_333_not__None__eq,axiom,
! [X4: option_c_d_set_a] :
( ( X4 != none_c_d_set_a )
= ( ? [Y5: ( c > d ) > set_a] :
( X4
= ( some_c_d_set_a @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_334_not__None__eq,axiom,
! [X4: option_a] :
( ( X4 != none_a )
= ( ? [Y5: a] :
( X4
= ( some_a @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_335_not__Some__eq,axiom,
! [X4: option_c_d_set_a] :
( ( ! [Y5: ( c > d ) > set_a] :
( X4
!= ( some_c_d_set_a @ Y5 ) ) )
= ( X4 = none_c_d_set_a ) ) ).
% not_Some_eq
thf(fact_336_not__Some__eq,axiom,
! [X4: option_a] :
( ( ! [Y5: a] :
( X4
!= ( some_a @ Y5 ) ) )
= ( X4 = none_a ) ) ).
% not_Some_eq
thf(fact_337_preorder__class_Obdd__above__empty,axiom,
condit3373647341569784514_set_a @ bot_bot_set_set_a ).
% preorder_class.bdd_above_empty
thf(fact_338_preorder__class_Obdd__above__empty,axiom,
condit7392869265169887891_set_a @ bot_bo738396921950161403_set_a ).
% preorder_class.bdd_above_empty
thf(fact_339_UnionE,axiom,
! [A6: set_a,C3: set_set_set_a] :
( ( member_set_a @ A6 @ ( comple3958522678809307947_set_a @ C3 ) )
=> ~ ! [X7: set_set_a] :
( ( member_set_a @ A6 @ X7 )
=> ~ ( member_set_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_340_UnionE,axiom,
! [A6: set_c_d_set_a,C3: set_se3584202636623819855_set_a] :
( ( member_set_c_d_set_a @ A6 @ ( comple2723893648999256284_set_a @ C3 ) )
=> ~ ! [X7: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ A6 @ X7 )
=> ~ ( member6574826897039512728_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_341_UnionE,axiom,
! [A6: option_a,C3: set_set_option_a] :
( ( member_option_a @ A6 @ ( comple4629436989800923665tion_a @ C3 ) )
=> ~ ! [X7: set_option_a] :
( ( member_option_a @ A6 @ X7 )
=> ~ ( member_set_option_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_342_UnionE,axiom,
! [A6: option_c_d_set_a,C3: set_se1522906970093639477_set_a] :
( ( member4306893881663408030_set_a @ A6 @ ( comple6619285321642111682_set_a @ C3 ) )
=> ~ ! [X7: set_option_c_d_set_a] :
( ( member4306893881663408030_set_a @ A6 @ X7 )
=> ~ ( member8260580452227636350_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_343_UnionE,axiom,
! [A6: a,C3: set_set_a] :
( ( member_a @ A6 @ ( comple2307003609928055243_set_a @ C3 ) )
=> ~ ! [X7: set_a] :
( ( member_a @ A6 @ X7 )
=> ~ ( member_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_344_UnionE,axiom,
! [A6: ( c > d ) > set_a,C3: set_set_c_d_set_a] :
( ( member_c_d_set_a @ A6 @ ( comple6131501996466690428_set_a @ C3 ) )
=> ~ ! [X7: set_c_d_set_a] :
( ( member_c_d_set_a @ A6 @ X7 )
=> ~ ( member_set_c_d_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_345_preorder__class_Obdd__above_OE,axiom,
! [A6: set_set_c_d_set_a] :
( ( condit2151140496889778675_set_a @ A6 )
=> ~ ! [M2: set_c_d_set_a] :
~ ! [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A6 )
=> ( ord_le5982164083705284911_set_a @ X @ M2 ) ) ) ).
% preorder_class.bdd_above.E
thf(fact_346_preorder__class_Obdd__above_OE,axiom,
! [A6: set_set_set_a] :
( ( condit5548584133349953570_set_a @ A6 )
=> ~ ! [M2: set_set_a] :
~ ! [X: set_set_a] :
( ( member_set_set_a @ X @ A6 )
=> ( ord_le3724670747650509150_set_a @ X @ M2 ) ) ) ).
% preorder_class.bdd_above.E
thf(fact_347_preorder__class_Obdd__above_OE,axiom,
! [A6: set_set_a] :
( ( condit3373647341569784514_set_a @ A6 )
=> ~ ! [M2: set_a] :
~ ! [X: set_a] :
( ( member_set_a @ X @ A6 )
=> ( ord_less_eq_set_a @ X @ M2 ) ) ) ).
% preorder_class.bdd_above.E
thf(fact_348_preorder__class_Obdd__above_OE,axiom,
! [A6: set_c_d_set_a] :
( ( condit7392869265169887891_set_a @ A6 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
=> ( ord_le8464990428230162895_set_a @ X @ M2 ) ) ) ).
% preorder_class.bdd_above.E
thf(fact_349_preorder__class_Obdd__above_Ounfold,axiom,
( condit2151140496889778675_set_a
= ( ^ [A7: set_set_c_d_set_a] :
? [M3: set_c_d_set_a] :
! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A7 )
=> ( ord_le5982164083705284911_set_a @ X3 @ M3 ) ) ) ) ).
% preorder_class.bdd_above.unfold
thf(fact_350_preorder__class_Obdd__above_Ounfold,axiom,
( condit5548584133349953570_set_a
= ( ^ [A7: set_set_set_a] :
? [M3: set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A7 )
=> ( ord_le3724670747650509150_set_a @ X3 @ M3 ) ) ) ) ).
% preorder_class.bdd_above.unfold
thf(fact_351_preorder__class_Obdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A7: set_set_a] :
? [M3: set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( ord_less_eq_set_a @ X3 @ M3 ) ) ) ) ).
% preorder_class.bdd_above.unfold
thf(fact_352_preorder__class_Obdd__above_Ounfold,axiom,
( condit7392869265169887891_set_a
= ( ^ [A7: set_c_d_set_a] :
? [M3: ( c > d ) > set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A7 )
=> ( ord_le8464990428230162895_set_a @ X3 @ M3 ) ) ) ) ).
% preorder_class.bdd_above.unfold
thf(fact_353_preorder__class_Obdd__above__mono,axiom,
! [B5: set_c_d_set_a,A6: set_c_d_set_a] :
( ( condit7392869265169887891_set_a @ B5 )
=> ( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( condit7392869265169887891_set_a @ A6 ) ) ) ).
% preorder_class.bdd_above_mono
thf(fact_354_preorder__class_Obdd__above__mono,axiom,
! [B5: set_set_a,A6: set_set_a] :
( ( condit3373647341569784514_set_a @ B5 )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( condit3373647341569784514_set_a @ A6 ) ) ) ).
% preorder_class.bdd_above_mono
thf(fact_355_option_Odistinct_I1_J,axiom,
! [X22: ( c > d ) > set_a] :
( none_c_d_set_a
!= ( some_c_d_set_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_356_option_Odistinct_I1_J,axiom,
! [X22: a] :
( none_a
!= ( some_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_357_option_OdiscI,axiom,
! [Option: option_c_d_set_a,X22: ( c > d ) > set_a] :
( ( Option
= ( some_c_d_set_a @ X22 ) )
=> ( Option != none_c_d_set_a ) ) ).
% option.discI
thf(fact_358_option_OdiscI,axiom,
! [Option: option_a,X22: a] :
( ( Option
= ( some_a @ X22 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_359_option_Oexhaust,axiom,
! [Y: option_c_d_set_a] :
( ( Y != none_c_d_set_a )
=> ~ ! [X23: ( c > d ) > set_a] :
( Y
!= ( some_c_d_set_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_360_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X23: a] :
( Y
!= ( some_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_361_split__option__ex,axiom,
( ( ^ [P2: option_c_d_set_a > $o] :
? [X8: option_c_d_set_a] : ( P2 @ X8 ) )
= ( ^ [P3: option_c_d_set_a > $o] :
( ( P3 @ none_c_d_set_a )
| ? [X3: ( c > d ) > set_a] : ( P3 @ ( some_c_d_set_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_362_split__option__ex,axiom,
( ( ^ [P2: option_a > $o] :
? [X8: option_a] : ( P2 @ X8 ) )
= ( ^ [P3: option_a > $o] :
( ( P3 @ none_a )
| ? [X3: a] : ( P3 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_363_split__option__all,axiom,
( ( ^ [P2: option_c_d_set_a > $o] :
! [X8: option_c_d_set_a] : ( P2 @ X8 ) )
= ( ^ [P3: option_c_d_set_a > $o] :
( ( P3 @ none_c_d_set_a )
& ! [X3: ( c > d ) > set_a] : ( P3 @ ( some_c_d_set_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_364_split__option__all,axiom,
( ( ^ [P2: option_a > $o] :
! [X8: option_a] : ( P2 @ X8 ) )
= ( ^ [P3: option_a > $o] :
( ( P3 @ none_a )
& ! [X3: a] : ( P3 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_365_combine__options__cases,axiom,
! [X4: option_a,P: option_a > option_c_d_set_a > $o,Y: option_c_d_set_a] :
( ( ( X4 = none_a )
=> ( P @ X4 @ Y ) )
=> ( ( ( Y = none_c_d_set_a )
=> ( P @ X4 @ Y ) )
=> ( ! [A2: a,B2: ( c > d ) > set_a] :
( ( X4
= ( some_a @ A2 ) )
=> ( ( Y
= ( some_c_d_set_a @ B2 ) )
=> ( P @ X4 @ Y ) ) )
=> ( P @ X4 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_366_combine__options__cases,axiom,
! [X4: option_c_d_set_a,P: option_c_d_set_a > option_a > $o,Y: option_a] :
( ( ( X4 = none_c_d_set_a )
=> ( P @ X4 @ Y ) )
=> ( ( ( Y = none_a )
=> ( P @ X4 @ Y ) )
=> ( ! [A2: ( c > d ) > set_a,B2: a] :
( ( X4
= ( some_c_d_set_a @ A2 ) )
=> ( ( Y
= ( some_a @ B2 ) )
=> ( P @ X4 @ Y ) ) )
=> ( P @ X4 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_367_combine__options__cases,axiom,
! [X4: option_c_d_set_a,P: option_c_d_set_a > option_c_d_set_a > $o,Y: option_c_d_set_a] :
( ( ( X4 = none_c_d_set_a )
=> ( P @ X4 @ Y ) )
=> ( ( ( Y = none_c_d_set_a )
=> ( P @ X4 @ Y ) )
=> ( ! [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( X4
= ( some_c_d_set_a @ A2 ) )
=> ( ( Y
= ( some_c_d_set_a @ B2 ) )
=> ( P @ X4 @ Y ) ) )
=> ( P @ X4 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_368_combine__options__cases,axiom,
! [X4: option_a,P: option_a > option_a > $o,Y: option_a] :
( ( ( X4 = none_a )
=> ( P @ X4 @ Y ) )
=> ( ( ( Y = none_a )
=> ( P @ X4 @ Y ) )
=> ( ! [A2: a,B2: a] :
( ( X4
= ( some_a @ A2 ) )
=> ( ( Y
= ( some_a @ B2 ) )
=> ( P @ X4 @ Y ) ) )
=> ( P @ X4 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_369_pre__logic_Ocompatible__def,axiom,
( pre_co1390589184961732201_set_a
= ( ^ [Plus: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > option_c_d_set_a,A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( Plus @ A3 @ B3 )
!= none_c_d_set_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_370_pre__logic_Ocompatible__def,axiom,
( pre_compatible_a
= ( ^ [Plus: a > a > option_a,A3: a,B3: a] :
( ( Plus @ A3 @ B3 )
!= none_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_371_cSup__upper,axiom,
! [X4: set_set_a,X6: set_set_set_a] :
( ( member_set_set_a @ X4 @ X6 )
=> ( ( condit5548584133349953570_set_a @ X6 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( comple3958522678809307947_set_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_372_cSup__upper,axiom,
! [X4: set_c_d_set_a,X6: set_set_c_d_set_a] :
( ( member_set_c_d_set_a @ X4 @ X6 )
=> ( ( condit2151140496889778675_set_a @ X6 )
=> ( ord_le5982164083705284911_set_a @ X4 @ ( comple6131501996466690428_set_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_373_cSup__upper,axiom,
! [X4: ( ( c > d ) > set_a ) > $o,X6: set_c_d_set_a_o] :
( ( member_c_d_set_a_o @ X4 @ X6 )
=> ( ( condit6526574527931036330et_a_o @ X6 )
=> ( ord_le961293222253252206et_a_o @ X4 @ ( comple5290581719055393889et_a_o @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_374_cSup__upper,axiom,
! [X4: a > $o,X6: set_a_o] :
( ( member_a_o @ X4 @ X6 )
=> ( ( condit5969422546283407003ve_a_o @ X6 )
=> ( ord_less_eq_a_o @ X4 @ ( complete_Sup_Sup_a_o @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_375_cSup__upper,axiom,
! [X4: set_a,X6: set_set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( condit3373647341569784514_set_a @ X6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_376_cSup__upper,axiom,
! [X4: ( c > d ) > set_a,X6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ X6 )
=> ( ( condit7392869265169887891_set_a @ X6 )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( comple3834726295627996700_set_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_377_cSup__upper2,axiom,
! [X4: set_set_a,X6: set_set_set_a,Y: set_set_a] :
( ( member_set_set_a @ X4 @ X6 )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( ( condit5548584133349953570_set_a @ X6 )
=> ( ord_le3724670747650509150_set_a @ Y @ ( comple3958522678809307947_set_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_378_cSup__upper2,axiom,
! [X4: set_c_d_set_a,X6: set_set_c_d_set_a,Y: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X4 @ X6 )
=> ( ( ord_le5982164083705284911_set_a @ Y @ X4 )
=> ( ( condit2151140496889778675_set_a @ X6 )
=> ( ord_le5982164083705284911_set_a @ Y @ ( comple6131501996466690428_set_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_379_cSup__upper2,axiom,
! [X4: ( ( c > d ) > set_a ) > $o,X6: set_c_d_set_a_o,Y: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X4 @ X6 )
=> ( ( ord_le961293222253252206et_a_o @ Y @ X4 )
=> ( ( condit6526574527931036330et_a_o @ X6 )
=> ( ord_le961293222253252206et_a_o @ Y @ ( comple5290581719055393889et_a_o @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_380_cSup__upper2,axiom,
! [X4: a > $o,X6: set_a_o,Y: a > $o] :
( ( member_a_o @ X4 @ X6 )
=> ( ( ord_less_eq_a_o @ Y @ X4 )
=> ( ( condit5969422546283407003ve_a_o @ X6 )
=> ( ord_less_eq_a_o @ Y @ ( complete_Sup_Sup_a_o @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_381_cSup__upper2,axiom,
! [X4: set_a,X6: set_set_a,Y: set_a] :
( ( member_set_a @ X4 @ X6 )
=> ( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( condit3373647341569784514_set_a @ X6 )
=> ( ord_less_eq_set_a @ Y @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_382_cSup__upper2,axiom,
! [X4: ( c > d ) > set_a,X6: set_c_d_set_a,Y: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ X6 )
=> ( ( ord_le8464990428230162895_set_a @ Y @ X4 )
=> ( ( condit7392869265169887891_set_a @ X6 )
=> ( ord_le8464990428230162895_set_a @ Y @ ( comple3834726295627996700_set_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_383_cSup__mono,axiom,
! [B5: set_set_set_a,A6: set_set_set_a] :
( ( B5 != bot_bo3380559777022489994_set_a )
=> ( ( condit5548584133349953570_set_a @ A6 )
=> ( ! [B2: set_set_a] :
( ( member_set_set_a @ B2 @ B5 )
=> ? [X: set_set_a] :
( ( member_set_set_a @ X @ A6 )
& ( ord_le3724670747650509150_set_a @ B2 @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ B5 ) @ ( comple3958522678809307947_set_a @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_384_cSup__mono,axiom,
! [B5: set_set_c_d_set_a,A6: set_set_c_d_set_a] :
( ( B5 != bot_bo58555506362910043_set_a )
=> ( ( condit2151140496889778675_set_a @ A6 )
=> ( ! [B2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ B2 @ B5 )
=> ? [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ A6 )
& ( ord_le5982164083705284911_set_a @ B2 @ X ) ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ B5 ) @ ( comple6131501996466690428_set_a @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_385_cSup__mono,axiom,
! [B5: set_c_d_set_a_o,A6: set_c_d_set_a_o] :
( ( B5 != bot_bo848287272940216920et_a_o )
=> ( ( condit6526574527931036330et_a_o @ A6 )
=> ( ! [B2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ B2 @ B5 )
=> ? [X: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X @ A6 )
& ( ord_le961293222253252206et_a_o @ B2 @ X ) ) )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ B5 ) @ ( comple5290581719055393889et_a_o @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_386_cSup__mono,axiom,
! [B5: set_a_o,A6: set_a_o] :
( ( B5 != bot_bot_set_a_o2 )
=> ( ( condit5969422546283407003ve_a_o @ A6 )
=> ( ! [B2: a > $o] :
( ( member_a_o @ B2 @ B5 )
=> ? [X: a > $o] :
( ( member_a_o @ X @ A6 )
& ( ord_less_eq_a_o @ B2 @ X ) ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ B5 ) @ ( complete_Sup_Sup_a_o @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_387_cSup__mono,axiom,
! [B5: set_set_a,A6: set_set_a] :
( ( B5 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A6 )
=> ( ! [B2: set_a] :
( ( member_set_a @ B2 @ B5 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A6 )
& ( ord_less_eq_set_a @ B2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B5 ) @ ( comple2307003609928055243_set_a @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_388_cSup__mono,axiom,
! [B5: set_c_d_set_a,A6: set_c_d_set_a] :
( ( B5 != bot_bo738396921950161403_set_a )
=> ( ( condit7392869265169887891_set_a @ A6 )
=> ( ! [B2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B2 @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( ord_le8464990428230162895_set_a @ B2 @ X ) ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ B5 ) @ ( comple3834726295627996700_set_a @ A6 ) ) ) ) ) ).
% cSup_mono
thf(fact_389_cSup__le__iff,axiom,
! [S3: set_set_set_a,A: set_set_a] :
( ( S3 != bot_bo3380559777022489994_set_a )
=> ( ( condit5548584133349953570_set_a @ S3 )
=> ( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ S3 ) @ A )
= ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ S3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_390_cSup__le__iff,axiom,
! [S3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( S3 != bot_bo58555506362910043_set_a )
=> ( ( condit2151140496889778675_set_a @ S3 )
=> ( ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ S3 ) @ A )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ S3 )
=> ( ord_le5982164083705284911_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_391_cSup__le__iff,axiom,
! [S3: set_c_d_set_a_o,A: ( ( c > d ) > set_a ) > $o] :
( ( S3 != bot_bo848287272940216920et_a_o )
=> ( ( condit6526574527931036330et_a_o @ S3 )
=> ( ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ S3 ) @ A )
= ( ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ S3 )
=> ( ord_le961293222253252206et_a_o @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_392_cSup__le__iff,axiom,
! [S3: set_a_o,A: a > $o] :
( ( S3 != bot_bot_set_a_o2 )
=> ( ( condit5969422546283407003ve_a_o @ S3 )
=> ( ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ S3 ) @ A )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ S3 )
=> ( ord_less_eq_a_o @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_393_cSup__le__iff,axiom,
! [S3: set_set_a,A: set_a] :
( ( S3 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S3 )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S3 ) @ A )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ S3 )
=> ( ord_less_eq_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_394_cSup__le__iff,axiom,
! [S3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( S3 != bot_bo738396921950161403_set_a )
=> ( ( condit7392869265169887891_set_a @ S3 )
=> ( ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ S3 ) @ A )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ S3 )
=> ( ord_le8464990428230162895_set_a @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_395_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_c_d_set_a,Y5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y5 )
& ( ord_le5982164083705284911_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_396_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
& ( ord_le3724670747650509150_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_397_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_398_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: ( c > d ) > set_a,Y5: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y5 )
& ( ord_le8464990428230162895_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_399_ord__class_Oord__eq__le__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C2: set_c_d_set_a] :
( ( A = B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C2 )
=> ( ord_le5982164083705284911_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_400_ord__class_Oord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_401_ord__class_Oord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_402_ord__class_Oord__eq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( A = B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ord_le8464990428230162895_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_403_ord__class_Oord__le__eq__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le5982164083705284911_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_404_ord__class_Oord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_405_ord__class_Oord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_406_ord__class_Oord__le__eq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le8464990428230162895_set_a @ A @ C2 ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_407_order__class_Oorder__antisym,axiom,
! [X4: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X4 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_class.order_antisym
thf(fact_408_order__class_Oorder__antisym,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_class.order_antisym
thf(fact_409_order__class_Oorder__antisym,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_class.order_antisym
thf(fact_410_order__class_Oorder__antisym,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_class.order_antisym
thf(fact_411_preorder__class_Oorder_Otrans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C2 )
=> ( ord_le5982164083705284911_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.trans
thf(fact_412_preorder__class_Oorder_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.trans
thf(fact_413_preorder__class_Oorder_Otrans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.trans
thf(fact_414_preorder__class_Oorder_Otrans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ord_le8464990428230162895_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.trans
thf(fact_415_preorder__class_Oorder__trans,axiom,
! [X4: set_c_d_set_a,Y: set_c_d_set_a,Z: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X4 @ Y )
=> ( ( ord_le5982164083705284911_set_a @ Y @ Z )
=> ( ord_le5982164083705284911_set_a @ X4 @ Z ) ) ) ).
% preorder_class.order_trans
thf(fact_416_preorder__class_Oorder__trans,axiom,
! [X4: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X4 @ Z ) ) ) ).
% preorder_class.order_trans
thf(fact_417_preorder__class_Oorder__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% preorder_class.order_trans
thf(fact_418_preorder__class_Oorder__trans,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z )
=> ( ord_le8464990428230162895_set_a @ X4 @ Z ) ) ) ).
% preorder_class.order_trans
thf(fact_419_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B3 @ A3 )
& ( ord_le5982164083705284911_set_a @ A3 @ B3 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_420_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_421_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_422_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B3 @ A3 )
& ( ord_le8464990428230162895_set_a @ A3 @ B3 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_423_order__class_Odual__order_Oantisym,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( A = B ) ) ) ).
% order_class.dual_order.antisym
thf(fact_424_order__class_Odual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% order_class.dual_order.antisym
thf(fact_425_order__class_Odual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% order_class.dual_order.antisym
thf(fact_426_order__class_Odual__order_Oantisym,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( A = B ) ) ) ).
% order_class.dual_order.antisym
thf(fact_427_preorder__class_Odual__order_Otrans,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ C2 @ B )
=> ( ord_le5982164083705284911_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_428_preorder__class_Odual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_429_preorder__class_Odual__order_Otrans,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_430_preorder__class_Odual__order_Otrans,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C2 @ B )
=> ( ord_le8464990428230162895_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_431_antisym,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_432_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_433_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_434_antisym,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_435_le__funD,axiom,
! [F: ( c > d ) > set_a,G: ( c > d ) > set_a,X4: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_436_le__funE,axiom,
! [F: ( c > d ) > set_a,G: ( c > d ) > set_a,X4: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_437_le__funI,axiom,
! [F: ( c > d ) > set_a,G: ( c > d ) > set_a] :
( ! [X2: c > d] : ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le8464990428230162895_set_a @ F @ G ) ) ).
% le_funI
thf(fact_438_le__fun__def,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [F2: ( c > d ) > set_a,G2: ( c > d ) > set_a] :
! [X3: c > d] : ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_439_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_c_d_set_a,Z2: set_c_d_set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_c_d_set_a,B3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B3 )
& ( ord_le5982164083705284911_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_440_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_441_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_442_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y4 = Z2 ) )
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A3 @ B3 )
& ( ord_le8464990428230162895_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_443_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_444_order__subst1,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_445_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C2: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_446_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_447_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_448_order__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_449_order__subst1,axiom,
! [A: set_set_a,F: set_set_a > set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_450_order__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C2 )
=> ( ! [X2: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_451_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_set_a > ( c > d ) > set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_452_order__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C2: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_453_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_454_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_455_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C2: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_456_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_457_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_458_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_459_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_460_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_461_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_set_a,C2: set_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_462_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C2: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_463_order__eq__refl,axiom,
! [X4: set_c_d_set_a,Y: set_c_d_set_a] :
( ( X4 = Y )
=> ( ord_le5982164083705284911_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_464_order__eq__refl,axiom,
! [X4: set_set_a,Y: set_set_a] :
( ( X4 = Y )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_465_order__eq__refl,axiom,
! [X4: set_a,Y: set_a] :
( ( X4 = Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_466_order__eq__refl,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( X4 = Y )
=> ( ord_le8464990428230162895_set_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_467_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_468_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_469_ord__eq__le__subst,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_470_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_471_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_472_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C2: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_473_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_set_a > set_set_a,B: set_set_a,C2: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_474_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C2: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_475_ord__eq__le__subst,axiom,
! [A: set_set_a,F: ( ( c > d ) > set_a ) > set_set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_476_ord__eq__le__subst,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C2: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C2 )
=> ( ! [X2: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_477_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_478_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_479_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C2: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_481_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C2: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_482_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_483_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_484_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C2: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_485_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_set_a,C2: set_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_486_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C2: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_487_order__antisym__conv,axiom,
! [Y: set_c_d_set_a,X4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y @ X4 )
=> ( ( ord_le5982164083705284911_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
! [Y: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_489_order__antisym__conv,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_490_order__antisym__conv,axiom,
! [Y: ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ X4 )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_491_bot__fun__def,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_492_bot__fun__def,axiom,
( bot_bot_c_d_set_a
= ( ^ [X3: c > d] : bot_bot_set_a ) ) ).
% bot_fun_def
thf(fact_493_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_494_ccpo_OadmissibleD,axiom,
! [Lub: set_set_c_d_set_a > set_c_d_set_a,Ord: set_c_d_set_a > set_c_d_set_a > $o,P: set_c_d_set_a > $o,A6: set_set_c_d_set_a] :
( ( comple2100625987831660124_set_a @ Lub @ Ord @ P )
=> ( ( comple2185443536470187199_set_a @ Ord @ A6 )
=> ( ( A6 != bot_bo58555506362910043_set_a )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_495_ccpo_OadmissibleD,axiom,
! [Lub: set_option_a > option_a,Ord: option_a > option_a > $o,P: option_a > $o,A6: set_option_a] :
( ( comple2995885364255664145tion_a @ Lub @ Ord @ P )
=> ( ( comple6692356590997824628tion_a @ Ord @ A6 )
=> ( ( A6 != bot_bot_set_option_a )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_496_ccpo_OadmissibleD,axiom,
! [Lub: set_option_c_d_set_a > option_c_d_set_a,Ord: option_c_d_set_a > option_c_d_set_a > $o,P: option_c_d_set_a > $o,A6: set_option_c_d_set_a] :
( ( comple3108528645285135042_set_a @ Lub @ Ord @ P )
=> ( ( comple2313665252861385893_set_a @ Ord @ A6 )
=> ( ( A6 != bot_bo6666349697208826049_set_a )
=> ( ! [X2: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_497_ccpo_OadmissibleD,axiom,
! [Lub: set_set_a > set_a,Ord: set_a > set_a > $o,P: set_a > $o,A6: set_set_a] :
( ( comple8887300225568239275_set_a @ Lub @ Ord @ P )
=> ( ( comple4316259127148425102_set_a @ Ord @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_498_ccpo_OadmissibleD,axiom,
! [Lub: set_a > a,Ord: a > a > $o,P: a > $o,A6: set_a] :
( ( comple72871723935627595ible_a @ Lub @ Ord @ P )
=> ( ( comple1697357536187991598hain_a @ Ord @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_499_ccpo_OadmissibleD,axiom,
! [Lub: set_c_d_set_a > ( c > d ) > set_a,Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,P: ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a] :
( ( comple1957918121334358780_set_a @ Lub @ Ord @ P )
=> ( ( comple7455786223818501471_set_a @ Ord @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( P @ X2 ) )
=> ( P @ ( Lub @ A6 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_500_ccpo_OadmissibleI,axiom,
! [Ord: a > a > $o,P: a > $o,Lub: set_a > a] :
( ! [A8: set_a] :
( ( comple1697357536187991598hain_a @ Ord @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( P @ X ) )
=> ( P @ ( Lub @ A8 ) ) ) ) )
=> ( comple72871723935627595ible_a @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_501_ccpo_OadmissibleI,axiom,
! [Ord: set_a > set_a > $o,P: set_a > $o,Lub: set_set_a > set_a] :
( ! [A8: set_set_a] :
( ( comple4316259127148425102_set_a @ Ord @ A8 )
=> ( ( A8 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A8 )
=> ( P @ X ) )
=> ( P @ ( Lub @ A8 ) ) ) ) )
=> ( comple8887300225568239275_set_a @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_502_ccpo_OadmissibleI,axiom,
! [Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,P: ( ( c > d ) > set_a ) > $o,Lub: set_c_d_set_a > ( c > d ) > set_a] :
( ! [A8: set_c_d_set_a] :
( ( comple7455786223818501471_set_a @ Ord @ A8 )
=> ( ( A8 != bot_bo738396921950161403_set_a )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A8 )
=> ( P @ X ) )
=> ( P @ ( Lub @ A8 ) ) ) ) )
=> ( comple1957918121334358780_set_a @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_503_ccpo_Oadmissible__def,axiom,
( comple72871723935627595ible_a
= ( ^ [Lub2: set_a > a,Ord2: a > a > $o,P3: a > $o] :
! [A7: set_a] :
( ( comple1697357536187991598hain_a @ Ord2 @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( P3 @ X3 ) )
=> ( P3 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_504_ccpo_Oadmissible__def,axiom,
( comple8887300225568239275_set_a
= ( ^ [Lub2: set_set_a > set_a,Ord2: set_a > set_a > $o,P3: set_a > $o] :
! [A7: set_set_a] :
( ( comple4316259127148425102_set_a @ Ord2 @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( P3 @ X3 ) )
=> ( P3 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_505_ccpo_Oadmissible__def,axiom,
( comple1957918121334358780_set_a
= ( ^ [Lub2: set_c_d_set_a > ( c > d ) > set_a,Ord2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,P3: ( ( c > d ) > set_a ) > $o] :
! [A7: set_c_d_set_a] :
( ( comple7455786223818501471_set_a @ Ord2 @ A7 )
=> ( ( A7 != bot_bo738396921950161403_set_a )
=> ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A7 )
=> ( P3 @ X3 ) )
=> ( P3 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_506_ccpo__Sup__least,axiom,
! [A6: set_set_set_a,Z: set_set_a] :
( ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ A6 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_507_ccpo__Sup__least,axiom,
! [A6: set_set_c_d_set_a,Z: set_c_d_set_a] :
( ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ A6 )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X2 @ Z ) )
=> ( ord_le5982164083705284911_set_a @ ( comple6131501996466690428_set_a @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_508_ccpo__Sup__least,axiom,
! [A6: set_c_d_set_a_o,Z: ( ( c > d ) > set_a ) > $o] :
( ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ A6 )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ A6 )
=> ( ord_le961293222253252206et_a_o @ X2 @ Z ) )
=> ( ord_le961293222253252206et_a_o @ ( comple5290581719055393889et_a_o @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_509_ccpo__Sup__least,axiom,
! [A6: set_a_o,Z: a > $o] :
( ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ A6 )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ A6 )
=> ( ord_less_eq_a_o @ X2 @ Z ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_510_ccpo__Sup__least,axiom,
! [A6: set_set_a,Z: set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_511_ccpo__Sup__least,axiom,
! [A6: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ A6 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X2 @ Z ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ Z ) ) ) ).
% ccpo_Sup_least
thf(fact_512_ccpo__Sup__upper,axiom,
! [A6: set_set_set_a,X4: set_set_a] :
( ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ A6 )
=> ( ( member_set_set_a @ X4 @ A6 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( comple3958522678809307947_set_a @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_513_ccpo__Sup__upper,axiom,
! [A6: set_set_c_d_set_a,X4: set_c_d_set_a] :
( ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ A6 )
=> ( ( member_set_c_d_set_a @ X4 @ A6 )
=> ( ord_le5982164083705284911_set_a @ X4 @ ( comple6131501996466690428_set_a @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_514_ccpo__Sup__upper,axiom,
! [A6: set_c_d_set_a_o,X4: ( ( c > d ) > set_a ) > $o] :
( ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ A6 )
=> ( ( member_c_d_set_a_o @ X4 @ A6 )
=> ( ord_le961293222253252206et_a_o @ X4 @ ( comple5290581719055393889et_a_o @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_515_ccpo__Sup__upper,axiom,
! [A6: set_a_o,X4: a > $o] :
( ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ A6 )
=> ( ( member_a_o @ X4 @ A6 )
=> ( ord_less_eq_a_o @ X4 @ ( complete_Sup_Sup_a_o @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_516_ccpo__Sup__upper,axiom,
! [A6: set_set_a,X4: set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A6 )
=> ( ( member_set_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_517_ccpo__Sup__upper,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ A6 )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( comple3834726295627996700_set_a @ A6 ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_518_bot__empty__eq,axiom,
( bot_bo3591254198091563330et_a_o
= ( ^ [X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ bot_bo58555506362910043_set_a ) ) ) ).
% bot_empty_eq
thf(fact_519_bot__empty__eq,axiom,
( bot_bot_option_a_o
= ( ^ [X3: option_a] : ( member_option_a @ X3 @ bot_bot_set_option_a ) ) ) ).
% bot_empty_eq
thf(fact_520_bot__empty__eq,axiom,
( bot_bo2369851049062976348et_a_o
= ( ^ [X3: option_c_d_set_a] : ( member4306893881663408030_set_a @ X3 @ bot_bo6666349697208826049_set_a ) ) ) ).
% bot_empty_eq
thf(fact_521_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_522_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_523_bot__empty__eq,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ).
% bot_empty_eq
thf(fact_524_Collect__empty__eq__bot,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( P = bot_bot_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_525_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_526_Collect__empty__eq__bot,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( ( collect_c_d_set_a @ P )
= bot_bo738396921950161403_set_a )
= ( P = bot_bot_c_d_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_527_chain__subset,axiom,
! [Ord: set_a > set_a > $o,A6: set_set_a,B5: set_set_a] :
( ( comple4316259127148425102_set_a @ Ord @ A6 )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ A6 )
=> ( comple4316259127148425102_set_a @ Ord @ B5 ) ) ) ).
% chain_subset
thf(fact_528_chain__subset,axiom,
! [Ord: a > a > $o,A6: set_a,B5: set_a] :
( ( comple1697357536187991598hain_a @ Ord @ A6 )
=> ( ( ord_less_eq_set_a @ B5 @ A6 )
=> ( comple1697357536187991598hain_a @ Ord @ B5 ) ) ) ).
% chain_subset
thf(fact_529_chain__subset,axiom,
! [Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( comple7455786223818501471_set_a @ Ord @ A6 )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ A6 )
=> ( comple7455786223818501471_set_a @ Ord @ B5 ) ) ) ).
% chain_subset
thf(fact_530_chain__empty,axiom,
! [Ord: a > a > $o] : ( comple1697357536187991598hain_a @ Ord @ bot_bot_set_a ) ).
% chain_empty
thf(fact_531_chain__empty,axiom,
! [Ord: set_a > set_a > $o] : ( comple4316259127148425102_set_a @ Ord @ bot_bot_set_set_a ) ).
% chain_empty
thf(fact_532_chain__empty,axiom,
! [Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] : ( comple7455786223818501471_set_a @ Ord @ bot_bo738396921950161403_set_a ) ).
% chain_empty
thf(fact_533_chain__equality,axiom,
! [A6: set_a] :
( ( comple1697357536187991598hain_a
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ A6 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ A6 )
=> ( X3 = Y5 ) ) ) ) ) ).
% chain_equality
thf(fact_534_chain__equality,axiom,
! [A6: set_set_a] :
( ( comple4316259127148425102_set_a
@ ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 )
@ A6 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ A6 )
=> ( X3 = Y5 ) ) ) ) ) ).
% chain_equality
thf(fact_535_chain__equality,axiom,
! [A6: set_c_d_set_a] :
( ( comple7455786223818501471_set_a
@ ^ [Y4: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( Y4 = Z2 )
@ A6 )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ! [Y5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y5 @ A6 )
=> ( X3 = Y5 ) ) ) ) ) ).
% chain_equality
thf(fact_536_chain__def,axiom,
( comple1697357536187991598hain_a
= ( ^ [Ord2: a > a > $o,S5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S5 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ S5 )
=> ( ( Ord2 @ X3 @ Y5 )
| ( Ord2 @ Y5 @ X3 ) ) ) ) ) ) ).
% chain_def
thf(fact_537_chain__def,axiom,
( comple4316259127148425102_set_a
= ( ^ [Ord2: set_a > set_a > $o,S5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ S5 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ S5 )
=> ( ( Ord2 @ X3 @ Y5 )
| ( Ord2 @ Y5 @ X3 ) ) ) ) ) ) ).
% chain_def
thf(fact_538_chain__def,axiom,
( comple7455786223818501471_set_a
= ( ^ [Ord2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S5: set_c_d_set_a] :
! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ S5 )
=> ! [Y5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y5 @ S5 )
=> ( ( Ord2 @ X3 @ Y5 )
| ( Ord2 @ Y5 @ X3 ) ) ) ) ) ) ).
% chain_def
thf(fact_539_chainI,axiom,
! [S3: set_set_c_d_set_a,Ord: set_c_d_set_a > set_c_d_set_a > $o] :
( ! [X2: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ S3 )
=> ( ( member_set_c_d_set_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple2185443536470187199_set_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_540_chainI,axiom,
! [S3: set_option_a,Ord: option_a > option_a > $o] :
( ! [X2: option_a,Y2: option_a] :
( ( member_option_a @ X2 @ S3 )
=> ( ( member_option_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple6692356590997824628tion_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_541_chainI,axiom,
! [S3: set_option_c_d_set_a,Ord: option_c_d_set_a > option_c_d_set_a > $o] :
( ! [X2: option_c_d_set_a,Y2: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ X2 @ S3 )
=> ( ( member4306893881663408030_set_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple2313665252861385893_set_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_542_chainI,axiom,
! [S3: set_set_a,Ord: set_a > set_a > $o] :
( ! [X2: set_a,Y2: set_a] :
( ( member_set_a @ X2 @ S3 )
=> ( ( member_set_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple4316259127148425102_set_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_543_chainI,axiom,
! [S3: set_a,Ord: a > a > $o] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ S3 )
=> ( ( member_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple1697357536187991598hain_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_544_chainI,axiom,
! [S3: set_c_d_set_a,Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ S3 )
=> ( ( member_c_d_set_a @ Y2 @ S3 )
=> ( ( Ord @ X2 @ Y2 )
| ( Ord @ Y2 @ X2 ) ) ) )
=> ( comple7455786223818501471_set_a @ Ord @ S3 ) ) ).
% chainI
thf(fact_545_chainE,axiom,
! [Ord: set_c_d_set_a > set_c_d_set_a > $o,S3: set_set_c_d_set_a,X4: set_c_d_set_a,Y: set_c_d_set_a] :
( ( comple2185443536470187199_set_a @ Ord @ S3 )
=> ( ( member_set_c_d_set_a @ X4 @ S3 )
=> ( ( member_set_c_d_set_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_546_chainE,axiom,
! [Ord: option_a > option_a > $o,S3: set_option_a,X4: option_a,Y: option_a] :
( ( comple6692356590997824628tion_a @ Ord @ S3 )
=> ( ( member_option_a @ X4 @ S3 )
=> ( ( member_option_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_547_chainE,axiom,
! [Ord: option_c_d_set_a > option_c_d_set_a > $o,S3: set_option_c_d_set_a,X4: option_c_d_set_a,Y: option_c_d_set_a] :
( ( comple2313665252861385893_set_a @ Ord @ S3 )
=> ( ( member4306893881663408030_set_a @ X4 @ S3 )
=> ( ( member4306893881663408030_set_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_548_chainE,axiom,
! [Ord: set_a > set_a > $o,S3: set_set_a,X4: set_a,Y: set_a] :
( ( comple4316259127148425102_set_a @ Ord @ S3 )
=> ( ( member_set_a @ X4 @ S3 )
=> ( ( member_set_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_549_chainE,axiom,
! [Ord: a > a > $o,S3: set_a,X4: a,Y: a] :
( ( comple1697357536187991598hain_a @ Ord @ S3 )
=> ( ( member_a @ X4 @ S3 )
=> ( ( member_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_550_chainE,axiom,
! [Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S3: set_c_d_set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ Ord @ S3 )
=> ( ( member_c_d_set_a @ X4 @ S3 )
=> ( ( member_c_d_set_a @ Y @ S3 )
=> ( ~ ( Ord @ X4 @ Y )
=> ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainE
thf(fact_551_chainD,axiom,
! [Ord: set_c_d_set_a > set_c_d_set_a > $o,S3: set_set_c_d_set_a,X4: set_c_d_set_a,Y: set_c_d_set_a] :
( ( comple2185443536470187199_set_a @ Ord @ S3 )
=> ( ( member_set_c_d_set_a @ X4 @ S3 )
=> ( ( member_set_c_d_set_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_552_chainD,axiom,
! [Ord: option_a > option_a > $o,S3: set_option_a,X4: option_a,Y: option_a] :
( ( comple6692356590997824628tion_a @ Ord @ S3 )
=> ( ( member_option_a @ X4 @ S3 )
=> ( ( member_option_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_553_chainD,axiom,
! [Ord: option_c_d_set_a > option_c_d_set_a > $o,S3: set_option_c_d_set_a,X4: option_c_d_set_a,Y: option_c_d_set_a] :
( ( comple2313665252861385893_set_a @ Ord @ S3 )
=> ( ( member4306893881663408030_set_a @ X4 @ S3 )
=> ( ( member4306893881663408030_set_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_554_chainD,axiom,
! [Ord: set_a > set_a > $o,S3: set_set_a,X4: set_a,Y: set_a] :
( ( comple4316259127148425102_set_a @ Ord @ S3 )
=> ( ( member_set_a @ X4 @ S3 )
=> ( ( member_set_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_555_chainD,axiom,
! [Ord: a > a > $o,S3: set_a,X4: a,Y: a] :
( ( comple1697357536187991598hain_a @ Ord @ S3 )
=> ( ( member_a @ X4 @ S3 )
=> ( ( member_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_556_chainD,axiom,
! [Ord: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S3: set_c_d_set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ Ord @ S3 )
=> ( ( member_c_d_set_a @ X4 @ S3 )
=> ( ( member_c_d_set_a @ Y @ S3 )
=> ( ( Ord @ X4 @ Y )
| ( Ord @ Y @ X4 ) ) ) ) ) ).
% chainD
thf(fact_557_cSUP__subset__mono,axiom,
! [A6: set_a,G: a > set_a,B5: set_a,F: a > set_a] :
( ( A6 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ G @ B5 ) )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_558_cSUP__subset__mono,axiom,
! [A6: set_a,G: a > ( c > d ) > set_a,B5: set_a,F: a > ( c > d ) > set_a] :
( ( A6 != bot_bot_set_a )
=> ( ( condit7392869265169887891_set_a @ ( image_a_c_d_set_a @ G @ B5 ) )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) ) @ ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_559_cSUP__subset__mono,axiom,
! [A6: set_c_d_set_a,G: ( ( c > d ) > set_a ) > set_a,B5: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ G @ B5 ) )
=> ( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_560_cSUP__subset__mono,axiom,
! [A6: set_c_d_set_a,G: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B5: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ G @ B5 ) )
=> ( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_561_cSUP__subset__mono,axiom,
! [A6: set_a,G: a > set_set_a,B5: set_a,F: a > set_set_a] :
( ( A6 != bot_bot_set_a )
=> ( ( condit5548584133349953570_set_a @ ( image_a_set_set_a @ G @ B5 ) )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ F @ A6 ) ) @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_562_cSUP__subset__mono,axiom,
! [A6: set_a,G: a > a > $o,B5: set_a,F: a > a > $o] :
( ( A6 != bot_bot_set_a )
=> ( ( condit5969422546283407003ve_a_o @ ( image_a_a_o @ G @ B5 ) )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_less_eq_a_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_a_o @ ( complete_Sup_Sup_a_o @ ( image_a_a_o @ F @ A6 ) ) @ ( complete_Sup_Sup_a_o @ ( image_a_a_o @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_563_cSUP__subset__mono,axiom,
! [A6: set_option_a,G: option_a > set_a,B5: set_option_a,F: option_a > set_a] :
( ( A6 != bot_bot_set_option_a )
=> ( ( condit3373647341569784514_set_a @ ( image_option_a_set_a @ G @ B5 ) )
=> ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_option_a_set_a @ F @ A6 ) ) @ ( comple2307003609928055243_set_a @ ( image_option_a_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_564_cSUP__subset__mono,axiom,
! [A6: set_set_a,G: set_a > set_a,B5: set_set_a,F: set_a > set_a] :
( ( A6 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_set_a_set_a @ G @ B5 ) )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A6 ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_565_cSUP__subset__mono,axiom,
! [A6: set_option_a,G: option_a > set_set_a,B5: set_option_a,F: option_a > set_set_a] :
( ( A6 != bot_bot_set_option_a )
=> ( ( condit5548584133349953570_set_a @ ( image_4141439534252489855_set_a @ G @ B5 ) )
=> ( ( ord_le1955136853071979460tion_a @ A6 @ B5 )
=> ( ! [X2: option_a] :
( ( member_option_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_4141439534252489855_set_a @ F @ A6 ) ) @ ( comple3958522678809307947_set_a @ ( image_4141439534252489855_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_566_cSUP__subset__mono,axiom,
! [A6: set_set_a,G: set_a > set_set_a,B5: set_set_a,F: set_a > set_set_a] :
( ( A6 != bot_bot_set_set_a )
=> ( ( condit5548584133349953570_set_a @ ( image_4955109552351689957_set_a @ G @ B5 ) )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( image_4955109552351689957_set_a @ F @ A6 ) ) @ ( comple3958522678809307947_set_a @ ( image_4955109552351689957_set_a @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_567_iteratesp_Osimps,axiom,
( comple4561909299121069501_set_a
= ( ^ [F2: set_set_a > set_set_a,A3: set_set_a] :
( ? [X3: set_set_a] :
( ( A3
= ( F2 @ X3 ) )
& ( comple4561909299121069501_set_a @ F2 @ X3 ) )
| ? [M3: set_set_set_a] :
( ( A3
= ( comple3958522678809307947_set_a @ M3 ) )
& ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ M3 )
& ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M3 )
=> ( comple4561909299121069501_set_a @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_568_iteratesp_Osimps,axiom,
( comple8844751925773657358_set_a
= ( ^ [F2: set_c_d_set_a > set_c_d_set_a,A3: set_c_d_set_a] :
( ? [X3: set_c_d_set_a] :
( ( A3
= ( F2 @ X3 ) )
& ( comple8844751925773657358_set_a @ F2 @ X3 ) )
| ? [M3: set_set_c_d_set_a] :
( ( A3
= ( comple6131501996466690428_set_a @ M3 ) )
& ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ M3 )
& ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ M3 )
=> ( comple8844751925773657358_set_a @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_569_iteratesp_Osimps,axiom,
( comple235802229605493583et_a_o
= ( ^ [F2: ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o,A3: ( ( c > d ) > set_a ) > $o] :
( ? [X3: ( ( c > d ) > set_a ) > $o] :
( ( A3
= ( F2 @ X3 ) )
& ( comple235802229605493583et_a_o @ F2 @ X3 ) )
| ? [M3: set_c_d_set_a_o] :
( ( A3
= ( comple5290581719055393889et_a_o @ M3 ) )
& ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ M3 )
& ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ M3 )
=> ( comple235802229605493583et_a_o @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_570_iteratesp_Osimps,axiom,
( comple1243119448443558080sp_a_o
= ( ^ [F2: ( a > $o ) > a > $o,A3: a > $o] :
( ? [X3: a > $o] :
( ( A3
= ( F2 @ X3 ) )
& ( comple1243119448443558080sp_a_o @ F2 @ X3 ) )
| ? [M3: set_a_o] :
( ( A3
= ( complete_Sup_Sup_a_o @ M3 ) )
& ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ M3 )
& ! [X3: a > $o] :
( ( member_a_o @ X3 @ M3 )
=> ( comple1243119448443558080sp_a_o @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_571_iteratesp_Osimps,axiom,
( comple8134540176031052893_set_a
= ( ^ [F2: set_a > set_a,A3: set_a] :
( ? [X3: set_a] :
( ( A3
= ( F2 @ X3 ) )
& ( comple8134540176031052893_set_a @ F2 @ X3 ) )
| ? [M3: set_set_a] :
( ( A3
= ( comple2307003609928055243_set_a @ M3 ) )
& ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M3 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ( comple8134540176031052893_set_a @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_572_iteratesp_Osimps,axiom,
( comple8462753965213938094_set_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ? [X3: ( c > d ) > set_a] :
( ( A3
= ( F2 @ X3 ) )
& ( comple8462753965213938094_set_a @ F2 @ X3 ) )
| ? [M3: set_c_d_set_a] :
( ( A3
= ( comple3834726295627996700_set_a @ M3 ) )
& ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M3 )
& ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ M3 )
=> ( comple8462753965213938094_set_a @ F2 @ X3 ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_573_iteratesp_Ocases,axiom,
! [F: set_set_a > set_set_a,A: set_set_a] :
( ( comple4561909299121069501_set_a @ F @ A )
=> ( ! [X2: set_set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple4561909299121069501_set_a @ F @ X2 ) )
=> ~ ! [M2: set_set_set_a] :
( ( A
= ( comple3958522678809307947_set_a @ M2 ) )
=> ( ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ M2 )
=> ~ ! [X: set_set_a] :
( ( member_set_set_a @ X @ M2 )
=> ( comple4561909299121069501_set_a @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_574_iteratesp_Ocases,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,A: set_c_d_set_a] :
( ( comple8844751925773657358_set_a @ F @ A )
=> ( ! [X2: set_c_d_set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple8844751925773657358_set_a @ F @ X2 ) )
=> ~ ! [M2: set_set_c_d_set_a] :
( ( A
= ( comple6131501996466690428_set_a @ M2 ) )
=> ( ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ M2 )
=> ~ ! [X: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X @ M2 )
=> ( comple8844751925773657358_set_a @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_575_iteratesp_Ocases,axiom,
! [F: ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o,A: ( ( c > d ) > set_a ) > $o] :
( ( comple235802229605493583et_a_o @ F @ A )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple235802229605493583et_a_o @ F @ X2 ) )
=> ~ ! [M2: set_c_d_set_a_o] :
( ( A
= ( comple5290581719055393889et_a_o @ M2 ) )
=> ( ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ M2 )
=> ~ ! [X: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X @ M2 )
=> ( comple235802229605493583et_a_o @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_576_iteratesp_Ocases,axiom,
! [F: ( a > $o ) > a > $o,A: a > $o] :
( ( comple1243119448443558080sp_a_o @ F @ A )
=> ( ! [X2: a > $o] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple1243119448443558080sp_a_o @ F @ X2 ) )
=> ~ ! [M2: set_a_o] :
( ( A
= ( complete_Sup_Sup_a_o @ M2 ) )
=> ( ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ M2 )
=> ~ ! [X: a > $o] :
( ( member_a_o @ X @ M2 )
=> ( comple1243119448443558080sp_a_o @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_577_iteratesp_Ocases,axiom,
! [F: set_a > set_a,A: set_a] :
( ( comple8134540176031052893_set_a @ F @ A )
=> ( ! [X2: set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple8134540176031052893_set_a @ F @ X2 ) )
=> ~ ! [M2: set_set_a] :
( ( A
= ( comple2307003609928055243_set_a @ M2 ) )
=> ( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M2 )
=> ~ ! [X: set_a] :
( ( member_set_a @ X @ M2 )
=> ( comple8134540176031052893_set_a @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_578_iteratesp_Ocases,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( comple8462753965213938094_set_a @ F @ A )
=> ( ! [X2: ( c > d ) > set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( comple8462753965213938094_set_a @ F @ X2 ) )
=> ~ ! [M2: set_c_d_set_a] :
( ( A
= ( comple3834726295627996700_set_a @ M2 ) )
=> ( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M2 )
=> ~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ M2 )
=> ( comple8462753965213938094_set_a @ F @ X ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_579_iteratesp_OSup,axiom,
! [M: set_set_set_a,F: set_set_a > set_set_a] :
( ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ M )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ M )
=> ( comple4561909299121069501_set_a @ F @ X2 ) )
=> ( comple4561909299121069501_set_a @ F @ ( comple3958522678809307947_set_a @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_580_iteratesp_OSup,axiom,
! [M: set_set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ M )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ M )
=> ( comple8844751925773657358_set_a @ F @ X2 ) )
=> ( comple8844751925773657358_set_a @ F @ ( comple6131501996466690428_set_a @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_581_iteratesp_OSup,axiom,
! [M: set_c_d_set_a_o,F: ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o] :
( ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ M )
=> ( ! [X2: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X2 @ M )
=> ( comple235802229605493583et_a_o @ F @ X2 ) )
=> ( comple235802229605493583et_a_o @ F @ ( comple5290581719055393889et_a_o @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_582_iteratesp_OSup,axiom,
! [M: set_a_o,F: ( a > $o ) > a > $o] :
( ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ M )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ M )
=> ( comple1243119448443558080sp_a_o @ F @ X2 ) )
=> ( comple1243119448443558080sp_a_o @ F @ ( complete_Sup_Sup_a_o @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_583_iteratesp_OSup,axiom,
! [M: set_set_a,F: set_a > set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ( comple8134540176031052893_set_a @ F @ X2 ) )
=> ( comple8134540176031052893_set_a @ F @ ( comple2307003609928055243_set_a @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_584_iteratesp_OSup,axiom,
! [M: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ M )
=> ( comple8462753965213938094_set_a @ F @ X2 ) )
=> ( comple8462753965213938094_set_a @ F @ ( comple3834726295627996700_set_a @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_585_iterates_Osimps,axiom,
! [A: set_set_a,F: set_set_a > set_set_a] :
( ( member_set_set_a @ A @ ( comple1698038292816285837_set_a @ F ) )
= ( ? [X3: set_set_a] :
( ( A
= ( F @ X3 ) )
& ( member_set_set_a @ X3 @ ( comple1698038292816285837_set_a @ F ) ) )
| ? [M3: set_set_set_a] :
( ( A
= ( comple3958522678809307947_set_a @ M3 ) )
& ( comple2480829807455835374_set_a @ ord_le3724670747650509150_set_a @ M3 )
& ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M3 )
=> ( member_set_set_a @ X3 @ ( comple1698038292816285837_set_a @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_586_iterates_Osimps,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( member_set_c_d_set_a @ A @ ( comple3976681931473715166_set_a @ F ) )
= ( ? [X3: set_c_d_set_a] :
( ( A
= ( F @ X3 ) )
& ( member_set_c_d_set_a @ X3 @ ( comple3976681931473715166_set_a @ F ) ) )
| ? [M3: set_set_c_d_set_a] :
( ( A
= ( comple6131501996466690428_set_a @ M3 ) )
& ( comple2185443536470187199_set_a @ ord_le5982164083705284911_set_a @ M3 )
& ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ M3 )
=> ( member_set_c_d_set_a @ X3 @ ( comple3976681931473715166_set_a @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_587_iterates_Osimps,axiom,
! [A: ( ( c > d ) > set_a ) > $o,F: ( ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ A @ ( comple8961357239255020671et_a_o @ F ) )
= ( ? [X3: ( ( c > d ) > set_a ) > $o] :
( ( A
= ( F @ X3 ) )
& ( member_c_d_set_a_o @ X3 @ ( comple8961357239255020671et_a_o @ F ) ) )
| ? [M3: set_c_d_set_a_o] :
( ( A
= ( comple5290581719055393889et_a_o @ M3 ) )
& ( comple4742651485759770334et_a_o @ ord_le961293222253252206et_a_o @ M3 )
& ! [X3: ( ( c > d ) > set_a ) > $o] :
( ( member_c_d_set_a_o @ X3 @ M3 )
=> ( member_c_d_set_a_o @ X3 @ ( comple8961357239255020671et_a_o @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_588_iterates_Osimps,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o] :
( ( member_a_o @ A @ ( comple4809514086394661360es_a_o @ F ) )
= ( ? [X3: a > $o] :
( ( A
= ( F @ X3 ) )
& ( member_a_o @ X3 @ ( comple4809514086394661360es_a_o @ F ) ) )
| ? [M3: set_a_o] :
( ( A
= ( complete_Sup_Sup_a_o @ M3 ) )
& ( comple1735739171376666831in_a_o @ ord_less_eq_a_o @ M3 )
& ! [X3: a > $o] :
( ( member_a_o @ X3 @ M3 )
=> ( member_a_o @ X3 @ ( comple4809514086394661360es_a_o @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_589_iterates_Osimps,axiom,
! [A: set_a,F: set_a > set_a] :
( ( member_set_a @ A @ ( comple4964449497533277997_set_a @ F ) )
= ( ? [X3: set_a] :
( ( A
= ( F @ X3 ) )
& ( member_set_a @ X3 @ ( comple4964449497533277997_set_a @ F ) ) )
| ? [M3: set_set_a] :
( ( A
= ( comple2307003609928055243_set_a @ M3 ) )
& ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M3 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ( member_set_a @ X3 @ ( comple4964449497533277997_set_a @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_590_iterates_Osimps,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ A @ ( comple4855714899335171198_set_a @ F ) )
= ( ? [X3: ( c > d ) > set_a] :
( ( A
= ( F @ X3 ) )
& ( member_c_d_set_a @ X3 @ ( comple4855714899335171198_set_a @ F ) ) )
| ? [M3: set_c_d_set_a] :
( ( A
= ( comple3834726295627996700_set_a @ M3 ) )
& ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M3 )
& ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ M3 )
=> ( member_c_d_set_a @ X3 @ ( comple4855714899335171198_set_a @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_591_local_OSUP__eq__const,axiom,
! [I: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bo58555506362910043_set_a )
=> ( ! [I2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_212549500329102437_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_592_local_OSUP__eq__const,axiom,
! [I: set_option_a,F: option_a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bot_set_option_a )
=> ( ! [I2: option_a] :
( ( member_option_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_317793290637937008_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_593_local_OSUP__eq__const,axiom,
! [I: set_option_c_d_set_a,F: option_c_d_set_a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bo6666349697208826049_set_a )
=> ( ! [I2: option_c_d_set_a] :
( ( member4306893881663408030_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_6768572723705120255_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_594_local_OSUP__eq__const,axiom,
! [I: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bo738396921950161403_set_a )
=> ( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_595_local_OSUP__eq__const,axiom,
! [I: set_set_a,F: set_a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bot_set_set_a )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_1482592857945081046_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_596_local_OSUP__eq__const,axiom,
! [I: set_a,F: a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( sup_c_d_a @ ( image_a_c_d_set_a @ F @ I ) )
= X4 ) ) ) ).
% local.SUP_eq_const
thf(fact_597_image__eqI,axiom,
! [B: a,F: a > a,X4: a,A6: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( member_a @ B @ ( image_a_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_598_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: a > ( c > d ) > set_a,X4: a,A6: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_599_image__eqI,axiom,
! [B: a,F: ( ( c > d ) > set_a ) > a,X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_600_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_601_image__eqI,axiom,
! [B: set_a,F: a > set_a,X4: a,A6: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_602_image__eqI,axiom,
! [B: option_a,F: a > option_a,X4: a,A6: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( member_option_a @ B @ ( image_a_option_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_603_image__eqI,axiom,
! [B: a,F: set_a > a,X4: set_a,A6: set_set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A6 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_604_image__eqI,axiom,
! [B: a,F: option_a > a,X4: option_a,A6: set_option_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_option_a @ X4 @ A6 )
=> ( member_a @ B @ ( image_option_a_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_605_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X4: set_a,A6: set_set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A6 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_606_image__eqI,axiom,
! [B: option_a,F: set_a > option_a,X4: set_a,A6: set_set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_set_a @ X4 @ A6 )
=> ( member_option_a @ B @ ( image_set_a_option_a @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_607_ball__UN,axiom,
! [B5: ( a > $o ) > set_a,A6: set_a_o,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ B5 @ A6 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_608_ball__UN,axiom,
! [B5: a > set_a,A6: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A6 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_609_bex__UN,axiom,
! [B5: ( a > $o ) > set_a,A6: set_a_o,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ B5 @ A6 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
& ? [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_610_bex__UN,axiom,
! [B5: a > set_a,A6: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A6 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ? [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_611_UN__ball__bex__simps_I2_J,axiom,
! [B5: ( a > $o ) > set_a,A6: set_a_o,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ B5 @ A6 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_612_UN__ball__bex__simps_I2_J,axiom,
! [B5: a > set_a,A6: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A6 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_613_UN__ball__bex__simps_I4_J,axiom,
! [B5: ( a > $o ) > set_a,A6: set_a_o,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ B5 @ A6 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
& ? [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_614_UN__ball__bex__simps_I4_J,axiom,
! [B5: a > set_a,A6: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A6 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ? [Y5: a] :
( ( member_a @ Y5 @ ( B5 @ X3 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_615_sup__empty,axiom,
( ( sup_c_d_a @ bot_bo738396921950161403_set_a )
= empty_interp_c_d_a ) ).
% sup_empty
thf(fact_616_image__empty,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( image_5710119992958135237_set_a @ F @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% image_empty
thf(fact_617_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_618_image__empty,axiom,
! [F: set_a > a] :
( ( image_set_a_a @ F @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_619_image__empty,axiom,
! [F: a > option_a] :
( ( image_a_option_a @ F @ bot_bot_set_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_620_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_621_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_622_image__empty,axiom,
! [F: ( a > $o ) > set_a] :
( ( image_a_o_set_a @ F @ bot_bot_set_a_o2 )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_623_image__empty,axiom,
! [F: ( ( c > d ) > set_a ) > a] :
( ( image_c_d_set_a_a @ F @ bot_bo738396921950161403_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_624_image__empty,axiom,
! [F: a > ( c > d ) > set_a] :
( ( image_a_c_d_set_a @ F @ bot_bot_set_a )
= bot_bo738396921950161403_set_a ) ).
% image_empty
thf(fact_625_image__empty,axiom,
! [F: ( ( c > d ) > set_a ) > set_a] :
( ( image_5050625251388476148_set_a @ F @ bot_bo738396921950161403_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_626_empty__is__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_5710119992958135237_set_a @ F @ A6 ) )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% empty_is_image
thf(fact_627_empty__is__image,axiom,
! [F: a > a,A6: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A6 ) )
= ( A6 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_628_empty__is__image,axiom,
! [F: a > option_a,A6: set_a] :
( ( bot_bot_set_option_a
= ( image_a_option_a @ F @ A6 ) )
= ( A6 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_629_empty__is__image,axiom,
! [F: a > set_a,A6: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A6 ) )
= ( A6 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_630_empty__is__image,axiom,
! [F: set_a > a,A6: set_set_a] :
( ( bot_bot_set_a
= ( image_set_a_a @ F @ A6 ) )
= ( A6 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_631_empty__is__image,axiom,
! [F: set_a > set_a,A6: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a @ F @ A6 ) )
= ( A6 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_632_empty__is__image,axiom,
! [F: ( a > $o ) > set_a,A6: set_a_o] :
( ( bot_bot_set_set_a
= ( image_a_o_set_a @ F @ A6 ) )
= ( A6 = bot_bot_set_a_o2 ) ) ).
% empty_is_image
thf(fact_633_empty__is__image,axiom,
! [F: a > ( c > d ) > set_a,A6: set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_a_c_d_set_a @ F @ A6 ) )
= ( A6 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_634_empty__is__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A6: set_c_d_set_a] :
( ( bot_bot_set_a
= ( image_c_d_set_a_a @ F @ A6 ) )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% empty_is_image
thf(fact_635_empty__is__image,axiom,
! [F: set_a > ( c > d ) > set_a,A6: set_set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_1482592857945081046_set_a @ F @ A6 ) )
= ( A6 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_636_image__is__empty,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( ( image_5710119992958135237_set_a @ F @ A6 )
= bot_bo738396921950161403_set_a )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% image_is_empty
thf(fact_637_image__is__empty,axiom,
! [F: a > a,A6: set_a] :
( ( ( image_a_a @ F @ A6 )
= bot_bot_set_a )
= ( A6 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_638_image__is__empty,axiom,
! [F: a > option_a,A6: set_a] :
( ( ( image_a_option_a @ F @ A6 )
= bot_bot_set_option_a )
= ( A6 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_639_image__is__empty,axiom,
! [F: a > set_a,A6: set_a] :
( ( ( image_a_set_a @ F @ A6 )
= bot_bot_set_set_a )
= ( A6 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_640_image__is__empty,axiom,
! [F: set_a > a,A6: set_set_a] :
( ( ( image_set_a_a @ F @ A6 )
= bot_bot_set_a )
= ( A6 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_641_image__is__empty,axiom,
! [F: set_a > set_a,A6: set_set_a] :
( ( ( image_set_a_set_a @ F @ A6 )
= bot_bot_set_set_a )
= ( A6 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_642_image__is__empty,axiom,
! [F: ( a > $o ) > set_a,A6: set_a_o] :
( ( ( image_a_o_set_a @ F @ A6 )
= bot_bot_set_set_a )
= ( A6 = bot_bot_set_a_o2 ) ) ).
% image_is_empty
thf(fact_643_image__is__empty,axiom,
! [F: a > ( c > d ) > set_a,A6: set_a] :
( ( ( image_a_c_d_set_a @ F @ A6 )
= bot_bo738396921950161403_set_a )
= ( A6 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_644_image__is__empty,axiom,
! [F: ( ( c > d ) > set_a ) > a,A6: set_c_d_set_a] :
( ( ( image_c_d_set_a_a @ F @ A6 )
= bot_bot_set_a )
= ( A6 = bot_bo738396921950161403_set_a ) ) ).
% image_is_empty
thf(fact_645_image__is__empty,axiom,
! [F: set_a > ( c > d ) > set_a,A6: set_set_a] :
( ( ( image_1482592857945081046_set_a @ F @ A6 )
= bot_bo738396921950161403_set_a )
= ( A6 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_646_iteratesp__iterates__eq,axiom,
( comple8844751925773657358_set_a
= ( ^ [F2: set_c_d_set_a > set_c_d_set_a,X3: set_c_d_set_a] : ( member_set_c_d_set_a @ X3 @ ( comple3976681931473715166_set_a @ F2 ) ) ) ) ).
% iteratesp_iterates_eq
thf(fact_647_iteratesp__iterates__eq,axiom,
( comple8134540176031052893_set_a
= ( ^ [F2: set_a > set_a,X3: set_a] : ( member_set_a @ X3 @ ( comple4964449497533277997_set_a @ F2 ) ) ) ) ).
% iteratesp_iterates_eq
thf(fact_648_iteratesp__iterates__eq,axiom,
( comple8462753965213938094_set_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ ( comple4855714899335171198_set_a @ F2 ) ) ) ) ).
% iteratesp_iterates_eq
thf(fact_649_iteratesp_Ostep,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( comple8462753965213938094_set_a @ F @ X4 )
=> ( comple8462753965213938094_set_a @ F @ ( F @ X4 ) ) ) ).
% iteratesp.step
thf(fact_650_iteratesp_Ostep,axiom,
! [F: set_a > set_a,X4: set_a] :
( ( comple8134540176031052893_set_a @ F @ X4 )
=> ( comple8134540176031052893_set_a @ F @ ( F @ X4 ) ) ) ).
% iteratesp.step
thf(fact_651_iterates_Ostep,axiom,
! [X4: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( member_set_c_d_set_a @ X4 @ ( comple3976681931473715166_set_a @ F ) )
=> ( member_set_c_d_set_a @ ( F @ X4 ) @ ( comple3976681931473715166_set_a @ F ) ) ) ).
% iterates.step
thf(fact_652_iterates_Ostep,axiom,
! [X4: set_a,F: set_a > set_a] :
( ( member_set_a @ X4 @ ( comple4964449497533277997_set_a @ F ) )
=> ( member_set_a @ ( F @ X4 ) @ ( comple4964449497533277997_set_a @ F ) ) ) ).
% iterates.step
thf(fact_653_iterates_Ostep,axiom,
! [X4: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ ( comple4855714899335171198_set_a @ F ) )
=> ( member_c_d_set_a @ ( F @ X4 ) @ ( comple4855714899335171198_set_a @ F ) ) ) ).
% iterates.step
thf(fact_654_rev__image__eqI,axiom,
! [X4: a,A6: set_a,B: a,F: a > a] :
( ( member_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_655_rev__image__eqI,axiom,
! [X4: a,A6: set_a,B: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_656_rev__image__eqI,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B: a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_657_rev__image__eqI,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_658_rev__image__eqI,axiom,
! [X4: a,A6: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_659_rev__image__eqI,axiom,
! [X4: a,A6: set_a,B: option_a,F: a > option_a] :
( ( member_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_option_a @ B @ ( image_a_option_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_660_rev__image__eqI,axiom,
! [X4: set_a,A6: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_661_rev__image__eqI,axiom,
! [X4: option_a,A6: set_option_a,B: a,F: option_a > a] :
( ( member_option_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_option_a_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_662_rev__image__eqI,axiom,
! [X4: set_a,A6: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_663_rev__image__eqI,axiom,
! [X4: set_a,A6: set_set_a,B: option_a,F: set_a > option_a] :
( ( member_set_a @ X4 @ A6 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_option_a @ B @ ( image_set_a_option_a @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_664_ball__imageD,axiom,
! [F: ( a > $o ) > set_a,A6: set_a_o,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_o_set_a @ F @ A6 ) )
=> ( P @ X2 ) )
=> ! [X: a > $o] :
( ( member_a_o @ X @ A6 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_665_ball__imageD,axiom,
! [F: a > option_a,A6: set_a,P: option_a > $o] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ ( image_a_option_a @ F @ A6 ) )
=> ( P @ X2 ) )
=> ! [X: a] :
( ( member_a @ X @ A6 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_666_ball__imageD,axiom,
! [F: a > set_a,A6: set_a,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A6 ) )
=> ( P @ X2 ) )
=> ! [X: a] :
( ( member_a @ X @ A6 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_667_ball__imageD,axiom,
! [F: a > ( c > d ) > set_a,A6: set_a,P: ( ( c > d ) > set_a ) > $o] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ ( image_a_c_d_set_a @ F @ A6 ) )
=> ( P @ X2 ) )
=> ! [X: a] :
( ( member_a @ X @ A6 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_668_ball__imageD,axiom,
! [F: a > a,A6: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A6 ) )
=> ( P @ X2 ) )
=> ! [X: a] :
( ( member_a @ X @ A6 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_669_image__cong,axiom,
! [M: set_a_o,N: set_a_o,F: ( a > $o ) > set_a,G: ( a > $o ) > set_a] :
( ( M = N )
=> ( ! [X2: a > $o] :
( ( member_a_o @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_o_set_a @ F @ M )
= ( image_a_o_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_670_image__cong,axiom,
! [M: set_a,N: set_a,F: a > option_a,G: a > option_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_option_a @ F @ M )
= ( image_a_option_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_671_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_a,G: a > set_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_672_image__cong,axiom,
! [M: set_a,N: set_a,F: a > ( c > d ) > set_a,G: a > ( c > d ) > set_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_c_d_set_a @ F @ M )
= ( image_a_c_d_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_673_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_674_bex__imageD,axiom,
! [F: ( a > $o ) > set_a,A6: set_a_o,P: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_a_o_set_a @ F @ A6 ) )
& ( P @ X ) )
=> ? [X2: a > $o] :
( ( member_a_o @ X2 @ A6 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_675_bex__imageD,axiom,
! [F: a > option_a,A6: set_a,P: option_a > $o] :
( ? [X: option_a] :
( ( member_option_a @ X @ ( image_a_option_a @ F @ A6 ) )
& ( P @ X ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_676_bex__imageD,axiom,
! [F: a > set_a,A6: set_a,P: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_a_set_a @ F @ A6 ) )
& ( P @ X ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_677_bex__imageD,axiom,
! [F: a > ( c > d ) > set_a,A6: set_a,P: ( ( c > d ) > set_a ) > $o] :
( ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( image_a_c_d_set_a @ F @ A6 ) )
& ( P @ X ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_678_bex__imageD,axiom,
! [F: a > a,A6: set_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A6 ) )
& ( P @ X ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A6 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_679_image__iff,axiom,
! [Z: a,F: a > a,A6: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A6 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_680_image__iff,axiom,
! [Z: ( c > d ) > set_a,F: a > ( c > d ) > set_a,A6: set_a] :
( ( member_c_d_set_a @ Z @ ( image_a_c_d_set_a @ F @ A6 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_681_image__iff,axiom,
! [Z: set_a,F: ( a > $o ) > set_a,A6: set_a_o] :
( ( member_set_a @ Z @ ( image_a_o_set_a @ F @ A6 ) )
= ( ? [X3: a > $o] :
( ( member_a_o @ X3 @ A6 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_682_image__iff,axiom,
! [Z: set_a,F: a > set_a,A6: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A6 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_683_image__iff,axiom,
! [Z: option_a,F: a > option_a,A6: set_a] :
( ( member_option_a @ Z @ ( image_a_option_a @ F @ A6 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_684_imageI,axiom,
! [X4: option_c_d_set_a,A6: set_option_c_d_set_a,F: option_c_d_set_a > option_a] :
( ( member4306893881663408030_set_a @ X4 @ A6 )
=> ( member_option_a @ ( F @ X4 ) @ ( image_9195261480396307732tion_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_685_imageI,axiom,
! [X4: option_c_d_set_a,A6: set_option_c_d_set_a,F: option_c_d_set_a > option_c_d_set_a] :
( ( member4306893881663408030_set_a @ X4 @ A6 )
=> ( member4306893881663408030_set_a @ ( F @ X4 ) @ ( image_928718206969431365_set_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_686_imageI,axiom,
! [X4: a,A6: set_a,F: a > a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_687_imageI,axiom,
! [X4: a,A6: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X4 @ A6 )
=> ( member_c_d_set_a @ ( F @ X4 ) @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_688_imageI,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( member_a @ ( F @ X4 ) @ ( image_c_d_set_a_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_689_imageI,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( member_c_d_set_a @ ( F @ X4 ) @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ).
% imageI
thf(fact_690_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S5: set_a_o,X3: a] : ( member_a @ X3 @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_691_Sup__SUP__eq,axiom,
( comple5290581719055393889et_a_o
= ( ^ [S5: set_c_d_set_a_o,X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ ( comple6131501996466690428_set_a @ ( image_8552787320881293370_set_a @ collect_c_d_set_a @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_692_image__mono,axiom,
! [A6: set_a,B5: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A6 ) @ ( image_a_a @ F @ B5 ) ) ) ).
% image_mono
thf(fact_693_image__subsetI,axiom,
! [A6: set_a,F: a > ( c > d ) > set_a,B5: set_c_d_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_c_d_set_a @ ( F @ X2 ) @ B5 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_694_image__subsetI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B5: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( member_c_d_set_a @ ( F @ X2 ) @ B5 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_695_image__subsetI,axiom,
! [A6: set_a,F: a > a,B5: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( member_a @ ( F @ X2 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_696_image__subsetI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,B5: set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( member_a @ ( F @ X2 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_697_subset__imageE,axiom,
! [B5: set_a,F: a > a,A6: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A6 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A6 )
=> ( B5
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_698_subset__image__iff,axiom,
! [B5: set_a,F: a > a,A6: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A6 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A6 )
& ( B5
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_699_Sup__class_OSUP__cong,axiom,
! [A6: set_a,B5: set_a,C3: a > ( c > d ) > set_a,D: a > ( c > d ) > set_a] :
( ( A6 = B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B5 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ C3 @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ D @ B5 ) ) ) ) ) ).
% Sup_class.SUP_cong
thf(fact_700_Sup__class_OSUP__cong,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,C3: ( ( c > d ) > set_a ) > ( c > d ) > set_a,D: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( A6 = B5 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ B5 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ C3 @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ D @ B5 ) ) ) ) ) ).
% Sup_class.SUP_cong
thf(fact_701_chain__imageI,axiom,
! [Le_a: a > a > $o,Y6: set_a,Le_b: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: a > ( c > d ) > set_a] :
( ( comple1697357536187991598hain_a @ Le_a @ Y6 )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ Y6 )
=> ( ( member_a @ Y2 @ Y6 )
=> ( ( Le_a @ X2 @ Y2 )
=> ( Le_b @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( comple7455786223818501471_set_a @ Le_b @ ( image_a_c_d_set_a @ F @ Y6 ) ) ) ) ).
% chain_imageI
thf(fact_702_chain__imageI,axiom,
! [Le_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Y6: set_c_d_set_a,Le_b: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ Le_a @ Y6 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ Y6 )
=> ( ( member_c_d_set_a @ Y2 @ Y6 )
=> ( ( Le_a @ X2 @ Y2 )
=> ( Le_b @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( comple7455786223818501471_set_a @ Le_b @ ( image_5710119992958135237_set_a @ F @ Y6 ) ) ) ) ).
% chain_imageI
thf(fact_703_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_a,B5: set_a,F: a > set_a,G: a > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A6 )
=> ? [X: a] :
( ( member_a @ X @ B5 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B5 )
=> ? [X: a] :
( ( member_a @ X @ A6 )
& ( ord_less_eq_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_704_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_a,B5: set_c_d_set_a,F: a > set_a,G: ( ( c > d ) > set_a ) > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B5 )
=> ? [X: a] :
( ( member_a @ X @ A6 )
& ( ord_less_eq_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) )
= ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_705_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_c_d_set_a,B5: set_a,F: ( ( c > d ) > set_a ) > set_a,G: a > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A6 )
=> ? [X: a] :
( ( member_a @ X @ B5 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( ord_less_eq_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_706_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,G: ( ( c > d ) > set_a ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( ord_less_eq_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( ord_less_eq_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) )
= ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_707_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_a,B5: set_a,F: a > ( c > d ) > set_a,G: a > ( c > d ) > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A6 )
=> ? [X: a] :
( ( member_a @ X @ B5 )
& ( ord_le8464990428230162895_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B5 )
=> ? [X: a] :
( ( member_a @ X @ A6 )
& ( ord_le8464990428230162895_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_708_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_a,B5: set_c_d_set_a,F: a > ( c > d ) > set_a,G: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( ord_le8464990428230162895_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B5 )
=> ? [X: a] :
( ( member_a @ X @ A6 )
& ( ord_le8464990428230162895_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_709_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_c_d_set_a,B5: set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G: a > ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A6 )
=> ? [X: a] :
( ( member_a @ X @ B5 )
& ( ord_le8464990428230162895_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( ord_le8464990428230162895_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_710_complete__lattice__class_OSUP__eq,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( ord_le8464990428230162895_set_a @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( ord_le8464990428230162895_set_a @ ( G @ J ) @ ( F @ X ) ) ) )
=> ( ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) )
= ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ G @ B5 ) ) ) ) ) ).
% complete_lattice_class.SUP_eq
thf(fact_711_complete__lattice__class_OSUP__eq__const,axiom,
! [I: set_a,F: a > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ I ) )
= X4 ) ) ) ).
% complete_lattice_class.SUP_eq_const
thf(fact_712_complete__lattice__class_OSUP__eq__const,axiom,
! [I: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( I != bot_bo738396921950161403_set_a )
=> ( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X4 ) )
=> ( ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ I ) )
= X4 ) ) ) ).
% complete_lattice_class.SUP_eq_const
thf(fact_713_preorder__class_Obdd__above_OI2,axiom,
! [A6: set_a,F: a > set_a,M: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A6 ) ) ) ).
% preorder_class.bdd_above.I2
thf(fact_714_preorder__class_Obdd__above_OI2,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,M: set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) ) ).
% preorder_class.bdd_above.I2
thf(fact_715_preorder__class_Obdd__above_OI2,axiom,
! [A6: set_a,F: a > ( c > d ) > set_a,M: ( c > d ) > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ M ) )
=> ( condit7392869265169887891_set_a @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ).
% preorder_class.bdd_above.I2
thf(fact_716_preorder__class_Obdd__above_OI2,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,M: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ M ) )
=> ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ).
% preorder_class.bdd_above.I2
thf(fact_717_None__notin__image__Some,axiom,
! [A6: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A6 ) ) ).
% None_notin_image_Some
thf(fact_718_bot__in__iterates,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] : ( member_c_d_set_a @ ( comple3834726295627996700_set_a @ bot_bo738396921950161403_set_a ) @ ( comple4855714899335171198_set_a @ F ) ) ).
% bot_in_iterates
thf(fact_719_complete__lattice__class_OSUP__eq__iff,axiom,
! [I: set_a,C2: set_a,F: a > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% complete_lattice_class.SUP_eq_iff
thf(fact_720_complete__lattice__class_OSUP__eq__iff,axiom,
! [I: set_c_d_set_a,C2: set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( I != bot_bo738396921950161403_set_a )
=> ( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ I )
=> ( ord_less_eq_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ I ) )
= C2 )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ I )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% complete_lattice_class.SUP_eq_iff
thf(fact_721_complete__lattice__class_OSUP__eq__iff,axiom,
! [I: set_a,C2: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le8464990428230162895_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ I ) )
= C2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% complete_lattice_class.SUP_eq_iff
thf(fact_722_complete__lattice__class_OSUP__eq__iff,axiom,
! [I: set_c_d_set_a,C2: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( I != bot_bo738396921950161403_set_a )
=> ( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ I )
=> ( ord_le8464990428230162895_set_a @ C2 @ ( F @ I2 ) ) )
=> ( ( ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ I ) )
= C2 )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ I )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% complete_lattice_class.SUP_eq_iff
thf(fact_723_cSUP__least,axiom,
! [A6: set_a,F: a > set_a,M: set_a] :
( ( A6 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_724_cSUP__least,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,M: set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_725_cSUP__least,axiom,
! [A6: set_a,F: a > ( c > d ) > set_a,M: ( c > d ) > set_a] :
( ( A6 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ M ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_726_cSUP__least,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,M: ( c > d ) > set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ M ) )
=> ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_727_cSUP__upper,axiom,
! [X4: a,A6: set_a,F: a > set_a] :
( ( member_a @ X4 @ A6 )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A6 ) )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) ) ) ) ) ).
% cSUP_upper
thf(fact_728_cSUP__upper,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) ) ) ) ).
% cSUP_upper
thf(fact_729_cSUP__upper,axiom,
! [X4: a,A6: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X4 @ A6 )
=> ( ( condit7392869265169887891_set_a @ ( image_a_c_d_set_a @ F @ A6 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ) ) ).
% cSUP_upper
thf(fact_730_cSUP__upper,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ) ) ).
% cSUP_upper
thf(fact_731_cSUP__upper2,axiom,
! [F: a > set_a,A6: set_a,X4: a,U: set_a] :
( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A6 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ X4 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_732_cSUP__upper2,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A6: set_c_d_set_a,X4: ( c > d ) > set_a,U: set_a] :
( ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ X4 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_733_cSUP__upper2,axiom,
! [F: a > ( c > d ) > set_a,A6: set_a,X4: a,U: ( c > d ) > set_a] :
( ( condit7392869265169887891_set_a @ ( image_a_c_d_set_a @ F @ A6 ) )
=> ( ( member_a @ X4 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ U @ ( F @ X4 ) )
=> ( ord_le8464990428230162895_set_a @ U @ ( comple3834726295627996700_set_a @ ( image_a_c_d_set_a @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_734_cSUP__upper2,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: set_c_d_set_a,X4: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ U @ ( F @ X4 ) )
=> ( ord_le8464990428230162895_set_a @ U @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_735_cSUP__le__iff,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,U: set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) @ U )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_736_cSUP__le__iff,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) )
=> ( ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) @ U )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_737_iterates_OSup,axiom,
! [M: set_set_a,F: set_a > set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ M )
=> ( member_set_a @ X2 @ ( comple4964449497533277997_set_a @ F ) ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ M ) @ ( comple4964449497533277997_set_a @ F ) ) ) ) ).
% iterates.Sup
thf(fact_738_iterates_OSup,axiom,
! [M: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ M )
=> ( member_c_d_set_a @ X2 @ ( comple4855714899335171198_set_a @ F ) ) )
=> ( member_c_d_set_a @ ( comple3834726295627996700_set_a @ M ) @ ( comple4855714899335171198_set_a @ F ) ) ) ) ).
% iterates.Sup
thf(fact_739_iterates_Ocases,axiom,
! [A: set_a,F: set_a > set_a] :
( ( member_set_a @ A @ ( comple4964449497533277997_set_a @ F ) )
=> ( ! [X2: set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( member_set_a @ X2 @ ( comple4964449497533277997_set_a @ F ) ) )
=> ~ ! [M2: set_set_a] :
( ( A
= ( comple2307003609928055243_set_a @ M2 ) )
=> ( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ M2 )
=> ~ ! [X: set_a] :
( ( member_set_a @ X @ M2 )
=> ( member_set_a @ X @ ( comple4964449497533277997_set_a @ F ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_740_iterates_Ocases,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ A @ ( comple4855714899335171198_set_a @ F ) )
=> ( ! [X2: ( c > d ) > set_a] :
( ( A
= ( F @ X2 ) )
=> ~ ( member_c_d_set_a @ X2 @ ( comple4855714899335171198_set_a @ F ) ) )
=> ~ ! [M2: set_c_d_set_a] :
( ( A
= ( comple3834726295627996700_set_a @ M2 ) )
=> ( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ M2 )
=> ~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ M2 )
=> ( member_c_d_set_a @ X @ ( comple4855714899335171198_set_a @ F ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_741_all__subset__image,axiom,
! [F: a > a,A6: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A6 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A6 )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_742_fixp__def,axiom,
( comple2361085228800170300_set_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] : ( comple3834726295627996700_set_a @ ( comple4855714899335171198_set_a @ F2 ) ) ) ) ).
% fixp_def
thf(fact_743_local_OSup__eqI,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ! [Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y2 @ A6 )
=> ( smaller_interp_c_d_a @ Y2 @ X4 ) )
=> ( ! [Y2: ( c > d ) > set_a] :
( ! [Z3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Z3 @ A6 )
=> ( smaller_interp_c_d_a @ Z3 @ Y2 ) )
=> ( smaller_interp_c_d_a @ X4 @ Y2 ) )
=> ( ( sup_c_d_a @ A6 )
= X4 ) ) ) ).
% local.Sup_eqI
thf(fact_744_local_OSup__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ! [A2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A2 @ A6 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ B5 )
& ( smaller_interp_c_d_a @ A2 @ X ) ) )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A6 ) @ ( sup_c_d_a @ B5 ) ) ) ).
% local.Sup_mono
thf(fact_745_local_OSup__upper,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( smaller_interp_c_d_a @ X4 @ ( sup_c_d_a @ A6 ) ) ) ).
% local.Sup_upper
thf(fact_746_local_OSup__upper2,axiom,
! [U: ( c > d ) > set_a,A6: set_c_d_set_a,V2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ U @ A6 )
=> ( ( smaller_interp_c_d_a @ V2 @ U )
=> ( smaller_interp_c_d_a @ V2 @ ( sup_c_d_a @ A6 ) ) ) ) ).
% local.Sup_upper2
thf(fact_747_test__axiom__sup,axiom,
! [A6: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( smaller_interp_c_d_a @ X2 @ Z ) )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A6 ) @ Z ) ) ).
% test_axiom_sup
thf(fact_748_local_Oless__eq__Sup,axiom,
! [A6: set_c_d_set_a,U: ( c > d ) > set_a] :
( ! [V: ( c > d ) > set_a] :
( ( member_c_d_set_a @ V @ A6 )
=> ( smaller_interp_c_d_a @ U @ V ) )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ U @ ( sup_c_d_a @ A6 ) ) ) ) ).
% local.less_eq_Sup
thf(fact_749_local_Onot__empty__eq__Iic__eq__empty,axiom,
! [H: ( c > d ) > set_a] :
( bot_bo738396921950161403_set_a
!= ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ H ) ) ).
% local.not_empty_eq_Iic_eq_empty
thf(fact_750_local_OatMost__iff,axiom,
! [I3: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ K ) )
= ( smaller_interp_c_d_a @ I3 @ K ) ) ).
% local.atMost_iff
thf(fact_751_local_Onot__empty__eq__Ici__eq__empty,axiom,
! [L: ( c > d ) > set_a] :
( bot_bo738396921950161403_set_a
!= ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ L ) ) ).
% local.not_empty_eq_Ici_eq_empty
thf(fact_752_test__axiom__inf,axiom,
! [A6: set_c_d_set_a,Z: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( smaller_interp_c_d_a @ Z @ X2 ) )
=> ( smaller_interp_c_d_a @ Z @ ( inf_c_d_a @ A6 ) ) ) ).
% test_axiom_inf
thf(fact_753_local_OInf__mono,axiom,
! [B5: set_c_d_set_a,A6: set_c_d_set_a] :
( ! [B2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B2 @ B5 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
& ( smaller_interp_c_d_a @ X @ B2 ) ) )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A6 ) @ ( inf_c_d_a @ B5 ) ) ) ).
% local.Inf_mono
thf(fact_754_local_OInf__lower2,axiom,
! [U: ( c > d ) > set_a,A6: set_c_d_set_a,V2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ U @ A6 )
=> ( ( smaller_interp_c_d_a @ U @ V2 )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A6 ) @ V2 ) ) ) ).
% local.Inf_lower2
thf(fact_755_local_OInf__lower,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A6 ) @ X4 ) ) ).
% local.Inf_lower
thf(fact_756_local_OInf__eqI,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A6 )
=> ( smaller_interp_c_d_a @ X4 @ I2 ) )
=> ( ! [Y2: ( c > d ) > set_a] :
( ! [I4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I4 @ A6 )
=> ( smaller_interp_c_d_a @ Y2 @ I4 ) )
=> ( smaller_interp_c_d_a @ Y2 @ X4 ) )
=> ( ( inf_c_d_a @ A6 )
= X4 ) ) ) ).
% local.Inf_eqI
thf(fact_757_local_Obdd__below_OE,axiom,
! [A6: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A6 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
=> ( smaller_interp_c_d_a @ M2 @ X ) ) ) ).
% local.bdd_below.E
thf(fact_758_local_Obdd__above_OE,axiom,
! [A6: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A6 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
=> ( smaller_interp_c_d_a @ X @ M2 ) ) ) ).
% local.bdd_above.E
thf(fact_759_local_OInf__less__eq,axiom,
! [A6: set_c_d_set_a,U: ( c > d ) > set_a] :
( ! [V: ( c > d ) > set_a] :
( ( member_c_d_set_a @ V @ A6 )
=> ( smaller_interp_c_d_a @ V @ U ) )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A6 ) @ U ) ) ) ).
% local.Inf_less_eq
thf(fact_760_local_OInf__le__Sup,axiom,
! [A6: set_c_d_set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A6 ) @ ( sup_c_d_a @ A6 ) ) ) ).
% local.Inf_le_Sup
thf(fact_761_local_Obdd__belowI,axiom,
! [A6: set_c_d_set_a,M4: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( smaller_interp_c_d_a @ M4 @ X2 ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A6 ) ) ).
% local.bdd_belowI
thf(fact_762_local_Obdd__below_OI,axiom,
! [A6: set_c_d_set_a,M: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( smaller_interp_c_d_a @ M @ X2 ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A6 ) ) ).
% local.bdd_below.I
thf(fact_763_local_Obdd__aboveI,axiom,
! [A6: set_c_d_set_a,M: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( smaller_interp_c_d_a @ X2 @ M ) )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A6 ) ) ).
% local.bdd_aboveI
thf(fact_764_local_OatLeast__iff,axiom,
! [I3: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ K ) )
= ( smaller_interp_c_d_a @ K @ I3 ) ) ).
% local.atLeast_iff
thf(fact_765_local_Obdd__below__empty,axiom,
condit9007271454129256903_set_a @ smaller_interp_c_d_a @ bot_bo738396921950161403_set_a ).
% local.bdd_below_empty
thf(fact_766_local_Obdd__above__empty,axiom,
condit6926915774301931483_set_a @ smaller_interp_c_d_a @ bot_bo738396921950161403_set_a ).
% local.bdd_above_empty
thf(fact_767_preorder__class_Oorder_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% preorder_class.order.partial_preordering_axioms
thf(fact_768_preorder__class_Oorder_Opartial__preordering__axioms,axiom,
partia701112543150332005_set_a @ ord_le8464990428230162895_set_a ).
% preorder_class.order.partial_preordering_axioms
thf(fact_769_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_770_UNIV__I,axiom,
! [X4: ( c > d ) > set_a] : ( member_c_d_set_a @ X4 @ top_to4267977599310771935_set_a ) ).
% UNIV_I
thf(fact_771_inf__empty,axiom,
( ( inf_c_d_a @ bot_bo738396921950161403_set_a )
= full_interp_c_d_a ) ).
% inf_empty
thf(fact_772_local_OgreaterThan__iff,axiom,
! [I3: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_gr5532796609634356233_set_a @ less_c_d_a @ K ) )
= ( less_c_d_a @ K @ I3 ) ) ).
% local.greaterThan_iff
thf(fact_773_complete__lattice__class_OSup__UNIV,axiom,
( ( comple3834726295627996700_set_a @ top_to4267977599310771935_set_a )
= top_top_c_d_set_a ) ).
% complete_lattice_class.Sup_UNIV
thf(fact_774_preordering__bdd_Oempty,axiom,
! [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Less: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ( condit5292637031048566470_set_a @ Less_eq @ Less )
=> ( condit8154225043310684324_set_a @ Less_eq @ bot_bo738396921950161403_set_a ) ) ).
% preordering_bdd.empty
thf(fact_775_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_776_UNIV__witness,axiom,
? [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ top_to4267977599310771935_set_a ) ).
% UNIV_witness
thf(fact_777_UNIV__eq__I,axiom,
! [A6: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A6 )
=> ( top_top_set_a = A6 ) ) ).
% UNIV_eq_I
thf(fact_778_UNIV__eq__I,axiom,
! [A6: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ A6 )
=> ( top_to4267977599310771935_set_a = A6 ) ) ).
% UNIV_eq_I
thf(fact_779_preordering__bdd_OI,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A6: set_a,M: a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( Less_eq @ X2 @ M ) )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ A6 ) ) ) ).
% preordering_bdd.I
thf(fact_780_preordering__bdd_OI,axiom,
! [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Less: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a,M: ( c > d ) > set_a] :
( ( condit5292637031048566470_set_a @ Less_eq @ Less )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( Less_eq @ X2 @ M ) )
=> ( condit8154225043310684324_set_a @ Less_eq @ A6 ) ) ) ).
% preordering_bdd.I
thf(fact_781_preordering__bdd_OE,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,A6: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ A6 )
=> ~ ! [M2: a] :
~ ! [X: a] :
( ( member_a @ X @ A6 )
=> ( Less_eq @ X @ M2 ) ) ) ) ).
% preordering_bdd.E
thf(fact_782_preordering__bdd_OE,axiom,
! [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Less: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A6: set_c_d_set_a] :
( ( condit5292637031048566470_set_a @ Less_eq @ Less )
=> ( ( condit8154225043310684324_set_a @ Less_eq @ A6 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A6 )
=> ( Less_eq @ X @ M2 ) ) ) ) ).
% preordering_bdd.E
thf(fact_783_preordering__bdd_Omono,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,B5: set_a,A6: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ B5 )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ A6 ) ) ) ) ).
% preordering_bdd.mono
thf(fact_784_order__top__class_Otop__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% order_top_class.top_greatest
thf(fact_785_order__top__class_Otop__greatest,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ top_top_c_d_set_a ) ).
% order_top_class.top_greatest
thf(fact_786_order__top__class_Otop_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% order_top_class.top.extremum_unique
thf(fact_787_order__top__class_Otop_Oextremum__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ top_top_c_d_set_a @ A )
= ( A = top_top_c_d_set_a ) ) ).
% order_top_class.top.extremum_unique
thf(fact_788_order__top__class_Otop_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% order_top_class.top.extremum_uniqueI
thf(fact_789_order__top__class_Otop_Oextremum__uniqueI,axiom,
! [A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ top_top_c_d_set_a @ A )
=> ( A = top_top_c_d_set_a ) ) ).
% order_top_class.top.extremum_uniqueI
thf(fact_790_empty__not__UNIV,axiom,
bot_bo738396921950161403_set_a != top_to4267977599310771935_set_a ).
% empty_not_UNIV
thf(fact_791_subset__UNIV,axiom,
! [A6: set_a] : ( ord_less_eq_set_a @ A6 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_792_preorder__class_Obdd__above__primitive__def,axiom,
( condit3373647341569784514_set_a
= ( condit4774827555938943059_set_a @ ord_less_eq_set_a ) ) ).
% preorder_class.bdd_above_primitive_def
thf(fact_793_preorder__class_Obdd__above__primitive__def,axiom,
( condit7392869265169887891_set_a
= ( condit8154225043310684324_set_a @ ord_le8464990428230162895_set_a ) ) ).
% preorder_class.bdd_above_primitive_def
thf(fact_794_notin__range__Some,axiom,
! [X4: option_a] :
( ( ~ ( member_option_a @ X4 @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
= ( X4 = none_a ) ) ).
% notin_range_Some
thf(fact_795_local_OatLeastAtMost__iff,axiom,
! [I3: ( c > d ) > set_a,L: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L @ U ) )
= ( ( smaller_interp_c_d_a @ L @ I3 )
& ( smaller_interp_c_d_a @ I3 @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_796_local_OatLeastatMost__empty__iff2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( bot_bo738396921950161403_set_a
= ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) )
= ( ~ ( smaller_interp_c_d_a @ A @ B ) ) ) ).
% local.atLeastatMost_empty_iff2
thf(fact_797_local_OatLeastatMost__empty__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B )
= bot_bo738396921950161403_set_a )
= ( ~ ( smaller_interp_c_d_a @ A @ B ) ) ) ).
% local.atLeastatMost_empty_iff
thf(fact_798_local_OatLeastatMost__empty,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( less_c_d_a @ B @ A )
=> ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B )
= bot_bo738396921950161403_set_a ) ) ).
% local.atLeastatMost_empty
thf(fact_799_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_800_top__empty__eq,axiom,
( top_top_c_d_set_a_o
= ( ^ [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ top_to4267977599310771935_set_a ) ) ) ).
% top_empty_eq
thf(fact_801_Int__iff,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A6 @ B5 ) )
= ( ( member_a @ C2 @ A6 )
& ( member_a @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_802_Int__iff,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) )
= ( ( member_c_d_set_a @ C2 @ A6 )
& ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_803_IntI,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ A6 )
=> ( ( member_a @ C2 @ B5 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% IntI
thf(fact_804_IntI,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ A6 )
=> ( ( member_c_d_set_a @ C2 @ B5 )
=> ( member_c_d_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ) ).
% IntI
thf(fact_805_Int__subset__iff,axiom,
! [C3: set_a,A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A6 @ B5 ) )
= ( ( ord_less_eq_set_a @ C3 @ A6 )
& ( ord_less_eq_set_a @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_806_psubsetI,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( A6 != B5 )
=> ( ord_less_set_a @ A6 @ B5 ) ) ) ).
% psubsetI
thf(fact_807_local_OlessThan__iff,axiom,
! [I3: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_le5418582716766741598_set_a @ less_c_d_a @ K ) )
= ( less_c_d_a @ I3 @ K ) ) ).
% local.lessThan_iff
thf(fact_808_local_OatLeastLessThan__iff,axiom,
! [I3: ( c > d ) > set_a,L: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ L @ U ) )
= ( ( smaller_interp_c_d_a @ L @ I3 )
& ( less_c_d_a @ I3 @ U ) ) ) ).
% local.atLeastLessThan_iff
thf(fact_809_local_OatLeastLessThan__empty,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ A @ B )
= bot_bo738396921950161403_set_a ) ) ).
% local.atLeastLessThan_empty
thf(fact_810_local_OatLeastLessThan__empty__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ A @ B )
= bot_bo738396921950161403_set_a )
= ( ~ ( less_c_d_a @ A @ B ) ) ) ).
% local.atLeastLessThan_empty_iff
thf(fact_811_local_OatLeastLessThan__empty__iff2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( bot_bo738396921950161403_set_a
= ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ A @ B ) )
= ( ~ ( less_c_d_a @ A @ B ) ) ) ).
% local.atLeastLessThan_empty_iff2
thf(fact_812_Union__Int__subset,axiom,
! [A6: set_set_a,B5: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A6 @ B5 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% Union_Int_subset
thf(fact_813_psubsetD,axiom,
! [A6: set_a,B5: set_a,C2: a] :
( ( ord_less_set_a @ A6 @ B5 )
=> ( ( member_a @ C2 @ A6 )
=> ( member_a @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_814_psubsetD,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,C2: ( c > d ) > set_a] :
( ( ord_le3685282097655362107_set_a @ A6 @ B5 )
=> ( ( member_c_d_set_a @ C2 @ A6 )
=> ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_815_IntD2,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A6 @ B5 ) )
=> ( member_a @ C2 @ B5 ) ) ).
% IntD2
thf(fact_816_IntD2,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) )
=> ( member_c_d_set_a @ C2 @ B5 ) ) ).
% IntD2
thf(fact_817_IntD1,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A6 @ B5 ) )
=> ( member_a @ C2 @ A6 ) ) ).
% IntD1
thf(fact_818_IntD1,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) )
=> ( member_c_d_set_a @ C2 @ A6 ) ) ).
% IntD1
thf(fact_819_IntE,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A6 @ B5 ) )
=> ~ ( ( member_a @ C2 @ A6 )
=> ~ ( member_a @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_820_IntE,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) )
=> ~ ( ( member_c_d_set_a @ C2 @ A6 )
=> ~ ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_821_complete__lattice__class_OSup__inter__less__eq,axiom,
! [A6: set_set_a,B5: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A6 @ B5 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ ( comple2307003609928055243_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_inter_less_eq
thf(fact_822_complete__lattice__class_OSup__inter__less__eq,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] : ( ord_le8464990428230162895_set_a @ ( comple3834726295627996700_set_a @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) @ ( inf_inf_c_d_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ ( comple3834726295627996700_set_a @ B5 ) ) ) ).
% complete_lattice_class.Sup_inter_less_eq
thf(fact_823_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_set_a @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_824_subset__psubset__trans,axiom,
! [A6: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C3 )
=> ( ord_less_set_a @ A6 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_825_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_826_psubset__subset__trans,axiom,
! [A6: set_a,B5: set_a,C3: set_a] :
( ( ord_less_set_a @ A6 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ord_less_set_a @ A6 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_827_psubset__imp__subset,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ A6 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_828_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_829_psubsetE,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_set_a @ A6 @ B5 )
=> ~ ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ).
% psubsetE
thf(fact_830_not__psubset__empty,axiom,
! [A6: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A6 @ bot_bo738396921950161403_set_a ) ).
% not_psubset_empty
thf(fact_831_order__bot__class_Obot_Onot__eq__extremum,axiom,
! [A: set_c_d_set_a] :
( ( A != bot_bo738396921950161403_set_a )
= ( ord_le3685282097655362107_set_a @ bot_bo738396921950161403_set_a @ A ) ) ).
% order_bot_class.bot.not_eq_extremum
thf(fact_832_order__bot__class_Obot_Oextremum__strict,axiom,
! [A: set_c_d_set_a] :
~ ( ord_le3685282097655362107_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% order_bot_class.bot.extremum_strict
thf(fact_833_Int__Collect__mono,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A6 @ ( collect_c_d_set_a @ P ) ) @ ( inf_in754637537901350525_set_a @ B5 @ ( collect_c_d_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_834_Int__Collect__mono,axiom,
! [A6: set_a,B5: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A6 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B5 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_835_Int__greatest,axiom,
! [C3: set_a,A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C3 @ A6 )
=> ( ( ord_less_eq_set_a @ C3 @ B5 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_836_Int__absorb2,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( inf_inf_set_a @ A6 @ B5 )
= A6 ) ) ).
% Int_absorb2
thf(fact_837_Int__absorb1,axiom,
! [B5: set_a,A6: set_a] :
( ( ord_less_eq_set_a @ B5 @ A6 )
=> ( ( inf_inf_set_a @ A6 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_838_Int__lower2,axiom,
! [A6: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A6 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_839_Int__lower1,axiom,
! [A6: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A6 @ B5 ) @ A6 ) ).
% Int_lower1
thf(fact_840_Int__mono,axiom,
! [A6: set_a,C3: set_a,B5: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A6 @ C3 )
=> ( ( ord_less_eq_set_a @ B5 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A6 @ B5 ) @ ( inf_inf_set_a @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_841_order__le__imp__less__or__eq,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_842_order__le__imp__less__or__eq,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ord_less_c_d_set_a @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_843_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_844_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C2: set_a] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_845_order__less__le__subst1,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_846_order__less__le__subst1,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_847_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_848_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_849_order__le__less__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C2: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_850_order__le__less__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ ( F @ B ) @ C2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_c_d_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_851_order__less__le__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_852_order__less__le__trans,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X4 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ Y @ Z )
=> ( ord_less_c_d_set_a @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_853_order__le__less__trans,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_854_order__le__less__trans,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ord_less_c_d_set_a @ Y @ Z )
=> ( ord_less_c_d_set_a @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_855_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_856_order__neq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A != B )
=> ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ord_less_c_d_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_857_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_858_order__le__neq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_c_d_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_859_order__less__imp__le,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_860_order__less__imp__le,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X4 @ Y )
=> ( ord_le8464990428230162895_set_a @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_861_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_862_order__less__le,axiom,
( ord_less_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,Y5: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_863_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_set_a @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_864_order__le__less,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [X3: ( c > d ) > set_a,Y5: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_865_preorder__class_Odual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% preorder_class.dual_order.strict_implies_order
thf(fact_866_preorder__class_Odual__order_Ostrict__implies__order,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B @ A )
=> ( ord_le8464990428230162895_set_a @ B @ A ) ) ).
% preorder_class.dual_order.strict_implies_order
thf(fact_867_preorder__class_Oorder_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% preorder_class.order.strict_implies_order
thf(fact_868_preorder__class_Oorder_Ostrict__implies__order,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ B )
=> ( ord_le8464990428230162895_set_a @ A @ B ) ) ).
% preorder_class.order.strict_implies_order
thf(fact_869_preorder__class_Odual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% preorder_class.dual_order.strict_iff_not
thf(fact_870_preorder__class_Odual__order_Ostrict__iff__not,axiom,
( ord_less_c_d_set_a
= ( ^ [B3: ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B3 @ A3 )
& ~ ( ord_le8464990428230162895_set_a @ A3 @ B3 ) ) ) ) ).
% preorder_class.dual_order.strict_iff_not
thf(fact_871_preorder__class_Odual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.strict_trans2
thf(fact_872_preorder__class_Odual__order_Ostrict__trans2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C2 @ B )
=> ( ord_less_c_d_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.strict_trans2
thf(fact_873_preorder__class_Odual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C2 @ B )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.strict_trans1
thf(fact_874_preorder__class_Odual__order_Ostrict__trans1,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_less_c_d_set_a @ C2 @ B )
=> ( ord_less_c_d_set_a @ C2 @ A ) ) ) ).
% preorder_class.dual_order.strict_trans1
thf(fact_875_order__class_Odual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% order_class.dual_order.strict_iff_order
thf(fact_876_order__class_Odual__order_Ostrict__iff__order,axiom,
( ord_less_c_d_set_a
= ( ^ [B3: ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% order_class.dual_order.strict_iff_order
thf(fact_877_order__class_Odual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% order_class.dual_order.order_iff_strict
thf(fact_878_order__class_Odual__order_Oorder__iff__strict,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [B3: ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% order_class.dual_order.order_iff_strict
thf(fact_879_preorder__class_Oorder_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% preorder_class.order.strict_iff_not
thf(fact_880_preorder__class_Oorder_Ostrict__iff__not,axiom,
( ord_less_c_d_set_a
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A3 @ B3 )
& ~ ( ord_le8464990428230162895_set_a @ B3 @ A3 ) ) ) ) ).
% preorder_class.order.strict_iff_not
thf(fact_881_preorder__class_Oorder_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.strict_trans2
thf(fact_882_preorder__class_Oorder_Ostrict__trans2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ord_less_c_d_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.strict_trans2
thf(fact_883_preorder__class_Oorder_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.strict_trans1
thf(fact_884_preorder__class_Oorder_Ostrict__trans1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_c_d_set_a @ B @ C2 )
=> ( ord_less_c_d_set_a @ A @ C2 ) ) ) ).
% preorder_class.order.strict_trans1
thf(fact_885_order__class_Oorder_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order_class.order.strict_iff_order
thf(fact_886_order__class_Oorder_Ostrict__iff__order,axiom,
( ord_less_c_d_set_a
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order_class.order.strict_iff_order
thf(fact_887_order__class_Oorder_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order_class.order.order_iff_strict
thf(fact_888_order__class_Oorder_Oorder__iff__strict,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( ord_less_c_d_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order_class.order.order_iff_strict
thf(fact_889_preorder__class_Oless__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ~ ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).
% preorder_class.less_le_not_le
thf(fact_890_preorder__class_Oless__le__not__le,axiom,
( ord_less_c_d_set_a
= ( ^ [X3: ( c > d ) > set_a,Y5: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y5 )
& ~ ( ord_le8464990428230162895_set_a @ Y5 @ X3 ) ) ) ) ).
% preorder_class.less_le_not_le
thf(fact_891_order__class_Oantisym__conv2,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ~ ( ord_less_set_a @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% order_class.antisym_conv2
thf(fact_892_order__class_Oantisym__conv2,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ~ ( ord_less_c_d_set_a @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% order_class.antisym_conv2
thf(fact_893_order__class_Oantisym__conv1,axiom,
! [X4: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_class.antisym_conv1
thf(fact_894_order__class_Oantisym__conv1,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ~ ( ord_less_c_d_set_a @ X4 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_class.antisym_conv1
thf(fact_895_order__class_Onless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% order_class.nless_le
thf(fact_896_order__class_Onless__le,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ~ ( ord_less_c_d_set_a @ A @ B ) )
= ( ~ ( ord_le8464990428230162895_set_a @ A @ B )
| ( A = B ) ) ) ).
% order_class.nless_le
thf(fact_897_order__class_OleD,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ~ ( ord_less_set_a @ X4 @ Y ) ) ).
% order_class.leD
thf(fact_898_order__class_OleD,axiom,
! [Y: ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ X4 )
=> ~ ( ord_less_c_d_set_a @ X4 @ Y ) ) ).
% order_class.leD
thf(fact_899_disjoint__iff__not__equal,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ! [Y5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y5 @ B5 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_900_Int__empty__right,axiom,
! [A6: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A6 @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% Int_empty_right
thf(fact_901_Int__empty__left,axiom,
! [B5: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ B5 )
= bot_bo738396921950161403_set_a ) ).
% Int_empty_left
thf(fact_902_disjoint__iff,axiom,
! [A6: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A6 @ B5 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ~ ( member_a @ X3 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_903_disjoint__iff,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ~ ( member_c_d_set_a @ X3 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_904_Int__emptyI,axiom,
! [A6: set_a,B5: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A6 )
=> ~ ( member_a @ X2 @ B5 ) )
=> ( ( inf_inf_set_a @ A6 @ B5 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_905_Int__emptyI,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ~ ( member_c_d_set_a @ X2 @ B5 ) )
=> ( ( inf_in754637537901350525_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a ) ) ).
% Int_emptyI
thf(fact_906_Union__disjoint,axiom,
! [C3: set_set_c_d_set_a,A6: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ ( comple6131501996466690428_set_a @ C3 ) @ A6 )
= bot_bo738396921950161403_set_a )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ C3 )
=> ( ( inf_in754637537901350525_set_a @ X3 @ A6 )
= bot_bo738396921950161403_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_907_preorder__class_Oorder_Opreordering__axioms,axiom,
preordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% preorder_class.order.preordering_axioms
thf(fact_908_preorder__class_Oorder_Opreordering__axioms,axiom,
preord7021486942077351306_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% preorder_class.order.preordering_axioms
thf(fact_909_order__class_Oorder_Oordering__axioms,axiom,
ordering_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% order_class.order.ordering_axioms
thf(fact_910_order__class_Oorder_Oordering__axioms,axiom,
ordering_c_d_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% order_class.order.ordering_axioms
thf(fact_911_preorder__class_Obdd__above_Opreordering__bdd__axioms,axiom,
condit6315317455391067509_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% preorder_class.bdd_above.preordering_bdd_axioms
thf(fact_912_preorder__class_Obdd__above_Opreordering__bdd__axioms,axiom,
condit5292637031048566470_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% preorder_class.bdd_above.preordering_bdd_axioms
thf(fact_913_order__top__class_Otop_Oordering__top__axioms,axiom,
ordering_top_set_a @ ord_less_eq_set_a @ ord_less_set_a @ top_top_set_a ).
% order_top_class.top.ordering_top_axioms
thf(fact_914_order__top__class_Otop_Oordering__top__axioms,axiom,
orderi5785346111247480928_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a @ top_top_c_d_set_a ).
% order_top_class.top.ordering_top_axioms
thf(fact_915_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X4 @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_916_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ X4 )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_917_local_OgreaterThanLessThan__iff,axiom,
! [I3: ( c > d ) > set_a,L: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_gr2245648953767368143_set_a @ less_c_d_a @ L @ U ) )
= ( ( less_c_d_a @ L @ I3 )
& ( less_c_d_a @ I3 @ U ) ) ) ).
% local.greaterThanLessThan_iff
thf(fact_918_local_OgreaterThanAtMost__iff,axiom,
! [I3: ( c > d ) > set_a,L: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I3 @ ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ L @ U ) )
= ( ( less_c_d_a @ L @ I3 )
& ( smaller_interp_c_d_a @ I3 @ U ) ) ) ).
% local.greaterThanAtMost_iff
thf(fact_919_local_OgreaterThanAtMost__empty__iff2,axiom,
! [K: ( c > d ) > set_a,L: ( c > d ) > set_a] :
( ( bot_bo738396921950161403_set_a
= ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ K @ L ) )
= ( ~ ( less_c_d_a @ K @ L ) ) ) ).
% local.greaterThanAtMost_empty_iff2
thf(fact_920_local_OgreaterThanAtMost__empty__iff,axiom,
! [K: ( c > d ) > set_a,L: ( c > d ) > set_a] :
( ( ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ K @ L )
= bot_bo738396921950161403_set_a )
= ( ~ ( less_c_d_a @ K @ L ) ) ) ).
% local.greaterThanAtMost_empty_iff
thf(fact_921_local_OgreaterThanAtMost__empty,axiom,
! [L: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ L @ K )
=> ( ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ K @ L )
= bot_bo738396921950161403_set_a ) ) ).
% local.greaterThanAtMost_empty
thf(fact_922_local_OgreaterThanLessThan__empty,axiom,
! [L: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ L @ K )
=> ( ( set_gr2245648953767368143_set_a @ less_c_d_a @ K @ L )
= bot_bo738396921950161403_set_a ) ) ).
% local.greaterThanLessThan_empty
thf(fact_923_less__fun__def,axiom,
( ord_less_c_d_set_a
= ( ^ [F2: ( c > d ) > set_a,G2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ F2 @ G2 )
& ~ ( ord_le8464990428230162895_set_a @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_924_bounded__lattice__bot__class_Oinf__bot__left,axiom,
! [X4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ X4 )
= bot_bo738396921950161403_set_a ) ).
% bounded_lattice_bot_class.inf_bot_left
thf(fact_925_bounded__lattice__bot__class_Oinf__bot__right,axiom,
! [X4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X4 @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% bounded_lattice_bot_class.inf_bot_right
thf(fact_926_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_927_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C2 ) )
= ( ( ord_le8464990428230162895_set_a @ A @ B )
& ( ord_le8464990428230162895_set_a @ A @ C2 ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_928_semilattice__inf__class_Ole__inf__iff,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X4 @ Y )
& ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_929_semilattice__inf__class_Ole__inf__iff,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ ( inf_inf_c_d_set_a @ Y @ Z ) )
= ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
& ( ord_le8464990428230162895_set_a @ X4 @ Z ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_930_semilattice__inf__class_Oinf_OcoboundedI2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 ) ) ).
% semilattice_inf_class.inf.coboundedI2
thf(fact_931_semilattice__inf__class_Oinf_OcoboundedI2,axiom,
! [B: ( c > d ) > set_a,C2: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ C2 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ C2 ) ) ).
% semilattice_inf_class.inf.coboundedI2
thf(fact_932_semilattice__inf__class_Oinf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 ) ) ).
% semilattice_inf_class.inf.coboundedI1
thf(fact_933_semilattice__inf__class_Oinf_OcoboundedI1,axiom,
! [A: ( c > d ) > set_a,C2: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ C2 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ C2 ) ) ).
% semilattice_inf_class.inf.coboundedI1
thf(fact_934_semilattice__inf__class_Oinf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff2
thf(fact_935_semilattice__inf__class_Oinf_Oabsorb__iff2,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [B3: ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff2
thf(fact_936_semilattice__inf__class_Oinf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff1
thf(fact_937_semilattice__inf__class_Oinf_Oabsorb__iff1,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff1
thf(fact_938_semilattice__inf__class_Oinf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% semilattice_inf_class.inf.cobounded2
thf(fact_939_semilattice__inf__class_Oinf_Ocobounded2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ B ) ).
% semilattice_inf_class.inf.cobounded2
thf(fact_940_semilattice__inf__class_Oinf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% semilattice_inf_class.inf.cobounded1
thf(fact_941_semilattice__inf__class_Oinf_Ocobounded1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ A ) ).
% semilattice_inf_class.inf.cobounded1
thf(fact_942_semilattice__inf__class_Oinf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% semilattice_inf_class.inf.order_iff
thf(fact_943_semilattice__inf__class_Oinf_Oorder__iff,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A3: ( c > d ) > set_a,B3: ( c > d ) > set_a] :
( A3
= ( inf_inf_c_d_set_a @ A3 @ B3 ) ) ) ) ).
% semilattice_inf_class.inf.order_iff
thf(fact_944_semilattice__inf__class_Oinf__greatest,axiom,
! [X4: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( ord_less_eq_set_a @ X4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% semilattice_inf_class.inf_greatest
thf(fact_945_semilattice__inf__class_Oinf__greatest,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Z )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( inf_inf_c_d_set_a @ Y @ Z ) ) ) ) ).
% semilattice_inf_class.inf_greatest
thf(fact_946_semilattice__inf__class_Oinf_OboundedI,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% semilattice_inf_class.inf.boundedI
thf(fact_947_semilattice__inf__class_Oinf_OboundedI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ A @ C2 )
=> ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C2 ) ) ) ) ).
% semilattice_inf_class.inf.boundedI
thf(fact_948_semilattice__inf__class_Oinf_OboundedE,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% semilattice_inf_class.inf.boundedE
thf(fact_949_semilattice__inf__class_Oinf_OboundedE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C2 ) )
=> ~ ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ~ ( ord_le8464990428230162895_set_a @ A @ C2 ) ) ) ).
% semilattice_inf_class.inf.boundedE
thf(fact_950_semilattice__inf__class_Oinf__absorb2,axiom,
! [Y: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y )
= Y ) ) ).
% semilattice_inf_class.inf_absorb2
thf(fact_951_semilattice__inf__class_Oinf__absorb2,axiom,
! [Y: ( c > d ) > set_a,X4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y @ X4 )
=> ( ( inf_inf_c_d_set_a @ X4 @ Y )
= Y ) ) ).
% semilattice_inf_class.inf_absorb2
thf(fact_952_semilattice__inf__class_Oinf__absorb1,axiom,
! [X4: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ( inf_inf_set_a @ X4 @ Y )
= X4 ) ) ).
% semilattice_inf_class.inf_absorb1
thf(fact_953_semilattice__inf__class_Oinf__absorb1,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ( inf_inf_c_d_set_a @ X4 @ Y )
= X4 ) ) ).
% semilattice_inf_class.inf_absorb1
thf(fact_954_semilattice__inf__class_Oinf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% semilattice_inf_class.inf.absorb2
thf(fact_955_semilattice__inf__class_Oinf_Oabsorb2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( inf_inf_c_d_set_a @ A @ B )
= B ) ) ).
% semilattice_inf_class.inf.absorb2
thf(fact_956_semilattice__inf__class_Oinf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% semilattice_inf_class.inf.absorb1
thf(fact_957_semilattice__inf__class_Oinf_Oabsorb1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( inf_inf_c_d_set_a @ A @ B )
= A ) ) ).
% semilattice_inf_class.inf.absorb1
thf(fact_958_semilattice__inf__class_Ole__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X3 @ Y5 )
= X3 ) ) ) ).
% semilattice_inf_class.le_iff_inf
thf(fact_959_semilattice__inf__class_Ole__iff__inf,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [X3: ( c > d ) > set_a,Y5: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X3 @ Y5 )
= X3 ) ) ) ).
% semilattice_inf_class.le_iff_inf
thf(fact_960_semilattice__inf__class_Oinf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_a,Y2: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z4 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% semilattice_inf_class.inf_unique
thf(fact_961_semilattice__inf__class_Oinf__unique,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ Z4 )
=> ( ord_le8464990428230162895_set_a @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_c_d_set_a @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% semilattice_inf_class.inf_unique
thf(fact_962_semilattice__inf__class_Oinf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% semilattice_inf_class.inf.orderI
thf(fact_963_semilattice__inf__class_Oinf_OorderI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A
= ( inf_inf_c_d_set_a @ A @ B ) )
=> ( ord_le8464990428230162895_set_a @ A @ B ) ) ).
% semilattice_inf_class.inf.orderI
thf(fact_964_semilattice__inf__class_Oinf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% semilattice_inf_class.inf.orderE
thf(fact_965_semilattice__inf__class_Oinf_OorderE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( A
= ( inf_inf_c_d_set_a @ A @ B ) ) ) ).
% semilattice_inf_class.inf.orderE
thf(fact_966_semilattice__inf__class_Ole__infI2,axiom,
! [B: set_a,X4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% semilattice_inf_class.le_infI2
thf(fact_967_semilattice__inf__class_Ole__infI2,axiom,
! [B: ( c > d ) > set_a,X4: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ X4 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ X4 ) ) ).
% semilattice_inf_class.le_infI2
thf(fact_968_semilattice__inf__class_Ole__infI1,axiom,
! [A: set_a,X4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X4 ) ) ).
% semilattice_inf_class.le_infI1
thf(fact_969_semilattice__inf__class_Ole__infI1,axiom,
! [A: ( c > d ) > set_a,X4: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ X4 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ X4 ) ) ).
% semilattice_inf_class.le_infI1
thf(fact_970_semilattice__inf__class_Oinf__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% semilattice_inf_class.inf_mono
thf(fact_971_semilattice__inf__class_Oinf__mono,axiom,
! [A: ( c > d ) > set_a,C2: ( c > d ) > set_a,B: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ C2 )
=> ( ( ord_le8464990428230162895_set_a @ B @ D2 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ ( inf_inf_c_d_set_a @ C2 @ D2 ) ) ) ) ).
% semilattice_inf_class.inf_mono
thf(fact_972_semilattice__inf__class_Ole__infI,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ( ord_less_eq_set_a @ X4 @ B )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% semilattice_inf_class.le_infI
thf(fact_973_semilattice__inf__class_Ole__infI,axiom,
! [X4: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ A )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ B )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( inf_inf_c_d_set_a @ A @ B ) ) ) ) ).
% semilattice_inf_class.le_infI
thf(fact_974_semilattice__inf__class_Ole__infE,axiom,
! [X4: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A )
=> ~ ( ord_less_eq_set_a @ X4 @ B ) ) ) ).
% semilattice_inf_class.le_infE
thf(fact_975_semilattice__inf__class_Ole__infE,axiom,
! [X4: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X4 @ ( inf_inf_c_d_set_a @ A @ B ) )
=> ~ ( ( ord_le8464990428230162895_set_a @ X4 @ A )
=> ~ ( ord_le8464990428230162895_set_a @ X4 @ B ) ) ) ).
% semilattice_inf_class.le_infE
thf(fact_976_semilattice__inf__class_Oinf__le2,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Y ) ).
% semilattice_inf_class.inf_le2
thf(fact_977_semilattice__inf__class_Oinf__le2,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X4 @ Y ) @ Y ) ).
% semilattice_inf_class.inf_le2
thf(fact_978_semilattice__inf__class_Oinf__le1,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ X4 ) ).
% semilattice_inf_class.inf_le1
thf(fact_979_semilattice__inf__class_Oinf__le1,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X4 @ Y ) @ X4 ) ).
% semilattice_inf_class.inf_le1
thf(fact_980_lattice__class_Oinf__sup__ord_I1_J,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ X4 ) ).
% lattice_class.inf_sup_ord(1)
thf(fact_981_lattice__class_Oinf__sup__ord_I1_J,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X4 @ Y ) @ X4 ) ).
% lattice_class.inf_sup_ord(1)
thf(fact_982_lattice__class_Oinf__sup__ord_I2_J,axiom,
! [X4: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y ) @ Y ) ).
% lattice_class.inf_sup_ord(2)
thf(fact_983_lattice__class_Oinf__sup__ord_I2_J,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X4 @ Y ) @ Y ) ).
% lattice_class.inf_sup_ord(2)
thf(fact_984_Sup__inf__eq__bot__iff,axiom,
! [B5: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ ( comple6131501996466690428_set_a @ B5 ) @ A )
= bot_bo738396921950161403_set_a )
= ( ! [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ B5 )
=> ( ( inf_in754637537901350525_set_a @ X3 @ A )
= bot_bo738396921950161403_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_985_Sup__inf__eq__bot__iff,axiom,
! [B5: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ( inf_inf_c_d_set_a @ ( comple3834726295627996700_set_a @ B5 ) @ A )
= bot_bot_c_d_set_a )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ B5 )
=> ( ( inf_inf_c_d_set_a @ X3 @ A )
= bot_bot_c_d_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_986_inf__Sup,axiom,
! [A: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( inf_inf_c_d_set_a @ A @ ( comple3834726295627996700_set_a @ B5 ) )
= ( comple3834726295627996700_set_a @ ( image_5710119992958135237_set_a @ ( inf_inf_c_d_set_a @ A ) @ B5 ) ) ) ).
% inf_Sup
thf(fact_987_local_OIio__Int__singleton,axiom,
! [X4: ( c > d ) > set_a,K: ( c > d ) > set_a] :
( ( ( less_c_d_a @ X4 @ K )
=> ( ( inf_in754637537901350525_set_a @ ( set_le5418582716766741598_set_a @ less_c_d_a @ K ) @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) )
& ( ~ ( less_c_d_a @ X4 @ K )
=> ( ( inf_in754637537901350525_set_a @ ( set_le5418582716766741598_set_a @ less_c_d_a @ K ) @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a ) ) ) ).
% local.Iio_Int_singleton
thf(fact_988_local_OatLeastAtMost__singleton_H,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A = B )
=> ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B )
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ).
% local.atLeastAtMost_singleton'
thf(fact_989_insert__iff,axiom,
! [A: a,B: a,A6: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A6 ) )
= ( ( A = B )
| ( member_a @ A @ A6 ) ) ) ).
% insert_iff
thf(fact_990_insert__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A6 ) )
= ( ( A = B )
| ( member_c_d_set_a @ A @ A6 ) ) ) ).
% insert_iff
thf(fact_991_insertCI,axiom,
! [A: a,B5: set_a,B: a] :
( ( ~ ( member_a @ A @ B5 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_992_insertCI,axiom,
! [A: ( c > d ) > set_a,B5: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ~ ( member_c_d_set_a @ A @ B5 )
=> ( A = B ) )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_993_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_994_singletonI,axiom,
! [A: ( c > d ) > set_a] : ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ).
% singletonI
thf(fact_995_insert__subset,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ X4 @ A6 ) @ B5 )
= ( ( member_c_d_set_a @ X4 @ B5 )
& ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ) ).
% insert_subset
thf(fact_996_insert__subset,axiom,
! [X4: a,A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A6 ) @ B5 )
= ( ( member_a @ X4 @ B5 )
& ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ).
% insert_subset
thf(fact_997_Int__insert__right__if1,axiom,
! [A: a,A6: set_a,B5: set_a] :
( ( member_a @ A @ A6 )
=> ( ( inf_inf_set_a @ A6 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_998_Int__insert__right__if1,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A6 )
=> ( ( inf_in754637537901350525_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_999_Int__insert__right__if0,axiom,
! [A: a,A6: set_a,B5: set_a] :
( ~ ( member_a @ A @ A6 )
=> ( ( inf_inf_set_a @ A6 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1000_Int__insert__right__if0,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A6 )
=> ( ( inf_in754637537901350525_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_1001_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B5: set_a] :
( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1002_Int__insert__left__if1,axiom,
! [A: ( c > d ) > set_a,C3: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ C3 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B5 ) @ C3 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1003_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B5: set_a] :
( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C3 )
= ( inf_inf_set_a @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1004_Int__insert__left__if0,axiom,
! [A: ( c > d ) > set_a,C3: set_c_d_set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ C3 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B5 ) @ C3 )
= ( inf_in754637537901350525_set_a @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1005_singleton__insert__inj__eq,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a )
= ( insert_c_d_set_a @ A @ A6 ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1006_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A6: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A6 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1007_singleton__insert__inj__eq_H,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ A6 )
= ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1008_singleton__insert__inj__eq_H,axiom,
! [A: a,A6: set_a,B: a] :
( ( ( insert_a @ A @ A6 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A6 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1009_cSup__singleton,axiom,
! [X4: ( c > d ) > set_a] :
( ( comple3834726295627996700_set_a @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= X4 ) ).
% cSup_singleton
thf(fact_1010_ccpo__Sup__singleton,axiom,
! [X4: ( c > d ) > set_a] :
( ( comple3834726295627996700_set_a @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= X4 ) ).
% ccpo_Sup_singleton
thf(fact_1011_disjoint__insert_I2_J,axiom,
! [A6: set_a,B: a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A6 @ ( insert_a @ B @ B5 ) ) )
= ( ~ ( member_a @ B @ A6 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1012_disjoint__insert_I2_J,axiom,
! [A6: set_c_d_set_a,B: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A6 @ ( insert_c_d_set_a @ B @ B5 ) ) )
= ( ~ ( member_c_d_set_a @ B @ A6 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1013_disjoint__insert_I1_J,axiom,
! [B5: set_a,A: a,A6: set_a] :
( ( ( inf_inf_set_a @ B5 @ ( insert_a @ A @ A6 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ B5 @ A6 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1014_disjoint__insert_I1_J,axiom,
! [B5: set_c_d_set_a,A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ B5 @ ( insert_c_d_set_a @ A @ A6 ) )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B5 )
& ( ( inf_in754637537901350525_set_a @ B5 @ A6 )
= bot_bo738396921950161403_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1015_insert__disjoint_I2_J,axiom,
! [A: a,A6: set_a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A6 ) @ B5 ) )
= ( ~ ( member_a @ A @ B5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1016_insert__disjoint_I2_J,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A6 ) @ B5 ) )
= ( ~ ( member_c_d_set_a @ A @ B5 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1017_insert__disjoint_I1_J,axiom,
! [A: a,A6: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A6 ) @ B5 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B5 )
& ( ( inf_inf_set_a @ A6 @ B5 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1018_insert__disjoint_I1_J,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A6 ) @ B5 )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B5 )
& ( ( inf_in754637537901350525_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1019_the__elem__eq,axiom,
! [X4: ( c > d ) > set_a] :
( ( the_elem_c_d_set_a @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_1020_local_OatLeastAtMost__singleton__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B )
= ( insert_c_d_set_a @ C2 @ bot_bo738396921950161403_set_a ) )
= ( ( A = B )
& ( B = C2 ) ) ) ).
% local.atLeastAtMost_singleton_iff
thf(fact_1021_local_OatLeastAtMost__singleton,axiom,
! [A: ( c > d ) > set_a] :
( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ A )
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ).
% local.atLeastAtMost_singleton
thf(fact_1022_subset__singleton__iff,axiom,
! [X6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ X6 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
= ( ( X6 = bot_bo738396921950161403_set_a )
| ( X6
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1023_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1024_subset__singletonD,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
=> ( ( A6 = bot_bo738396921950161403_set_a )
| ( A6
= ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1025_subset__singletonD,axiom,
! [A6: set_a,X4: a] :
( ( ord_less_eq_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A6 = bot_bot_set_a )
| ( A6
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1026_Int__insert__right,axiom,
! [A: a,A6: set_a,B5: set_a] :
( ( ( member_a @ A @ A6 )
=> ( ( inf_inf_set_a @ A6 @ ( insert_a @ A @ B5 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A6 @ B5 ) ) ) )
& ( ~ ( member_a @ A @ A6 )
=> ( ( inf_inf_set_a @ A6 @ ( insert_a @ A @ B5 ) )
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1027_Int__insert__right,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ A6 )
=> ( ( inf_in754637537901350525_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ A6 )
=> ( ( inf_in754637537901350525_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( inf_in754637537901350525_set_a @ A6 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_1028_Int__insert__left,axiom,
! [A: a,C3: set_a,B5: set_a] :
( ( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B5 @ C3 ) ) ) )
& ( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B5 ) @ C3 )
= ( inf_inf_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1029_Int__insert__left,axiom,
! [A: ( c > d ) > set_a,C3: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ C3 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B5 ) @ C3 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B5 @ C3 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ C3 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B5 ) @ C3 )
= ( inf_in754637537901350525_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1030_mk__disjoint__insert,axiom,
! [A: a,A6: set_a] :
( ( member_a @ A @ A6 )
=> ? [B7: set_a] :
( ( A6
= ( insert_a @ A @ B7 ) )
& ~ ( member_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1031_mk__disjoint__insert,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A6 )
=> ? [B7: set_c_d_set_a] :
( ( A6
= ( insert_c_d_set_a @ A @ B7 ) )
& ~ ( member_c_d_set_a @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1032_insert__eq__iff,axiom,
! [A: a,A6: set_a,B: a,B5: set_a] :
( ~ ( member_a @ A @ A6 )
=> ( ~ ( member_a @ B @ B5 )
=> ( ( ( insert_a @ A @ A6 )
= ( insert_a @ B @ B5 ) )
= ( ( ( A = B )
=> ( A6 = B5 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A6
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B5
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1033_insert__eq__iff,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a,B: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A6 )
=> ( ~ ( member_c_d_set_a @ B @ B5 )
=> ( ( ( insert_c_d_set_a @ A @ A6 )
= ( insert_c_d_set_a @ B @ B5 ) )
= ( ( ( A = B )
=> ( A6 = B5 ) )
& ( ( A != B )
=> ? [C5: set_c_d_set_a] :
( ( A6
= ( insert_c_d_set_a @ B @ C5 ) )
& ~ ( member_c_d_set_a @ B @ C5 )
& ( B5
= ( insert_c_d_set_a @ A @ C5 ) )
& ~ ( member_c_d_set_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1034_insert__absorb,axiom,
! [A: a,A6: set_a] :
( ( member_a @ A @ A6 )
=> ( ( insert_a @ A @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_1035_insert__absorb,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A6 )
=> ( ( insert_c_d_set_a @ A @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_1036_insert__ident,axiom,
! [X4: a,A6: set_a,B5: set_a] :
( ~ ( member_a @ X4 @ A6 )
=> ( ~ ( member_a @ X4 @ B5 )
=> ( ( ( insert_a @ X4 @ A6 )
= ( insert_a @ X4 @ B5 ) )
= ( A6 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1037_insert__ident,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ~ ( member_c_d_set_a @ X4 @ B5 )
=> ( ( ( insert_c_d_set_a @ X4 @ A6 )
= ( insert_c_d_set_a @ X4 @ B5 ) )
= ( A6 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1038_Set_Oset__insert,axiom,
! [X4: a,A6: set_a] :
( ( member_a @ X4 @ A6 )
=> ~ ! [B7: set_a] :
( ( A6
= ( insert_a @ X4 @ B7 ) )
=> ( member_a @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1039_Set_Oset__insert,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ~ ! [B7: set_c_d_set_a] :
( ( A6
= ( insert_c_d_set_a @ X4 @ B7 ) )
=> ( member_c_d_set_a @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1040_insertI2,axiom,
! [A: a,B5: set_a,B: a] :
( ( member_a @ A @ B5 )
=> ( member_a @ A @ ( insert_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1041_insertI2,axiom,
! [A: ( c > d ) > set_a,B5: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A @ B5 )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1042_insertI1,axiom,
! [A: a,B5: set_a] : ( member_a @ A @ ( insert_a @ A @ B5 ) ) ).
% insertI1
thf(fact_1043_insertI1,axiom,
! [A: ( c > d ) > set_a,B5: set_c_d_set_a] : ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ A @ B5 ) ) ).
% insertI1
thf(fact_1044_insertE,axiom,
! [A: a,B: a,A6: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A6 ) )
=> ( ( A != B )
=> ( member_a @ A @ A6 ) ) ) ).
% insertE
thf(fact_1045_insertE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A6 ) )
=> ( ( A != B )
=> ( member_c_d_set_a @ A @ A6 ) ) ) ).
% insertE
thf(fact_1046_subset__insertI2,axiom,
! [A6: set_a,B5: set_a,B: a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ord_less_eq_set_a @ A6 @ ( insert_a @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_1047_subset__insertI,axiom,
! [B5: set_a,A: a] : ( ord_less_eq_set_a @ B5 @ ( insert_a @ A @ B5 ) ) ).
% subset_insertI
thf(fact_1048_subset__insert,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ B5 ) )
= ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ) ).
% subset_insert
thf(fact_1049_subset__insert,axiom,
! [X4: a,A6: set_a,B5: set_a] :
( ~ ( member_a @ X4 @ A6 )
=> ( ( ord_less_eq_set_a @ A6 @ ( insert_a @ X4 @ B5 ) )
= ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ).
% subset_insert
thf(fact_1050_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_1051_insert__subsetI,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,X6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( ord_le5982164083705284911_set_a @ X6 @ A6 )
=> ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ X4 @ X6 ) @ A6 ) ) ) ).
% insert_subsetI
thf(fact_1052_insert__subsetI,axiom,
! [X4: a,A6: set_a,X6: set_a] :
( ( member_a @ X4 @ A6 )
=> ( ( ord_less_eq_set_a @ X6 @ A6 )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X6 ) @ A6 ) ) ) ).
% insert_subsetI
thf(fact_1053_singleton__inject,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a )
= ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1054_insert__not__empty,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ A6 )
!= bot_bo738396921950161403_set_a ) ).
% insert_not_empty
thf(fact_1055_doubleton__eq__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C2: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
= ( insert_c_d_set_a @ C2 @ ( insert_c_d_set_a @ D2 @ bot_bo738396921950161403_set_a ) ) )
= ( ( ( A = C2 )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1056_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1057_singleton__iff,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1058_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1059_singletonD,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1060_chain__singleton,axiom,
! [X4: set_a] : ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ).
% chain_singleton
thf(fact_1061_chain__singleton,axiom,
! [X4: ( c > d ) > set_a] : ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) ).
% chain_singleton
thf(fact_1062_bounded__semilattice__inf__top__class_Oinf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817875450240012_set_a @ inf_inf_set_a @ top_top_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% bounded_semilattice_inf_top_class.inf_top.semilattice_neutr_order_axioms
thf(fact_1063_bounded__semilattice__inf__top__class_Oinf__top_Osemilattice__neutr__order__axioms,axiom,
semila6957839794703059165_set_a @ inf_inf_c_d_set_a @ top_top_c_d_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% bounded_semilattice_inf_top_class.inf_top.semilattice_neutr_order_axioms
thf(fact_1064_semilattice__inf__class_OInf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% semilattice_inf_class.Inf_fin.semilattice_order_set_axioms
thf(fact_1065_semilattice__inf__class_OInf__fin_Osemilattice__order__set__axioms,axiom,
lattic1995125144389820681_set_a @ inf_inf_c_d_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% semilattice_inf_class.Inf_fin.semilattice_order_set_axioms
thf(fact_1066_local_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ A @ B )
= ( minus_1665977719694084726_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ).
% local.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1067_DiffI,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ A6 )
=> ( ~ ( member_a @ C2 @ B5 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A6 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1068_DiffI,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ A6 )
=> ( ~ ( member_c_d_set_a @ C2 @ B5 )
=> ( member_c_d_set_a @ C2 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1069_Diff__iff,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A6 @ B5 ) )
= ( ( member_a @ C2 @ A6 )
& ~ ( member_a @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_1070_Diff__iff,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) )
= ( ( member_c_d_set_a @ C2 @ A6 )
& ~ ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_1071_Diff__empty,axiom,
! [A6: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A6 @ bot_bo738396921950161403_set_a )
= A6 ) ).
% Diff_empty
thf(fact_1072_empty__Diff,axiom,
! [A6: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ bot_bo738396921950161403_set_a @ A6 )
= bot_bo738396921950161403_set_a ) ).
% empty_Diff
thf(fact_1073_Diff__cancel,axiom,
! [A6: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A6 @ A6 )
= bot_bo738396921950161403_set_a ) ).
% Diff_cancel
thf(fact_1074_local_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ A @ B )
= ( minus_1665977719694084726_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ).
% local.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1075_insert__Diff1,axiom,
! [X4: a,B5: set_a,A6: set_a] :
( ( member_a @ X4 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A6 ) @ B5 )
= ( minus_minus_set_a @ A6 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1076_insert__Diff1,axiom,
! [X4: ( c > d ) > set_a,B5: set_c_d_set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ B5 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X4 @ A6 ) @ B5 )
= ( minus_1665977719694084726_set_a @ A6 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1077_Diff__insert0,axiom,
! [X4: a,A6: set_a,B5: set_a] :
( ~ ( member_a @ X4 @ A6 )
=> ( ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ B5 ) )
= ( minus_minus_set_a @ A6 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1078_Diff__insert0,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ B5 ) )
= ( minus_1665977719694084726_set_a @ A6 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1079_Diff__UNIV,axiom,
! [A6: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A6 @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% Diff_UNIV
thf(fact_1080_Diff__eq__empty__iff,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( minus_1665977719694084726_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a )
= ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_1081_Diff__eq__empty__iff,axiom,
! [A6: set_a,B5: set_a] :
( ( ( minus_minus_set_a @ A6 @ B5 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A6 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_1082_insert__Diff__single,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) )
= ( insert_c_d_set_a @ A @ A6 ) ) ).
% insert_Diff_single
thf(fact_1083_Diff__disjoint,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A6 @ ( minus_1665977719694084726_set_a @ B5 @ A6 ) )
= bot_bo738396921950161403_set_a ) ).
% Diff_disjoint
thf(fact_1084_Diff__insert,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ).
% Diff_insert
thf(fact_1085_insert__Diff,axiom,
! [A: a,A6: set_a] :
( ( member_a @ A @ A6 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A6 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A6 ) ) ).
% insert_Diff
thf(fact_1086_insert__Diff,axiom,
! [A: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A6 )
=> ( ( insert_c_d_set_a @ A @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) )
= A6 ) ) ).
% insert_Diff
thf(fact_1087_Diff__insert2,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ A @ B5 ) )
= ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_1088_Diff__insert__absorb,axiom,
! [X4: a,A6: set_a] :
( ~ ( member_a @ X4 @ A6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A6 ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= A6 ) ) ).
% Diff_insert_absorb
thf(fact_1089_Diff__insert__absorb,axiom,
! [X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X4 @ A6 ) @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= A6 ) ) ).
% Diff_insert_absorb
thf(fact_1090_in__image__insert__iff,axiom,
! [B5: set_set_a,X4: a,A6: set_a] :
( ! [C4: set_a] :
( ( member_set_a @ C4 @ B5 )
=> ~ ( member_a @ X4 @ C4 ) )
=> ( ( member_set_a @ A6 @ ( image_set_a_set_a @ ( insert_a @ X4 ) @ B5 ) )
= ( ( member_a @ X4 @ A6 )
& ( member_set_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1091_in__image__insert__iff,axiom,
! [B5: set_set_c_d_set_a,X4: ( c > d ) > set_a,A6: set_c_d_set_a] :
( ! [C4: set_c_d_set_a] :
( ( member_set_c_d_set_a @ C4 @ B5 )
=> ~ ( member_c_d_set_a @ X4 @ C4 ) )
=> ( ( member_set_c_d_set_a @ A6 @ ( image_5418612861375423429_set_a @ ( insert_c_d_set_a @ X4 ) @ B5 ) )
= ( ( member_c_d_set_a @ X4 @ A6 )
& ( member_set_c_d_set_a @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1092_insert__Diff__if,axiom,
! [X4: a,B5: set_a,A6: set_a] :
( ( ( member_a @ X4 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A6 ) @ B5 )
= ( minus_minus_set_a @ A6 @ B5 ) ) )
& ( ~ ( member_a @ X4 @ B5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A6 ) @ B5 )
= ( insert_a @ X4 @ ( minus_minus_set_a @ A6 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1093_insert__Diff__if,axiom,
! [X4: ( c > d ) > set_a,B5: set_c_d_set_a,A6: set_c_d_set_a] :
( ( ( member_c_d_set_a @ X4 @ B5 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X4 @ A6 ) @ B5 )
= ( minus_1665977719694084726_set_a @ A6 @ B5 ) ) )
& ( ~ ( member_c_d_set_a @ X4 @ B5 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X4 @ A6 ) @ B5 )
= ( insert_c_d_set_a @ X4 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1094_subset__Diff__insert,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a,X4: ( c > d ) > set_a,C3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ ( minus_1665977719694084726_set_a @ B5 @ ( insert_c_d_set_a @ X4 @ C3 ) ) )
= ( ( ord_le5982164083705284911_set_a @ A6 @ ( minus_1665977719694084726_set_a @ B5 @ C3 ) )
& ~ ( member_c_d_set_a @ X4 @ A6 ) ) ) ).
% subset_Diff_insert
thf(fact_1095_subset__Diff__insert,axiom,
! [A6: set_a,B5: set_a,X4: a,C3: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( minus_minus_set_a @ B5 @ ( insert_a @ X4 @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A6 @ ( minus_minus_set_a @ B5 @ C3 ) )
& ~ ( member_a @ X4 @ A6 ) ) ) ).
% subset_Diff_insert
thf(fact_1096_psubset__imp__ex__mem,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_set_a @ A6 @ B5 )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B5 @ A6 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1097_psubset__imp__ex__mem,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A6 @ B5 )
=> ? [B2: ( c > d ) > set_a] : ( member_c_d_set_a @ B2 @ ( minus_1665977719694084726_set_a @ B5 @ A6 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1098_diff__shunt__var,axiom,
! [X4: set_c_d_set_a,Y: set_c_d_set_a] :
( ( ( minus_1665977719694084726_set_a @ X4 @ Y )
= bot_bo738396921950161403_set_a )
= ( ord_le5982164083705284911_set_a @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_1099_diff__shunt__var,axiom,
! [X4: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X4 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_1100_diff__shunt__var,axiom,
! [X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( ( minus_6165026464846083862_set_a @ X4 @ Y )
= bot_bot_c_d_set_a )
= ( ord_le8464990428230162895_set_a @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_1101_Diff__mono,axiom,
! [A6: set_a,C3: set_a,D: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B5 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A6 @ B5 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1102_Diff__subset,axiom,
! [A6: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A6 @ B5 ) @ A6 ) ).
% Diff_subset
thf(fact_1103_double__diff,axiom,
! [A6: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ( minus_minus_set_a @ B5 @ ( minus_minus_set_a @ C3 @ A6 ) )
= A6 ) ) ) ).
% double_diff
thf(fact_1104_DiffE,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A6 @ B5 ) )
=> ~ ( ( member_a @ C2 @ A6 )
=> ( member_a @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_1105_DiffE,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) )
=> ~ ( ( member_c_d_set_a @ C2 @ A6 )
=> ( member_c_d_set_a @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_1106_DiffD1,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A6 @ B5 ) )
=> ( member_a @ C2 @ A6 ) ) ).
% DiffD1
thf(fact_1107_DiffD1,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) )
=> ( member_c_d_set_a @ C2 @ A6 ) ) ).
% DiffD1
thf(fact_1108_DiffD2,axiom,
! [C2: a,A6: set_a,B5: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A6 @ B5 ) )
=> ~ ( member_a @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_1109_DiffD2,axiom,
! [C2: ( c > d ) > set_a,A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( member_c_d_set_a @ C2 @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) )
=> ~ ( member_c_d_set_a @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_1110_Diff__triv,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A6 @ B5 )
= bot_bo738396921950161403_set_a )
=> ( ( minus_1665977719694084726_set_a @ A6 @ B5 )
= A6 ) ) ).
% Diff_triv
thf(fact_1111_Int__Diff__disjoint,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ A6 @ B5 ) @ ( minus_1665977719694084726_set_a @ A6 @ B5 ) )
= bot_bo738396921950161403_set_a ) ).
% Int_Diff_disjoint
thf(fact_1112_subset__insert__iff,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ B5 ) )
= ( ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) @ B5 ) )
& ( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1113_subset__insert__iff,axiom,
! [A6: set_a,X4: a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( insert_a @ X4 @ B5 ) )
= ( ( ( member_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1114_Diff__single__insert,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) @ B5 )
=> ( ord_le5982164083705284911_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1115_Diff__single__insert,axiom,
! [A6: set_a,X4: a,B5: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B5 )
=> ( ord_less_eq_set_a @ A6 @ ( insert_a @ X4 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1116_insert__partition,axiom,
! [X4: set_c_d_set_a,F3: set_set_c_d_set_a] :
( ~ ( member_set_c_d_set_a @ X4 @ F3 )
=> ( ! [X2: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X2 @ ( insert_set_c_d_set_a @ X4 @ F3 ) )
=> ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ ( insert_set_c_d_set_a @ X4 @ F3 ) )
=> ( ( X2 != Xa )
=> ( ( inf_in754637537901350525_set_a @ X2 @ Xa )
= bot_bo738396921950161403_set_a ) ) ) )
=> ( ( inf_in754637537901350525_set_a @ X4 @ ( comple6131501996466690428_set_a @ F3 ) )
= bot_bo738396921950161403_set_a ) ) ) ).
% insert_partition
thf(fact_1117_psubset__insert__iff,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ B5 ) )
= ( ( ( member_c_d_set_a @ X4 @ B5 )
=> ( ord_le3685282097655362107_set_a @ A6 @ B5 ) )
& ( ~ ( member_c_d_set_a @ X4 @ B5 )
=> ( ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le3685282097655362107_set_a @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) @ B5 ) )
& ( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ord_le5982164083705284911_set_a @ A6 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1118_psubset__insert__iff,axiom,
! [A6: set_a,X4: a,B5: set_a] :
( ( ord_less_set_a @ A6 @ ( insert_a @ X4 @ B5 ) )
= ( ( ( member_a @ X4 @ B5 )
=> ( ord_less_set_a @ A6 @ B5 ) )
& ( ~ ( member_a @ X4 @ B5 )
=> ( ( ( member_a @ X4 @ A6 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B5 ) )
& ( ~ ( member_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ A6 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1119_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_1120_these__insert__Some,axiom,
! [X4: a,A6: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X4 ) @ A6 ) )
= ( insert_a @ X4 @ ( these_a @ A6 ) ) ) ).
% these_insert_Some
thf(fact_1121_these__empty,axiom,
( ( these_c_d_set_a @ bot_bo6666349697208826049_set_a )
= bot_bo738396921950161403_set_a ) ).
% these_empty
thf(fact_1122_these__image__Some__eq,axiom,
! [A6: set_a] :
( ( these_a @ ( image_a_option_a @ some_a @ A6 ) )
= A6 ) ).
% these_image_Some_eq
thf(fact_1123_these__insert__None,axiom,
! [A6: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A6 ) )
= ( these_a @ A6 ) ) ).
% these_insert_None
thf(fact_1124_in__these__eq,axiom,
! [X4: ( c > d ) > set_a,A6: set_option_c_d_set_a] :
( ( member_c_d_set_a @ X4 @ ( these_c_d_set_a @ A6 ) )
= ( member4306893881663408030_set_a @ ( some_c_d_set_a @ X4 ) @ A6 ) ) ).
% in_these_eq
thf(fact_1125_in__these__eq,axiom,
! [X4: a,A6: set_option_a] :
( ( member_a @ X4 @ ( these_a @ A6 ) )
= ( member_option_a @ ( some_a @ X4 ) @ A6 ) ) ).
% in_these_eq
thf(fact_1126_these__not__empty__eq,axiom,
! [B5: set_option_a] :
( ( ( these_a @ B5 )
!= bot_bot_set_a )
= ( ( B5 != bot_bot_set_option_a )
& ( B5
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_1127_these__not__empty__eq,axiom,
! [B5: set_option_c_d_set_a] :
( ( ( these_c_d_set_a @ B5 )
!= bot_bo738396921950161403_set_a )
= ( ( B5 != bot_bo6666349697208826049_set_a )
& ( B5
!= ( insert1935891768494221125_set_a @ none_c_d_set_a @ bot_bo6666349697208826049_set_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_1128_these__empty__eq,axiom,
! [B5: set_option_a] :
( ( ( these_a @ B5 )
= bot_bot_set_a )
= ( ( B5 = bot_bot_set_option_a )
| ( B5
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_1129_these__empty__eq,axiom,
! [B5: set_option_c_d_set_a] :
( ( ( these_c_d_set_a @ B5 )
= bot_bo738396921950161403_set_a )
= ( ( B5 = bot_bo6666349697208826049_set_a )
| ( B5
= ( insert1935891768494221125_set_a @ none_c_d_set_a @ bot_bo6666349697208826049_set_a ) ) ) ) ).
% these_empty_eq
thf(fact_1130_local_Omono__onI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( smaller_interp_c_d_a @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A6 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F ) ) ).
% local.mono_onI
thf(fact_1131_local_Omono__onI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( smaller_interp_c_d_a @ R @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A6 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% local.mono_onI
thf(fact_1132_local_Omono__onD,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A6 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( smaller_interp_c_d_a @ R2 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_1133_local_Omono__onD,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A6 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( smaller_interp_c_d_a @ R2 @ S4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_1134_order__class_Ostrict__mono__mono,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_set_a @ ord_less_set_a @ F )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.strict_mono_mono
thf(fact_1135_order__class_Ostrict__mono__mono,axiom,
! [F: set_a > ( c > d ) > set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_set_a @ ord_less_c_d_set_a @ F )
=> ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.strict_mono_mono
thf(fact_1136_order__class_Ostrict__mono__mono,axiom,
! [F: ( ( c > d ) > set_a ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_less_c_d_set_a @ ord_less_set_a @ F )
=> ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.strict_mono_mono
thf(fact_1137_order__class_Ostrict__mono__mono,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_less_c_d_set_a @ ord_less_c_d_set_a @ F )
=> ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.strict_mono_mono
thf(fact_1138_order__class_Omono__imp__mono__on,axiom,
! [F: set_a > set_a,A6: set_set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( monoto7172710143293369831_set_a @ A6 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_1139_order__class_Omono__imp__mono__on,axiom,
! [F: set_a > ( c > d ) > set_a,A6: set_set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto2748056057003999288_set_a @ A6 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_1140_order__class_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A6: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( monoto6316088450447394390_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_1141_order__class_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto2937423850181994535_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_1142_order__class_OmonoI,axiom,
! [F: set_a > set_a] :
( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.monoI
thf(fact_1143_order__class_OmonoI,axiom,
! [F: set_a > ( c > d ) > set_a] :
( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.monoI
thf(fact_1144_order__class_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > set_a] :
( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.monoI
thf(fact_1145_order__class_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.monoI
thf(fact_1146_order__class_OmonoE,axiom,
! [F: set_a > set_a,X4: set_a,Y: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoE
thf(fact_1147_order__class_OmonoE,axiom,
! [F: set_a > ( c > d ) > set_a,X4: set_a,Y: set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoE
thf(fact_1148_order__class_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoE
thf(fact_1149_order__class_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoE
thf(fact_1150_order__class_OmonoD,axiom,
! [F: set_a > set_a,X4: set_a,Y: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoD
thf(fact_1151_order__class_OmonoD,axiom,
! [F: set_a > ( c > d ) > set_a,X4: set_a,Y: set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_less_eq_set_a @ X4 @ Y )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoD
thf(fact_1152_order__class_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoD
thf(fact_1153_order__class_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X4: ( c > d ) > set_a,Y: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X4 @ Y )
=> ( ord_le8464990428230162895_set_a @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% order_class.monoD
thf(fact_1154_ord_Omono__on__subset,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > set_a,B5: set_a] :
( ( monotone_on_a_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ B5 @ A6 )
=> ( monotone_on_a_set_a @ B5 @ Less_eq @ ord_less_eq_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_1155_ord_Omono__on__subset,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > ( c > d ) > set_a,B5: set_a] :
( ( monoto2502030104860647832_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_less_eq_set_a @ B5 @ A6 )
=> ( monoto2502030104860647832_set_a @ B5 @ Less_eq @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_1156_ord__class_Omono__on__subset,axiom,
! [A6: set_set_a,F: set_a > set_a,B5: set_set_a] :
( ( monoto7172710143293369831_set_a @ A6 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ A6 )
=> ( monoto7172710143293369831_set_a @ B5 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_1157_ord__class_Omono__on__subset,axiom,
! [A6: set_set_a,F: set_a > ( c > d ) > set_a,B5: set_set_a] :
( ( monoto2748056057003999288_set_a @ A6 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B5 @ A6 )
=> ( monoto2748056057003999288_set_a @ B5 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_1158_ord__class_Omono__on__subset,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,B5: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ A6 )
=> ( monoto6316088450447394390_set_a @ B5 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_1159_ord__class_Omono__on__subset,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B5: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ A6 )
=> ( monoto2937423850181994535_set_a @ B5 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_1160_ord_Omono__onD,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > set_a,R2: a,S4: a] :
( ( monotone_on_a_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( member_a @ R2 @ A6 )
=> ( ( member_a @ S4 @ A6 )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1161_ord_Omono__onD,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1162_ord_Omono__onD,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > ( c > d ) > set_a,R2: a,S4: a] :
( ( monoto2502030104860647832_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_a @ R2 @ A6 )
=> ( ( member_a @ S4 @ A6 )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1163_ord_Omono__onD,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1164_ord_Omono__onI,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > set_a] :
( ! [R: a,S: a] :
( ( member_a @ R @ A6 )
=> ( ( member_a @ S @ A6 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1165_ord_Omono__onI,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1166_ord_Omono__onI,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > ( c > d ) > set_a] :
( ! [R: a,S: a] :
( ( member_a @ R @ A6 )
=> ( ( member_a @ S @ A6 )
=> ( ( Less_eq @ R @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto2502030104860647832_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1167_ord_Omono__onI,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( Less_eq @ R @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_1168_ord_Omono__on__def,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > set_a] :
( ( monotone_on_a_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F )
= ( ! [R3: a,S2: a] :
( ( ( member_a @ R3 @ A6 )
& ( member_a @ S2 @ A6 )
& ( Less_eq @ R3 @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1169_ord_Omono__on__def,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a] :
( ( monoto6316088450447394390_set_a @ A6 @ Less_eq @ ord_less_eq_set_a @ F )
= ( ! [R3: ( c > d ) > set_a,S2: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ R3 @ A6 )
& ( member_c_d_set_a @ S2 @ A6 )
& ( Less_eq @ R3 @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1170_ord_Omono__on__def,axiom,
! [A6: set_a,Less_eq: a > a > $o,F: a > ( c > d ) > set_a] :
( ( monoto2502030104860647832_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F )
= ( ! [R3: a,S2: a] :
( ( ( member_a @ R3 @ A6 )
& ( member_a @ S2 @ A6 )
& ( Less_eq @ R3 @ S2 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1171_ord_Omono__on__def,axiom,
! [A6: set_c_d_set_a,Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A6 @ Less_eq @ ord_le8464990428230162895_set_a @ F )
= ( ! [R3: ( c > d ) > set_a,S2: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ R3 @ A6 )
& ( member_c_d_set_a @ S2 @ A6 )
& ( Less_eq @ R3 @ S2 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1172_ord__class_Omono__onD,axiom,
! [A6: set_set_a,F: set_a > set_a,R2: set_a,S4: set_a] :
( ( monoto7172710143293369831_set_a @ A6 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_set_a @ R2 @ A6 )
=> ( ( member_set_a @ S4 @ A6 )
=> ( ( ord_less_eq_set_a @ R2 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_1173_ord__class_Omono__onD,axiom,
! [A6: set_set_a,F: set_a > ( c > d ) > set_a,R2: set_a,S4: set_a] :
( ( monoto2748056057003999288_set_a @ A6 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_set_a @ R2 @ A6 )
=> ( ( member_set_a @ S4 @ A6 )
=> ( ( ord_less_eq_set_a @ R2 @ S4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_1174_ord__class_Omono__onD,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ R2 @ S4 )
=> ( ord_less_eq_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_1175_ord__class_Omono__onD,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R2: ( c > d ) > set_a,S4: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R2 @ A6 )
=> ( ( member_c_d_set_a @ S4 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ R2 @ S4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_1176_ord__class_Omono__onI,axiom,
! [A6: set_set_a,F: set_a > set_a] :
( ! [R: set_a,S: set_a] :
( ( member_set_a @ R @ A6 )
=> ( ( member_set_a @ S @ A6 )
=> ( ( ord_less_eq_set_a @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A6 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_1177_ord__class_Omono__onI,axiom,
! [A6: set_set_a,F: set_a > ( c > d ) > set_a] :
( ! [R: set_a,S: set_a] :
( ( member_set_a @ R @ A6 )
=> ( ( member_set_a @ S @ A6 )
=> ( ( ord_less_eq_set_a @ R @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto2748056057003999288_set_a @ A6 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_1178_ord__class_Omono__onI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_1179_ord__class_Omono__onI,axiom,
! [A6: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R @ A6 )
=> ( ( member_c_d_set_a @ S @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ R @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A6 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_1180_semilattice__inf__class_Omono__inf,axiom,
! [F: set_a > set_a,A6: set_a,B5: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A6 @ B5 ) ) @ ( inf_inf_set_a @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_1181_semilattice__inf__class_Omono__inf,axiom,
! [F: set_a > ( c > d ) > set_a,A6: set_a,B5: set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ ( F @ ( inf_inf_set_a @ A6 @ B5 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_1182_semilattice__inf__class_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A6: ( c > d ) > set_a,B5: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_c_d_set_a @ A6 @ B5 ) ) @ ( inf_inf_set_a @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_1183_semilattice__inf__class_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: ( c > d ) > set_a,B5: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ ( F @ ( inf_inf_c_d_set_a @ A6 @ B5 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_1184_iterates__le__f,axiom,
! [X4: set_a,F: set_a > set_a] :
( ( member_set_a @ X4 @ ( comple4964449497533277997_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ X4 @ ( F @ X4 ) ) ) ) ).
% iterates_le_f
thf(fact_1185_iterates__le__f,axiom,
! [X4: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ ( comple4855714899335171198_set_a @ F ) )
=> ( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( F @ X4 ) ) ) ) ).
% iterates_le_f
thf(fact_1186_fixp__unfold,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( comple6813827801316615403_set_a @ F )
= ( F @ ( comple6813827801316615403_set_a @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1187_fixp__unfold,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( comple2361085228800170300_set_a @ F )
= ( F @ ( comple2361085228800170300_set_a @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1188_fixp__lowerbound,axiom,
! [F: set_a > set_a,Z: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ ( F @ Z ) @ Z )
=> ( ord_less_eq_set_a @ ( comple6813827801316615403_set_a @ F ) @ Z ) ) ) ).
% fixp_lowerbound
thf(fact_1189_fixp__lowerbound,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Z: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ Z ) @ Z )
=> ( ord_le8464990428230162895_set_a @ ( comple2361085228800170300_set_a @ F ) @ Z ) ) ) ).
% fixp_lowerbound
thf(fact_1190_bdd__above__image__mono,axiom,
! [F: set_a > set_a,A6: set_set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( condit3373647341569784514_set_a @ A6 )
=> ( condit3373647341569784514_set_a @ ( image_set_a_set_a @ F @ A6 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1191_bdd__above__image__mono,axiom,
! [F: set_a > ( c > d ) > set_a,A6: set_set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( condit3373647341569784514_set_a @ A6 )
=> ( condit7392869265169887891_set_a @ ( image_1482592857945081046_set_a @ F @ A6 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1192_bdd__above__image__mono,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A6: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( condit7392869265169887891_set_a @ A6 )
=> ( condit3373647341569784514_set_a @ ( image_5050625251388476148_set_a @ F @ A6 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1193_bdd__above__image__mono,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A6: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( condit7392869265169887891_set_a @ A6 )
=> ( condit7392869265169887891_set_a @ ( image_5710119992958135237_set_a @ F @ A6 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1194_chain__iterates,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ ( comple4964449497533277997_set_a @ F ) ) ) ).
% chain_iterates
thf(fact_1195_chain__iterates,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ ( comple4855714899335171198_set_a @ F ) ) ) ).
% chain_iterates
thf(fact_1196_iterates__fixp,axiom,
! [F: set_a > set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( member_set_a @ ( comple6813827801316615403_set_a @ F ) @ ( comple4964449497533277997_set_a @ F ) ) ) ).
% iterates_fixp
thf(fact_1197_iterates__fixp,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( member_c_d_set_a @ ( comple2361085228800170300_set_a @ F ) @ ( comple4855714899335171198_set_a @ F ) ) ) ).
% iterates_fixp
thf(fact_1198_fixp__induct,axiom,
! [P: set_a > $o,F: set_a > set_a] :
( ( comple8887300225568239275_set_a @ comple2307003609928055243_set_a @ ord_less_eq_set_a @ P )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( P @ ( comple2307003609928055243_set_a @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( P @ ( comple6813827801316615403_set_a @ F ) ) ) ) ) ) ).
% fixp_induct
thf(fact_1199_fixp__induct,axiom,
! [P: ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( comple1957918121334358780_set_a @ comple3834726295627996700_set_a @ ord_le8464990428230162895_set_a @ P )
=> ( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( P @ ( comple3834726295627996700_set_a @ bot_bo738396921950161403_set_a ) )
=> ( ! [X2: ( c > d ) > set_a] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( P @ ( comple2361085228800170300_set_a @ F ) ) ) ) ) ) ).
% fixp_induct
thf(fact_1200_mono__Int,axiom,
! [F: set_a > set_a,A6: set_a,B5: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A6 @ B5 ) ) @ ( inf_inf_set_a @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ).
% mono_Int
thf(fact_1201_semilattice__inf__class_Oinf_Osemilattice__order__axioms,axiom,
semila4706084620769370446_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% semilattice_inf_class.inf.semilattice_order_axioms
thf(fact_1202_semilattice__inf__class_Oinf_Osemilattice__order__axioms,axiom,
semila1630236661048524575_set_a @ inf_inf_c_d_set_a @ ord_le8464990428230162895_set_a @ ord_less_c_d_set_a ).
% semilattice_inf_class.inf.semilattice_order_axioms
thf(fact_1203_is__singleton__the__elem,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A7: set_c_d_set_a] :
( A7
= ( insert_c_d_set_a @ ( the_elem_c_d_set_a @ A7 ) @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1204_is__singletonI,axiom,
! [X4: ( c > d ) > set_a] : ( is_sin6979784932356128547_set_a @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) ).
% is_singletonI
thf(fact_1205_is__singletonI_H,axiom,
! [A6: set_a] :
( ( A6 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A6 )
=> ( ( member_a @ Y2 @ A6 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A6 ) ) ) ).
% is_singletonI'
thf(fact_1206_is__singletonI_H,axiom,
! [A6: set_c_d_set_a] :
( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ( member_c_d_set_a @ Y2 @ A6 )
=> ( X2 = Y2 ) ) )
=> ( is_sin6979784932356128547_set_a @ A6 ) ) ) ).
% is_singletonI'
thf(fact_1207_empty__in__Fpow,axiom,
! [A6: set_c_d_set_a] : ( member_set_c_d_set_a @ bot_bo738396921950161403_set_a @ ( finite3010068450757450645_set_a @ A6 ) ) ).
% empty_in_Fpow
thf(fact_1208_Fpow__mono,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A6 ) @ ( finite_Fpow_a @ B5 ) ) ) ).
% Fpow_mono
thf(fact_1209_is__singleton__def,axiom,
( is_sin6979784932356128547_set_a
= ( ^ [A7: set_c_d_set_a] :
? [X3: ( c > d ) > set_a] :
( A7
= ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1210_is__singletonE,axiom,
! [A6: set_c_d_set_a] :
( ( is_sin6979784932356128547_set_a @ A6 )
=> ~ ! [X2: ( c > d ) > set_a] :
( A6
!= ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ).
% is_singletonE
thf(fact_1211_local_Ofinite__Inf__in,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
=> ( ( member_c_d_set_a @ Y2 @ A6 )
=> ( member_c_d_set_a @ ( inf_c_d_a2 @ X2 @ Y2 ) @ A6 ) ) )
=> ( member_c_d_set_a @ ( inf_c_d_a @ A6 ) @ A6 ) ) ) ) ).
% local.finite_Inf_in
thf(fact_1212_local_Ofinite__has__maximal2,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ A @ A6 )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ( smaller_interp_c_d_a @ A @ X2 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( smaller_interp_c_d_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% local.finite_has_maximal2
thf(fact_1213_local_Ofinite__has__minimal2,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ A @ A6 )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ( smaller_interp_c_d_a @ X2 @ A )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( smaller_interp_c_d_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% local.finite_has_minimal2
thf(fact_1214_local_Ofinite__has__maximal,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( smaller_interp_c_d_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% local.finite_has_maximal
thf(fact_1215_local_Ofinite__has__minimal,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( smaller_interp_c_d_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% local.finite_has_minimal
thf(fact_1216_infinite__remove,axiom,
! [S3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ~ ( finite3330819693523053784_set_a @ S3 )
=> ~ ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ S3 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1217_infinite__coinduct,axiom,
! [X6: set_c_d_set_a > $o,A6: set_c_d_set_a] :
( ( X6 @ A6 )
=> ( ! [A8: set_c_d_set_a] :
( ( X6 @ A8 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A8 )
& ( ( X6 @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) )
| ~ ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) ) ) )
=> ~ ( finite3330819693523053784_set_a @ A6 ) ) ) ).
% infinite_coinduct
thf(fact_1218_finite__empty__induct,axiom,
! [A6: set_a,P: set_a > $o] :
( ( finite_finite_a @ A6 )
=> ( ( P @ A6 )
=> ( ! [A2: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A2 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1219_finite__empty__induct,axiom,
! [A6: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( P @ A6 )
=> ( ! [A2: ( c > d ) > set_a,A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( member_c_d_set_a @ A2 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ A2 @ bot_bo738396921950161403_set_a ) ) ) ) ) )
=> ( P @ bot_bo738396921950161403_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1220_ex__min__if__finite,axiom,
! [S3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ S3 )
=> ( ( S3 != bot_bo738396921950161403_set_a )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ S3 )
& ~ ? [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ S3 )
& ( ord_less_c_d_set_a @ Xa2 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1221_order__class_Ofinite__has__minimal,axiom,
! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A6 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_minimal
thf(fact_1222_order__class_Ofinite__has__minimal,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_minimal
thf(fact_1223_order__class_Ofinite__has__maximal,axiom,
! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A6 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_maximal
thf(fact_1224_order__class_Ofinite__has__maximal,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_maximal
thf(fact_1225_infinite__imp__nonempty,axiom,
! [S3: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ S3 )
=> ( S3 != bot_bo738396921950161403_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1226_finite_OemptyI,axiom,
finite3330819693523053784_set_a @ bot_bo738396921950161403_set_a ).
% finite.emptyI
thf(fact_1227_finite_Ocases,axiom,
! [A: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A )
=> ( ( A != bot_bo738396921950161403_set_a )
=> ~ ! [A8: set_c_d_set_a] :
( ? [A2: ( c > d ) > set_a] :
( A
= ( insert_c_d_set_a @ A2 @ A8 ) )
=> ~ ( finite3330819693523053784_set_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1228_finite_Osimps,axiom,
( finite3330819693523053784_set_a
= ( ^ [A3: set_c_d_set_a] :
( ( A3 = bot_bo738396921950161403_set_a )
| ? [A7: set_c_d_set_a,B3: ( c > d ) > set_a] :
( ( A3
= ( insert_c_d_set_a @ B3 @ A7 ) )
& ( finite3330819693523053784_set_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_1229_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1230_finite__induct,axiom,
! [F3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ~ ( member_c_d_set_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1231_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1232_finite__ne__induct,axiom,
! [F3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( F3 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a] : ( P @ ( insert_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) )
=> ( ! [X2: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( F4 != bot_bo738396921950161403_set_a )
=> ( ~ ( member_c_d_set_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1233_infinite__finite__induct,axiom,
! [P: set_a > $o,A6: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1234_infinite__finite__induct,axiom,
! [P: set_c_d_set_a > $o,A6: set_c_d_set_a] :
( ! [A8: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ~ ( member_c_d_set_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1235_all__finite__subset__image,axiom,
! [F: a > a,A6: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A6 ) ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1236_ex__finite__subset__image,axiom,
! [F: a > a,A6: set_a,P: set_a > $o] :
( ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A6 ) )
& ( P @ B6 ) ) )
= ( ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 )
& ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1237_finite__subset__image,axiom,
! [B5: set_a,F: a > a,A6: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A6 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A6 )
& ( finite_finite_a @ C4 )
& ( B5
= ( image_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1238_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_1239_order__class_Ofinite__has__minimal2,axiom,
! [A6: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A @ A6 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A6 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
thf(fact_1240_order__class_Ofinite__has__minimal2,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ A @ A6 )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ( ord_le8464990428230162895_set_a @ X2 @ A )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
thf(fact_1241_order__class_Ofinite__has__maximal2,axiom,
! [A6: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A @ A6 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A6 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A6 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_maximal2
thf(fact_1242_order__class_Ofinite__has__maximal2,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ A @ A6 )
=> ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A6 )
& ( ord_le8464990428230162895_set_a @ A @ X2 )
& ! [Xa2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa2 @ A6 )
=> ( ( ord_le8464990428230162895_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% order_class.finite_has_maximal2
thf(fact_1243_rev__finite__subset,axiom,
! [B5: set_a,A6: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( finite_finite_a @ A6 ) ) ) ).
% rev_finite_subset
thf(fact_1244_infinite__super,axiom,
! [S3: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S3 @ T2 )
=> ( ~ ( finite_finite_a @ S3 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1245_finite__subset,axiom,
! [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
=> ( ( finite_finite_a @ B5 )
=> ( finite_finite_a @ A6 ) ) ) ).
% finite_subset
thf(fact_1246_le__cSup__finite,axiom,
! [X6: set_set_a,X4: set_a] :
( ( finite_finite_set_a @ X6 )
=> ( ( member_set_a @ X4 @ X6 )
=> ( ord_less_eq_set_a @ X4 @ ( comple2307003609928055243_set_a @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_1247_le__cSup__finite,axiom,
! [X6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ X6 )
=> ( ( member_c_d_set_a @ X4 @ X6 )
=> ( ord_le8464990428230162895_set_a @ X4 @ ( comple3834726295627996700_set_a @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_1248_finite__subset__Union,axiom,
! [A6: set_a,B8: set_set_a] :
( ( finite_finite_a @ A6 )
=> ( ( ord_less_eq_set_a @ A6 @ ( comple2307003609928055243_set_a @ B8 ) )
=> ~ ! [F5: set_set_a] :
( ( finite_finite_set_a @ F5 )
=> ( ( ord_le3724670747650509150_set_a @ F5 @ B8 )
=> ~ ( ord_less_eq_set_a @ A6 @ ( comple2307003609928055243_set_a @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1249_finite__subset__induct,axiom,
! [F3: set_c_d_set_a,A6: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A6 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A2: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( member_c_d_set_a @ A2 @ A6 )
=> ( ~ ( member_c_d_set_a @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ A2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1250_finite__subset__induct,axiom,
! [F3: set_a,A6: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A6 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A2 @ A6 )
=> ( ~ ( member_a @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1251_finite__subset__induct_H,axiom,
! [F3: set_c_d_set_a,A6: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A6 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A2: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( member_c_d_set_a @ A2 @ A6 )
=> ( ( ord_le5982164083705284911_set_a @ F4 @ A6 )
=> ( ~ ( member_c_d_set_a @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ A2 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1252_finite__subset__induct_H,axiom,
! [F3: set_a,A6: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A6 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A2 @ A6 )
=> ( ( ord_less_eq_set_a @ F4 @ A6 )
=> ( ~ ( member_a @ A2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A2 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1253_finite__remove__induct,axiom,
! [B5: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ B5 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( A8 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A8 @ B5 )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1254_finite__remove__induct,axiom,
! [B5: set_a,P: set_a > $o] :
( ( finite_finite_a @ B5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_1255_remove__induct,axiom,
! [P: set_c_d_set_a > $o,B5: set_c_d_set_a] :
( ( P @ bot_bo738396921950161403_set_a )
=> ( ( ~ ( finite3330819693523053784_set_a @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A8 )
=> ( ( A8 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ A8 @ B5 )
=> ( ! [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A8 )
=> ( P @ ( minus_1665977719694084726_set_a @ A8 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1256_remove__induct,axiom,
! [P: set_a > $o,B5: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B5 )
=> ( P @ B5 ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B5 )
=> ( ! [X: a] :
( ( member_a @ X @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B5 ) ) ) ) ).
% remove_induct
thf(fact_1257_finite__induct__select,axiom,
! [S3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ S3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [T3: set_c_d_set_a] :
( ( ord_le3685282097655362107_set_a @ T3 @ S3 )
=> ( ( P @ T3 )
=> ? [X: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( minus_1665977719694084726_set_a @ S3 @ T3 ) )
& ( P @ ( insert_c_d_set_a @ X @ T3 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_1258_admissible__chfin,axiom,
! [P: set_a > $o] :
( ! [S6: set_set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ S6 )
=> ( finite_finite_set_a @ S6 ) )
=> ( comple8887300225568239275_set_a @ comple2307003609928055243_set_a @ ord_less_eq_set_a @ P ) ) ).
% admissible_chfin
thf(fact_1259_admissible__chfin,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ! [S6: set_c_d_set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ S6 )
=> ( finite3330819693523053784_set_a @ S6 ) )
=> ( comple1957918121334358780_set_a @ comple3834726295627996700_set_a @ ord_le8464990428230162895_set_a @ P ) ) ).
% admissible_chfin
thf(fact_1260_in__chain__finite,axiom,
! [A6: set_set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A6 )
=> ( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ A6 ) @ A6 ) ) ) ) ).
% in_chain_finite
thf(fact_1261_in__chain__finite,axiom,
! [A6: set_c_d_set_a] :
( ( comple7455786223818501471_set_a @ ord_le8464990428230162895_set_a @ A6 )
=> ( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( member_c_d_set_a @ ( comple3834726295627996700_set_a @ A6 ) @ A6 ) ) ) ) ).
% in_chain_finite
thf(fact_1262_local_OInf__fin_Oremove,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 )
= X4 ) )
& ( ( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 )
= ( inf_c_d_a2 @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ) ).
% local.Inf_fin.remove
thf(fact_1263_local_OInf__fin_Oinsert__remove,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X4 @ A6 ) )
= X4 ) )
& ( ( ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X4 @ A6 ) )
= ( inf_c_d_a2 @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( minus_1665977719694084726_set_a @ A6 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ).
% local.Inf_fin.insert_remove
thf(fact_1264_local_OInf__fin_Oin__idem,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ X4 @ A6 )
=> ( ( inf_c_d_a2 @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) )
= ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ).
% local.Inf_fin.in_idem
thf(fact_1265_local_OInf__fin_OcoboundedI,axiom,
! [A6: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( member_c_d_set_a @ A @ A6 )
=> ( smaller_interp_c_d_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) @ A ) ) ) ).
% local.Inf_fin.coboundedI
thf(fact_1266_local_OInf__fin_Ohom__commute,axiom,
! [H: ( ( c > d ) > set_a ) > ( c > d ) > set_a,N: set_c_d_set_a] :
( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( H @ ( inf_c_d_a2 @ X2 @ Y2 ) )
= ( inf_c_d_a2 @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite3330819693523053784_set_a @ N )
=> ( ( N != bot_bo738396921950161403_set_a )
=> ( ( H @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ N ) )
= ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( image_5710119992958135237_set_a @ H @ N ) ) ) ) ) ) ).
% local.Inf_fin.hom_commute
thf(fact_1267_local_OInf__fin_OboundedE,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) )
=> ! [A9: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A9 @ A6 )
=> ( smaller_interp_c_d_a @ X4 @ A9 ) ) ) ) ) ).
% local.Inf_fin.boundedE
thf(fact_1268_local_OInf__fin_OboundedI,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [A2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A2 @ A6 )
=> ( smaller_interp_c_d_a @ X4 @ A2 ) )
=> ( smaller_interp_c_d_a @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ) ).
% local.Inf_fin.boundedI
thf(fact_1269_local_OInf__fin_Obounded__iff,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) )
= ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A6 )
=> ( smaller_interp_c_d_a @ X4 @ X3 ) ) ) ) ) ) ).
% local.Inf_fin.bounded_iff
thf(fact_1270_local_OInf__fin__Inf,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 )
= ( inf_c_d_a @ A6 ) ) ) ) ).
% local.Inf_fin_Inf
thf(fact_1271_local_OInf__fin_Oinsert__not__elem,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ~ ( member_c_d_set_a @ X4 @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X4 @ A6 ) )
= ( inf_c_d_a2 @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ) ) ).
% local.Inf_fin.insert_not_elem
thf(fact_1272_local_OInf__fin_Oclosed,axiom,
! [A6: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ! [X2: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( member_c_d_set_a @ ( inf_c_d_a2 @ X2 @ Y2 ) @ ( insert_c_d_set_a @ X2 @ ( insert_c_d_set_a @ Y2 @ bot_bo738396921950161403_set_a ) ) )
=> ( member_c_d_set_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) @ A6 ) ) ) ) ).
% local.Inf_fin.closed
thf(fact_1273_local_OInf__fin_Osubset,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( B5 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ B5 @ A6 )
=> ( ( inf_c_d_a2 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ B5 ) @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) )
= ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ) ).
% local.Inf_fin.subset
thf(fact_1274_local_OInf__fin_Osubset__imp,axiom,
! [A6: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A6 @ B5 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( finite3330819693523053784_set_a @ B5 )
=> ( smaller_interp_c_d_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ B5 ) @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ) ).
% local.Inf_fin.subset_imp
thf(fact_1275_local_OInf__fin_Osingleton,axiom,
! [X4: ( c > d ) > set_a] :
( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X4 @ bot_bo738396921950161403_set_a ) )
= X4 ) ).
% local.Inf_fin.singleton
thf(fact_1276_local_OInf__fin_Oinsert,axiom,
! [A6: set_c_d_set_a,X4: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A6 )
=> ( ( A6 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X4 @ A6 ) )
= ( inf_c_d_a2 @ X4 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A6 ) ) ) ) ) ).
% local.Inf_fin.insert
% Conjectures (1)
thf(conj_0,conjecture,
? [X: a] :
( ( pre_larger_a @ plus @ a3 @ X )
& ( pre_larger_a @ plus @ b @ X )
& ( member_a @ X @ ( comple3834726295627996700_set_a @ a2 @ s ) ) ) ).
%------------------------------------------------------------------------------