TPTP Problem File: SLH0097^1.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_01078_032874__6982506_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1405 ( 557 unt; 123 typ; 0 def)
% Number of atoms : 3911 (1040 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 10685 ( 234 ~; 23 |; 209 &;8641 @)
% ( 0 <=>;1578 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 4469 (4469 >; 0 *; 0 +; 0 <<)
% Number of symbols : 116 ( 113 usr; 13 con; 0-4 aty)
% Number of variables : 3475 ( 213 ^;3198 !; 64 ?;3475 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:47.636
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
set_Pr2676350728994116295_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
set_Su1130066786674581787_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_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_Itf__a_J_J,type,
set_set_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 (113)
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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple7316045241340859548_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_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_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_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_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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
finite1740182815655637662_set_a: set_option_c_d_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
finite2397556900044337168_set_a: set_Pr2676350728994116295_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
finite457288119118821432_set_a: set_set_c_d_set_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_001t__Sum____Type__Osum_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
finite5989733633321134460_set_a: set_Su1130066786674581787_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_FixedPoint_Ologic_OD_001tf__c_001tf__d_001tf__a,type,
d_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > set_c_d_set_a ).
thf(sy_c_FixedPoint_Ologic_OGFP_001tf__c_001tf__d_001tf__a,type,
gFP_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( c > d ) > set_a ).
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_Omonotonic_001tf__c_001tf__d_001tf__a,type,
monotonic_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_Osup_001tf__c_001tf__d_001tf__a,type,
sup_c_d_a2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
monoto6642458133393520519_set_a: set_c_d_set_a > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( ( ( c > d ) > set_a ) > set_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_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,
monoto5673664640695304391_set_a: set_set_c_d_set_a > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( set_c_d_set_a > ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_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,
monoto4733996707696316455_set_a: set_set_c_d_set_a > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_001t__Set__Oset_Itf__a_J,type,
monoto9091215303422693110_set_a: set_set_c_d_set_a > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_a > set_a > $o ) > ( set_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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
monoto7894950695950633880_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( set_a > set_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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
monoto4999900198720154872_set_a: set_a > ( a > a > $o ) > ( set_c_d_set_a > set_c_d_set_a > $o ) > ( a > set_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_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_Groups_Omonoid_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
monoid_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Groups_Omonoid_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
monoid_set_c_d_set_a: ( set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ) > set_c_d_set_a > $o ).
thf(sy_c_If_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
if_c_d_set_a: $o > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice_Ogfp_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple4132920576971123013_set_a: ( set_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 ) > ( c > d ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice_Olfp_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple5961674822413889664_set_a: ( set_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 ) > ( c > d ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
comple4054414736020850733_set_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
comple5772108289334984589_set_a: ( set_c_d_set_a > set_c_d_set_a ) > set_c_d_set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_Itf__a_J,type,
comple3341859861669737308_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_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
semila7616582506879544593_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
semila3717735699007493233_set_a: ( set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ) > set_c_d_set_a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
sup_sup_c_d_set_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_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,
sup_su3175602471750379875_set_a: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_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__sup_OSup__fin_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
lattic5849929604656016644_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_Orderings_Obot__class_Obot_001_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_J,type,
bot_bo919924463001950746et_a_o: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
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_Itf__a_M_Eo_J,type,
bot_bot_a_o: 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_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_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_OLeast_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
least_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > ( c > d ) > set_a ).
thf(sy_c_Orderings_Oord_Omax_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
max_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_Orderings_Oord_Omin_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
min_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
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_062_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_Eo_J_J,type,
ord_le1832228425591547726et_a_o: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > ( ( 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_I_Eo_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le252514701126353884_set_a: ( $o > ( c > d ) > set_a ) > ( $o > ( c > d ) > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
ord_le6704328240068426556_set_a: ( $o > set_c_d_set_a ) > ( $o > set_c_d_set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $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_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_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_Oorder_OGreatest_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
greatest_c_d_set_a: ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > ( ( ( c > d ) > set_a ) > $o ) > ( c > d ) > set_a ).
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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
partia1270112395057131461_set_a: ( set_c_d_set_a > set_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_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_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to1333438998097461157_set_a: set_option_c_d_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to3895570120271872023_set_a: set_Pr2676350728994116295_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_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,
top_to5717711934741766719_set_a: set_set_c_d_set_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_It__Sum____Type__Osum_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J_J,type,
top_to279427854467338187_set_a: set_Su1130066786674581787_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_Relation_Oantisymp__on_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J,type,
antisy1518167394357443548_set_a: set_c_d_set_a > ( ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ) > $o ).
thf(sy_c_Relation_Oantisymp__on_001t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
antisy2568922457103120188_set_a: set_set_c_d_set_a > ( set_c_d_set_a > set_c_d_set_a > $o ) > $o ).
thf(sy_c_Relation_Oantisymp__on_001t__Set__Oset_Itf__a_J,type,
antisymp_on_set_a: set_set_a > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Relation_Oantisymp__on_001tf__a,type,
antisymp_on_a: set_a > ( a > a > $o ) > $o ).
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_001tf__a,type,
collect_a: ( a > $o ) > 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_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_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_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_001tf__a,type,
insert_a: a > set_a > 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_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_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_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_f,type,
f: ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
% Relevant facts (1278)
thf(fact_0_local_Odual__order_Oantisym,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( smaller_interp_c_d_a @ A @ B )
=> ( A = B ) ) ) ).
% local.dual_order.antisym
thf(fact_1_local_Odual__order_Oeq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B2 @ A2 )
& ( smaller_interp_c_d_a @ A2 @ B2 ) ) ) ) ).
% local.dual_order.eq_iff
thf(fact_2_local_Odual__order_Otrans,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( smaller_interp_c_d_a @ C @ B )
=> ( smaller_interp_c_d_a @ C @ A ) ) ) ).
% local.dual_order.trans
thf(fact_3_local_Oorder_Oeq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A2 @ B2 )
& ( smaller_interp_c_d_a @ B2 @ A2 ) ) ) ) ).
% local.order.eq_iff
thf(fact_4_local_Oantisym__conv,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y2 @ X )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% local.antisym_conv
thf(fact_5_local_Oeq__refl,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( X = Y2 )
=> ( smaller_interp_c_d_a @ X @ Y2 ) ) ).
% local.eq_refl
thf(fact_6_local_Oord__eq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A = B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_7_local_Oord__le__eq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( B = C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_8_local_Oorder__antisym,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( smaller_interp_c_d_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% local.order_antisym
thf(fact_9_local_Oorder__eq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X2 @ Y3 )
& ( smaller_interp_c_d_a @ Y3 @ X2 ) ) ) ) ).
% local.order_eq_iff
thf(fact_10_local_Oorder__trans,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( smaller_interp_c_d_a @ Y2 @ Z2 )
=> ( smaller_interp_c_d_a @ X @ Z2 ) ) ) ).
% local.order_trans
thf(fact_11_smaller__interpI,axiom,
! [Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a] :
( ! [S: c > d,X3: a] :
( ( member_a @ X3 @ ( Delta @ S ) )
=> ( member_a @ X3 @ ( Delta2 @ S ) ) )
=> ( smaller_interp_c_d_a @ Delta @ Delta2 ) ) ).
% smaller_interpI
thf(fact_12_smaller__interp__antisym,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ A )
=> ( A = B ) ) ) ).
% smaller_interp_antisym
thf(fact_13_smaller__interp__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% smaller_interp_trans
thf(fact_14__092_060open_062smaller__interp_A_IGFP_Af_J_A_Icomplete__lattice__class_Ogfp_Af_J_092_060close_062,axiom,
smaller_interp_c_d_a @ ( gFP_c_d_a @ f ) @ ( comple4054414736020850733_set_a @ f ) ).
% \<open>smaller_interp (GFP f) (complete_lattice_class.gfp f)\<close>
thf(fact_15__092_060open_062smaller__interp_A_Icomplete__lattice__class_Ogfp_Af_J_A_IGFP_Af_J_092_060close_062,axiom,
smaller_interp_c_d_a @ ( comple4054414736020850733_set_a @ f ) @ ( gFP_c_d_a @ f ) ).
% \<open>smaller_interp (complete_lattice_class.gfp f) (GFP f)\<close>
thf(fact_16_GFP__greatest,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( ( F @ U )
= U )
=> ( smaller_interp_c_d_a @ U @ ( gFP_c_d_a @ F ) ) ) ).
% GFP_greatest
thf(fact_17__092_060open_062f_A_IGFP_Af_J_A_061_AGFP_Af_092_060close_062,axiom,
( ( f @ ( gFP_c_d_a @ f ) )
= ( gFP_c_d_a @ f ) ) ).
% \<open>f (GFP f) = GFP f\<close>
thf(fact_18__092_060open_062f_A_Icomplete__lattice__class_Ogfp_Af_J_A_061_Acomplete__lattice__class_Ogfp_Af_092_060close_062,axiom,
( ( f @ ( comple4054414736020850733_set_a @ f ) )
= ( comple4054414736020850733_set_a @ f ) ) ).
% \<open>f (complete_lattice_class.gfp f) = complete_lattice_class.gfp f\<close>
thf(fact_19_assms,axiom,
monotonic_c_d_a @ f ).
% assms
thf(fact_20_GFP__lub,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ ( d_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ X3 @ Y2 ) )
=> ( smaller_interp_c_d_a @ ( gFP_c_d_a @ F ) @ Y2 ) ) ).
% GFP_lub
thf(fact_21_smaller__interp__D,axiom,
! [X: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ ( d_c_d_a @ F ) )
=> ( smaller_interp_c_d_a @ X @ ( gFP_c_d_a @ F ) ) ) ).
% smaller_interp_D
thf(fact_22_local_Oorder_Orefl,axiom,
! [A: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ A @ A ) ).
% local.order.refl
thf(fact_23_local_Oorder__refl,axiom,
! [X: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ X @ X ) ).
% local.order_refl
thf(fact_24_smaller__interp__refl,axiom,
! [Delta: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ Delta @ Delta ) ).
% smaller_interp_refl
thf(fact_25_GFP__is__FP,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monotonic_c_d_a @ F )
=> ( ( F @ ( gFP_c_d_a @ F ) )
= ( gFP_c_d_a @ F ) ) ) ).
% GFP_is_FP
thf(fact_26_local_OGreatestI2__order,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a,Q: ( ( c > d ) > set_a ) > $o] :
( ( P @ X )
=> ( ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ Y4 @ X ) )
=> ( ! [X3: ( c > d ) > set_a] :
( ( P @ X3 )
=> ( ! [Y5: ( c > d ) > set_a] :
( ( P @ Y5 )
=> ( smaller_interp_c_d_a @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( greatest_c_d_set_a @ smaller_interp_c_d_a @ P ) ) ) ) ) ).
% local.GreatestI2_order
thf(fact_27_local_OGreatest__equality,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a] :
( ( P @ X )
=> ( ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ Y4 @ X ) )
=> ( ( greatest_c_d_set_a @ smaller_interp_c_d_a @ P )
= X ) ) ) ).
% local.Greatest_equality
thf(fact_28_local_OLeast1I,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ? [X4: ( c > d ) > set_a] :
( ( P @ X4 )
& ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ X4 @ Y4 ) )
& ! [Y4: ( c > d ) > set_a] :
( ( ( P @ Y4 )
& ! [Ya: ( c > d ) > set_a] :
( ( P @ Ya )
=> ( smaller_interp_c_d_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( P @ ( least_c_d_set_a @ smaller_interp_c_d_a @ P ) ) ) ).
% local.Least1I
thf(fact_29_local_OLeast1__le,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Z2: ( c > d ) > set_a] :
( ? [X4: ( c > d ) > set_a] :
( ( P @ X4 )
& ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ X4 @ Y4 ) )
& ! [Y4: ( c > d ) > set_a] :
( ( ( P @ Y4 )
& ! [Ya: ( c > d ) > set_a] :
( ( P @ Ya )
=> ( smaller_interp_c_d_a @ Y4 @ Ya ) ) )
=> ( Y4 = X4 ) ) )
=> ( ( P @ Z2 )
=> ( smaller_interp_c_d_a @ ( least_c_d_set_a @ smaller_interp_c_d_a @ P ) @ Z2 ) ) ) ).
% local.Least1_le
thf(fact_30_local_OLeastI2__order,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a,Q: ( ( c > d ) > set_a ) > $o] :
( ( P @ X )
=> ( ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ X @ Y4 ) )
=> ( ! [X3: ( c > d ) > set_a] :
( ( P @ X3 )
=> ( ! [Y5: ( c > d ) > set_a] :
( ( P @ Y5 )
=> ( smaller_interp_c_d_a @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( least_c_d_set_a @ smaller_interp_c_d_a @ P ) ) ) ) ) ).
% local.LeastI2_order
thf(fact_31_monotonicI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F @ Delta3 ) @ ( F @ Delta4 ) ) )
=> ( monotonic_c_d_a @ F ) ) ).
% monotonicI
thf(fact_32_monotonic__def,axiom,
( monotonic_c_d_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F2 @ Delta5 ) @ ( F2 @ Delta6 ) ) ) ) ) ).
% monotonic_def
thf(fact_33_local_OLeast__equality,axiom,
! [P: ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a] :
( ( P @ X )
=> ( ! [Y4: ( c > d ) > set_a] :
( ( P @ Y4 )
=> ( smaller_interp_c_d_a @ X @ Y4 ) )
=> ( ( least_c_d_set_a @ smaller_interp_c_d_a @ P )
= X ) ) ) ).
% local.Least_equality
thf(fact_34_local_Omin__def,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( smaller_interp_c_d_a @ A @ B )
=> ( ( min_c_d_set_a @ smaller_interp_c_d_a @ A @ B )
= A ) )
& ( ~ ( smaller_interp_c_d_a @ A @ B )
=> ( ( min_c_d_set_a @ smaller_interp_c_d_a @ A @ B )
= B ) ) ) ).
% local.min_def
thf(fact_35_local_Omax__def,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( smaller_interp_c_d_a @ A @ B )
=> ( ( max_c_d_set_a @ smaller_interp_c_d_a @ A @ B )
= B ) )
& ( ~ ( smaller_interp_c_d_a @ A @ B )
=> ( ( max_c_d_set_a @ smaller_interp_c_d_a @ A @ B )
= A ) ) ) ).
% local.max_def
thf(fact_36_local_Oorder_Opartial__preordering__axioms,axiom,
partia701112543150332005_set_a @ smaller_interp_c_d_a ).
% local.order.partial_preordering_axioms
thf(fact_37_order_OGreatest_Ocong,axiom,
greatest_c_d_set_a = greatest_c_d_set_a ).
% order.Greatest.cong
thf(fact_38_ord_OLeast_Ocong,axiom,
least_c_d_set_a = least_c_d_set_a ).
% ord.Least.cong
thf(fact_39_local_Oantisymp__on__le,axiom,
! [A3: set_c_d_set_a] : ( antisy1518167394357443548_set_a @ A3 @ smaller_interp_c_d_a ) ).
% local.antisymp_on_le
thf(fact_40_smaller__interp__def,axiom,
( smaller_interp_c_d_a
= ( ^ [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
! [S2: c > d] : ( ord_less_eq_set_a @ ( Delta5 @ S2 ) @ ( Delta6 @ S2 ) ) ) ) ).
% smaller_interp_def
thf(fact_41_local_Obdd__above__def,axiom,
! [A3: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 )
= ( ? [M: ( c > d ) > set_a] :
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ X2 @ M ) ) ) ) ).
% local.bdd_above_def
thf(fact_42_local_Obdd__above_OE,axiom,
! [A3: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A3 )
=> ( smaller_interp_c_d_a @ X4 @ M2 ) ) ) ).
% local.bdd_above.E
thf(fact_43_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_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_45_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_c_d_set_a] :
( ( collect_c_d_set_a
@ ^ [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_local_Obdd__below__def,axiom,
! [A3: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 )
= ( ? [M: ( c > d ) > set_a] :
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ M @ X2 ) ) ) ) ).
% local.bdd_below_def
thf(fact_48_local_Obdd__below_OE,axiom,
! [A3: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 )
=> ~ ! [M2: ( c > d ) > set_a] :
~ ! [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A3 )
=> ( smaller_interp_c_d_a @ M2 @ X4 ) ) ) ).
% local.bdd_below.E
thf(fact_49_preorder__class_Odual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_50_preorder__class_Odual__order_Orefl,axiom,
! [A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ A ) ).
% preorder_class.dual_order.refl
thf(fact_51_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_52_preorder__class_Oorder__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% preorder_class.order_refl
thf(fact_53_preorder__class_Oorder__refl,axiom,
! [X: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X @ X ) ).
% preorder_class.order_refl
thf(fact_54_preorder__class_Oorder__refl,axiom,
! [X: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X @ X ) ).
% preorder_class.order_refl
thf(fact_55_local_Obdd__below__bot,axiom,
! [A3: set_c_d_set_a] : ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ).
% local.bdd_below_bot
thf(fact_56_local_Obdd__belowI,axiom,
! [A3: set_c_d_set_a,M3: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M3 @ X3 ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_belowI
thf(fact_57_local_Obdd__below_OI,axiom,
! [A3: set_c_d_set_a,M4: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M4 @ X3 ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_below.I
thf(fact_58_local_Obdd__above__top,axiom,
! [A3: set_c_d_set_a] : ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 ) ).
% local.bdd_above_top
thf(fact_59_local_Obdd__aboveI,axiom,
! [A3: set_c_d_set_a,M4: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ X3 @ M4 ) )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_aboveI
thf(fact_60_partial__preordering__def,axiom,
( partia701112543150332005_set_a
= ( ^ [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [A2: ( c > d ) > set_a] : ( Less_eq @ A2 @ A2 )
& ! [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a,C2: ( c > d ) > set_a] :
( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ A2 @ C2 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_61_order__antisym__conv,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_62_order__antisym__conv,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y2 @ X )
=> ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_63_order__antisym__conv,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y2 @ X )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_64_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_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,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_order__eq__refl,axiom,
! [X: set_a,Y2: set_a] :
( ( X = Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_83_order__eq__refl,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( X = Y2 )
=> ( ord_le8464990428230162895_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_84_order__eq__refl,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( X = Y2 )
=> ( ord_le5982164083705284911_set_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_85_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A: set_a,F: ( ( c > d ) > set_a ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst1,axiom,
! [A: set_a,F: set_c_d_set_a > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_a > ( c > d ) > set_a,B: set_a,C: set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: ( c > d ) > set_a,F: set_c_d_set_a > ( c > d ) > set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le8464990428230162895_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: set_c_d_set_a,F: set_a > set_c_d_set_a,B: set_a,C: set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_order__subst1,axiom,
! [A: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5982164083705284911_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_104_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A2 @ B2 )
& ( ord_le8464990428230162895_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_105_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_c_d_set_a,Z: set_c_d_set_a] : ( Y = Z ) )
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A2 @ B2 )
& ( ord_le5982164083705284911_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_106_le__fun__def,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [F2: ( c > d ) > set_a,G: ( c > d ) > set_a] :
! [X2: c > d] : ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_107_le__funI,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a] :
( ! [X3: c > d] : ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le8464990428230162895_set_a @ F @ G2 ) ) ).
% le_funI
thf(fact_108_le__funE,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a,X: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_109_le__funD,axiom,
! [F: ( c > d ) > set_a,G2: ( c > d ) > set_a,X: c > d] :
( ( ord_le8464990428230162895_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_110_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_111_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_112_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_113_preorder__class_Oorder_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% preorder_class.order.partial_preordering_axioms
thf(fact_114_preorder__class_Oorder_Opartial__preordering__axioms,axiom,
partia1270112395057131461_set_a @ ord_le5982164083705284911_set_a ).
% preorder_class.order.partial_preordering_axioms
thf(fact_115_preorder__class_Oorder_Opartial__preordering__axioms,axiom,
partia701112543150332005_set_a @ ord_le8464990428230162895_set_a ).
% preorder_class.order.partial_preordering_axioms
thf(fact_116_preorder__class_Odual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_117_preorder__class_Odual__order_Otrans,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C @ B )
=> ( ord_le8464990428230162895_set_a @ C @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_118_preorder__class_Odual__order_Otrans,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ C @ B )
=> ( ord_le5982164083705284911_set_a @ C @ A ) ) ) ).
% preorder_class.dual_order.trans
thf(fact_119_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_120_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_121_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_122_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_123_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B2 @ A2 )
& ( ord_le8464990428230162895_set_a @ A2 @ B2 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_124_order__class_Odual__order_Oeq__iff,axiom,
( ( ^ [Y: set_c_d_set_a,Z: set_c_d_set_a] : ( Y = Z ) )
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B2 @ A2 )
& ( ord_le5982164083705284911_set_a @ A2 @ B2 ) ) ) ) ).
% order_class.dual_order.eq_iff
thf(fact_125_preorder__class_Oorder__trans,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% preorder_class.order_trans
thf(fact_126_preorder__class_Oorder__trans,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ( ord_le8464990428230162895_set_a @ Y2 @ Z2 )
=> ( ord_le8464990428230162895_set_a @ X @ Z2 ) ) ) ).
% preorder_class.order_trans
thf(fact_127_preorder__class_Oorder__trans,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ( ord_le5982164083705284911_set_a @ Y2 @ Z2 )
=> ( ord_le5982164083705284911_set_a @ X @ Z2 ) ) ) ).
% preorder_class.order_trans
thf(fact_128_preorder__class_Oorder_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% preorder_class.order.trans
thf(fact_129_preorder__class_Oorder_Otrans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% preorder_class.order.trans
thf(fact_130_preorder__class_Oorder_Otrans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% preorder_class.order.trans
thf(fact_131_partial__preordering_Otrans,axiom,
! [Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( partia701112543150332005_set_a @ Less_eq2 )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less_eq2 @ B @ C )
=> ( Less_eq2 @ A @ C ) ) ) ) ).
% partial_preordering.trans
thf(fact_132_partial__preordering_Ointro,axiom,
! [Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [A4: ( c > d ) > set_a] : ( Less_eq2 @ A4 @ A4 )
=> ( ! [A4: ( c > d ) > set_a,B3: ( c > d ) > set_a,C3: ( c > d ) > set_a] :
( ( Less_eq2 @ A4 @ B3 )
=> ( ( Less_eq2 @ B3 @ C3 )
=> ( Less_eq2 @ A4 @ C3 ) ) )
=> ( partia701112543150332005_set_a @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_133_order__class_Oorder__antisym,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_class.order_antisym
thf(fact_134_order__class_Oorder__antisym,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ( ord_le8464990428230162895_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_class.order_antisym
thf(fact_135_order__class_Oorder__antisym,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ( ord_le5982164083705284911_set_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_class.order_antisym
thf(fact_136_ord__class_Oord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_137_ord__class_Oord__le__eq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_138_ord__class_Oord__le__eq__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% ord_class.ord_le_eq_trans
thf(fact_139_ord__class_Oord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_140_ord__class_Oord__eq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( A = B )
=> ( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_141_ord__class_Oord__eq__le__trans,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( A = B )
=> ( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% ord_class.ord_eq_le_trans
thf(fact_142_partial__preordering_Orefl,axiom,
! [Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A: ( c > d ) > set_a] :
( ( partia701112543150332005_set_a @ Less_eq2 )
=> ( Less_eq2 @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_143_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_144_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) )
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X2 @ Y3 )
& ( ord_le8464990428230162895_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_145_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_c_d_set_a,Z: set_c_d_set_a] : ( Y = Z ) )
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X2 @ Y3 )
& ( ord_le5982164083705284911_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_146_ord_Omin_Ocong,axiom,
min_c_d_set_a = min_c_d_set_a ).
% ord.min.cong
thf(fact_147_ord_Omax_Ocong,axiom,
max_c_d_set_a = max_c_d_set_a ).
% ord.max.cong
thf(fact_148_ord_Omin__def,axiom,
( min_c_d_set_a
= ( ^ [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] : ( if_c_d_set_a @ ( Less_eq @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% ord.min_def
thf(fact_149_ord_Omax__def,axiom,
( max_c_d_set_a
= ( ^ [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] : ( if_c_d_set_a @ ( Less_eq @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% ord.max_def
thf(fact_150_local_Obdd__above__primitive__def,axiom,
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a )
= ( condit8154225043310684324_set_a @ smaller_interp_c_d_a ) ) ).
% local.bdd_above_primitive_def
thf(fact_151_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_152_subset__antisym,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_153_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_154_subsetI,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( member_c_d_set_a @ X3 @ B4 ) )
=> ( ord_le5982164083705284911_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_155_antisymp__on__subset,axiom,
! [A3: set_a,R: a > a > $o,B4: set_a] :
( ( antisymp_on_a @ A3 @ R )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( antisymp_on_a @ B4 @ R ) ) ) ).
% antisymp_on_subset
thf(fact_156_antisymp__on__subset,axiom,
! [A3: set_c_d_set_a,R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,B4: set_c_d_set_a] :
( ( antisy1518167394357443548_set_a @ A3 @ R )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( antisy1518167394357443548_set_a @ B4 @ R ) ) ) ).
% antisymp_on_subset
thf(fact_157_order__class_Oantisymp__on__le,axiom,
! [A3: set_set_a] : ( antisymp_on_set_a @ A3 @ ord_less_eq_set_a ) ).
% order_class.antisymp_on_le
thf(fact_158_order__class_Oantisymp__on__le,axiom,
! [A3: set_c_d_set_a] : ( antisy1518167394357443548_set_a @ A3 @ ord_le8464990428230162895_set_a ) ).
% order_class.antisymp_on_le
thf(fact_159_order__class_Oantisymp__on__le,axiom,
! [A3: set_set_c_d_set_a] : ( antisy2568922457103120188_set_a @ A3 @ ord_le5982164083705284911_set_a ) ).
% order_class.antisymp_on_le
thf(fact_160_complete__lattice__class_Ogfp__least,axiom,
! [F: set_a > set_a,X5: set_a] :
( ! [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ ( F @ U2 ) )
=> ( ord_less_eq_set_a @ U2 @ X5 ) )
=> ( ord_less_eq_set_a @ ( comple3341859861669737308_set_a @ F ) @ X5 ) ) ).
% complete_lattice_class.gfp_least
thf(fact_161_complete__lattice__class_Ogfp__least,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,X5: set_c_d_set_a] :
( ! [U2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ U2 @ ( F @ U2 ) )
=> ( ord_le5982164083705284911_set_a @ U2 @ X5 ) )
=> ( ord_le5982164083705284911_set_a @ ( comple5772108289334984589_set_a @ F ) @ X5 ) ) ).
% complete_lattice_class.gfp_least
thf(fact_162_complete__lattice__class_Ogfp__least,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X5: ( c > d ) > set_a] :
( ! [U2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ U2 @ ( F @ U2 ) )
=> ( ord_le8464990428230162895_set_a @ U2 @ X5 ) )
=> ( ord_le8464990428230162895_set_a @ ( comple4054414736020850733_set_a @ F ) @ X5 ) ) ).
% complete_lattice_class.gfp_least
thf(fact_163_complete__lattice__class_Ogfp__upperbound,axiom,
! [X5: set_a,F: set_a > set_a] :
( ( ord_less_eq_set_a @ X5 @ ( F @ X5 ) )
=> ( ord_less_eq_set_a @ X5 @ ( comple3341859861669737308_set_a @ F ) ) ) ).
% complete_lattice_class.gfp_upperbound
thf(fact_164_complete__lattice__class_Ogfp__upperbound,axiom,
! [X5: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X5 @ ( F @ X5 ) )
=> ( ord_le5982164083705284911_set_a @ X5 @ ( comple5772108289334984589_set_a @ F ) ) ) ).
% complete_lattice_class.gfp_upperbound
thf(fact_165_complete__lattice__class_Ogfp__upperbound,axiom,
! [X5: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X5 @ ( F @ X5 ) )
=> ( ord_le8464990428230162895_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) ) ).
% complete_lattice_class.gfp_upperbound
thf(fact_166_gfp__mono,axiom,
! [F: set_a > set_a,G2: set_a > set_a] :
( ! [Z3: set_a] : ( ord_less_eq_set_a @ ( F @ Z3 ) @ ( G2 @ Z3 ) )
=> ( ord_less_eq_set_a @ ( comple3341859861669737308_set_a @ F ) @ ( comple3341859861669737308_set_a @ G2 ) ) ) ).
% gfp_mono
thf(fact_167_gfp__mono,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,G2: set_c_d_set_a > set_c_d_set_a] :
( ! [Z3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( F @ Z3 ) @ ( G2 @ Z3 ) )
=> ( ord_le5982164083705284911_set_a @ ( comple5772108289334984589_set_a @ F ) @ ( comple5772108289334984589_set_a @ G2 ) ) ) ).
% gfp_mono
thf(fact_168_gfp__mono,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Z3: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( F @ Z3 ) @ ( G2 @ Z3 ) )
=> ( ord_le8464990428230162895_set_a @ ( comple4054414736020850733_set_a @ F ) @ ( comple4054414736020850733_set_a @ G2 ) ) ) ).
% gfp_mono
thf(fact_169_local_Obdd__above__mono,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 ) ) ) ).
% local.bdd_above_mono
thf(fact_170_local_Obdd__below__mono,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ) ) ).
% local.bdd_below_mono
thf(fact_171_weak__coinduct,axiom,
! [A: a,X5: set_a,F: set_a > set_a] :
( ( member_a @ A @ X5 )
=> ( ( ord_less_eq_set_a @ X5 @ ( F @ X5 ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_172_weak__coinduct,axiom,
! [A: ( c > d ) > set_a,X5: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( member_c_d_set_a @ A @ X5 )
=> ( ( ord_le5982164083705284911_set_a @ X5 @ ( F @ X5 ) )
=> ( member_c_d_set_a @ A @ ( comple5772108289334984589_set_a @ F ) ) ) ) ).
% weak_coinduct
thf(fact_173_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X6: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_174_le__rel__bool__arg__iff,axiom,
( ord_le252514701126353884_set_a
= ( ^ [X6: $o > ( c > d ) > set_a,Y6: $o > ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8464990428230162895_set_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_175_le__rel__bool__arg__iff,axiom,
( ord_le6704328240068426556_set_a
= ( ^ [X6: $o > set_c_d_set_a,Y6: $o > set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le5982164083705284911_set_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_176_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_177_in__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( member_c_d_set_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_178_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_179_subsetD,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_180_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_181_equalityE,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ~ ( ord_le5982164083705284911_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_183_subset__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A5 )
=> ( member_c_d_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_184_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_185_equalityD1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( A3 = B4 )
=> ( ord_le5982164083705284911_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_186_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_187_equalityD2,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( A3 = B4 )
=> ( ord_le5982164083705284911_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_188_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_189_subset__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
! [T: ( c > d ) > set_a] :
( ( member_c_d_set_a @ T @ A5 )
=> ( member_c_d_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_190_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_191_subset__refl,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_192_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_193_Collect__mono,axiom,
! [P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ! [X3: ( c > d ) > set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5982164083705284911_set_a @ ( collect_c_d_set_a @ P ) @ ( collect_c_d_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_194_subset__trans,axiom,
! [A3: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C4 )
=> ( ord_less_eq_set_a @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_195_subset__trans,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ C4 )
=> ( ord_le5982164083705284911_set_a @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_196_set__eq__subset,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_197_set__eq__subset,axiom,
( ( ^ [Y: set_c_d_set_a,Z: set_c_d_set_a] : ( Y = Z ) )
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A5 @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_198_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_199_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 ) )
= ( ! [X2: ( c > d ) > set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_200_antisymp__onD,axiom,
! [A3: set_a,R: a > a > $o,X: a,Y2: a] :
( ( antisymp_on_a @ A3 @ R )
=> ( ( member_a @ X @ A3 )
=> ( ( member_a @ Y2 @ A3 )
=> ( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_201_antisymp__onD,axiom,
! [A3: set_c_d_set_a,R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( antisy1518167394357443548_set_a @ A3 @ R )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ) ) ).
% antisymp_onD
thf(fact_202_antisymp__onI,axiom,
! [A3: set_a,R: a > a > $o] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A3 )
=> ( ( member_a @ Y4 @ A3 )
=> ( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisymp_on_a @ A3 @ R ) ) ).
% antisymp_onI
thf(fact_203_antisymp__onI,axiom,
! [A3: set_c_d_set_a,R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( ( member_c_d_set_a @ Y4 @ A3 )
=> ( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisy1518167394357443548_set_a @ A3 @ R ) ) ).
% antisymp_onI
thf(fact_204_antisymp__on__def,axiom,
( antisy1518167394357443548_set_a
= ( ^ [A5: set_c_d_set_a,R2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A5 )
=> ! [Y3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y3 @ A5 )
=> ( ( R2 @ X2 @ Y3 )
=> ( ( R2 @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ) ) ) ).
% antisymp_on_def
thf(fact_205_local_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,B4: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6316088450447394390_set_a @ B4 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F ) ) ) ).
% local.mono_on_subset
thf(fact_206_local_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto2937423850181994535_set_a @ B4 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% local.mono_on_subset
thf(fact_207_local_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6642458133393520519_set_a @ B4 @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% local.mono_on_subset
thf(fact_208_local_Obdd__above__finite,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_above_finite
thf(fact_209_local_Obdd__below__finite,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_below_finite
thf(fact_210_local_Obdd__aboveI2,axiom,
! [A3: set_a,F: a > ( c > d ) > set_a,M4: ( c > d ) > set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ ( F @ X3 ) @ M4 ) )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% local.bdd_aboveI2
thf(fact_211_local_Obdd__aboveI2,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,M4: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ ( F @ X3 ) @ M4 ) )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% local.bdd_aboveI2
thf(fact_212_local_Obdd__below_OI2,axiom,
! [A3: set_a,M4: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M4 @ ( F @ X3 ) ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% local.bdd_below.I2
thf(fact_213_local_Obdd__below_OI2,axiom,
! [A3: set_c_d_set_a,M4: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M4 @ ( F @ X3 ) ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% local.bdd_below.I2
thf(fact_214_local_Obdd__belowI2,axiom,
! [A3: set_a,M3: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M3 @ ( F @ X3 ) ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% local.bdd_belowI2
thf(fact_215_local_Obdd__belowI2,axiom,
! [A3: set_c_d_set_a,M3: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ M3 @ ( F @ X3 ) ) )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% local.bdd_belowI2
thf(fact_216_local_Obdd__above__Int1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% local.bdd_above_Int1
thf(fact_217_local_Obdd__above__Int2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ B4 )
=> ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% local.bdd_above_Int2
thf(fact_218_local_Obdd__below__Int1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% local.bdd_below_Int1
thf(fact_219_local_Obdd__below__Int2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ B4 )
=> ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% local.bdd_below_Int2
thf(fact_220_local_Ofinite__has__minimal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( smaller_interp_c_d_a @ X3 @ A )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( smaller_interp_c_d_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_minimal2
thf(fact_221_local_Ofinite__has__maximal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( smaller_interp_c_d_a @ A @ X3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( smaller_interp_c_d_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_maximal2
thf(fact_222_local_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( smaller_interp_c_d_a @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A3 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F ) ) ).
% local.mono_onI
thf(fact_223_local_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( smaller_interp_c_d_a @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A3 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% local.mono_onI
thf(fact_224_local_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( smaller_interp_c_d_a @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6642458133393520519_set_a @ A3 @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% local.mono_onI
thf(fact_225_local_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( smaller_interp_c_d_a @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_226_local_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( smaller_interp_c_d_a @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_227_local_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( smaller_interp_c_d_a @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% local.mono_onD
thf(fact_228_image__eqI,axiom,
! [B: a,F: a > a,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_229_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: a > ( c > d ) > set_a,X: a,A3: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A3 )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_230_image__eqI,axiom,
! [B: a,F: ( ( c > d ) > set_a ) > a,X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_231_image__eqI,axiom,
! [B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_232_IntI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_233_IntI,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( ( member_c_d_set_a @ C @ B4 )
=> ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_234_Int__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ( member_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_235_Int__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
= ( ( member_c_d_set_a @ C @ A3 )
& ( member_c_d_set_a @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_236_Int__subset__iff,axiom,
! [C4: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( ( ord_less_eq_set_a @ C4 @ A3 )
& ( ord_less_eq_set_a @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_237_Int__subset__iff,axiom,
! [C4: set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
= ( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
& ( ord_le5982164083705284911_set_a @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_238_image__Int__subset,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ ( image_c_d_set_a_a @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_239_image__Int__subset,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_240_IntE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ~ ( member_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_241_IntE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
=> ~ ( ( member_c_d_set_a @ C @ A3 )
=> ~ ( member_c_d_set_a @ C @ B4 ) ) ) ).
% IntE
thf(fact_242_IntD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% IntD1
thf(fact_243_IntD1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
=> ( member_c_d_set_a @ C @ A3 ) ) ).
% IntD1
thf(fact_244_IntD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ B4 ) ) ).
% IntD2
thf(fact_245_IntD2,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
=> ( member_c_d_set_a @ C @ B4 ) ) ).
% IntD2
thf(fact_246_imageI,axiom,
! [X: a,A3: set_a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_247_imageI,axiom,
! [X: a,A3: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X @ A3 )
=> ( member_c_d_set_a @ ( F @ X ) @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_248_imageI,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_a @ ( F @ X ) @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_249_imageI,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( member_c_d_set_a @ ( F @ X ) @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_250_Int__assoc,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ C4 )
= ( inf_in754637537901350525_set_a @ A3 @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) ).
% Int_assoc
thf(fact_251_Int__absorb,axiom,
! [A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_252_Int__commute,axiom,
( inf_in754637537901350525_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] : ( inf_in754637537901350525_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_253_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: a,F: a > a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_254_rev__image__eqI,axiom,
! [X: a,A3: set_a,B: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_c_d_set_a @ B @ ( image_a_c_d_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_255_rev__image__eqI,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B: a,F: ( ( c > d ) > set_a ) > a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_256_rev__image__eqI,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_257_Int__left__absorb,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_258_Int__left__commute,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) )
= ( inf_in754637537901350525_set_a @ B4 @ ( inf_in754637537901350525_set_a @ A3 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_259_subset__image__iff,axiom,
! [B4: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_260_subset__image__iff,axiom,
! [B4: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
= ( ? [AA: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ AA @ A3 )
& ( B4
= ( image_c_d_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_261_subset__image__iff,axiom,
! [B4: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_c_d_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_262_subset__image__iff,axiom,
! [B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( ? [AA: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ AA @ A3 )
& ( B4
= ( image_5710119992958135237_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_263_subset__imageE,axiom,
! [B4: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
=> ( B4
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_264_subset__imageE,axiom,
! [B4: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ~ ! [C5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C5 @ A3 )
=> ( B4
!= ( image_c_d_set_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_265_subset__imageE,axiom,
! [B4: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
=> ( B4
!= ( image_a_c_d_set_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_266_subset__imageE,axiom,
! [B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ~ ! [C5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C5 @ A3 )
=> ( B4
!= ( image_5710119992958135237_set_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_267_image__subsetI,axiom,
! [A3: set_a,F: a > a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_268_image__subsetI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a,B4: set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_269_image__subsetI,axiom,
! [A3: set_a,F: a > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_c_d_set_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_270_image__subsetI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( member_c_d_set_a @ ( F @ X3 ) @ B4 ) )
=> ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_271_image__mono,axiom,
! [A3: set_a,B4: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_272_image__mono,axiom,
! [A3: set_a,B4: set_a,F: a > ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ F @ A3 ) @ ( image_a_c_d_set_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_273_image__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ A3 ) @ ( image_c_d_set_a_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_274_image__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_275_weak__coinduct__image,axiom,
! [A: a,X5: set_a,G2: a > a,F: set_a > set_a] :
( ( member_a @ A @ X5 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ X5 ) @ ( F @ ( image_a_a @ G2 @ X5 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_276_weak__coinduct__image,axiom,
! [A: ( c > d ) > set_a,X5: set_c_d_set_a,G2: ( ( c > d ) > set_a ) > a,F: set_a > set_a] :
( ( member_c_d_set_a @ A @ X5 )
=> ( ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ G2 @ X5 ) @ ( F @ ( image_c_d_set_a_a @ G2 @ X5 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_277_weak__coinduct__image,axiom,
! [A: a,X5: set_a,G2: a > ( c > d ) > set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( member_a @ A @ X5 )
=> ( ( ord_le5982164083705284911_set_a @ ( image_a_c_d_set_a @ G2 @ X5 ) @ ( F @ ( image_a_c_d_set_a @ G2 @ X5 ) ) )
=> ( member_c_d_set_a @ ( G2 @ A ) @ ( comple5772108289334984589_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_278_weak__coinduct__image,axiom,
! [A: ( c > d ) > set_a,X5: set_c_d_set_a,G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( member_c_d_set_a @ A @ X5 )
=> ( ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ G2 @ X5 ) @ ( F @ ( image_5710119992958135237_set_a @ G2 @ X5 ) ) )
=> ( member_c_d_set_a @ ( G2 @ A ) @ ( comple5772108289334984589_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_279_Int__Collect__mono,axiom,
! [A3: set_a,B4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_280_Int__Collect__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o,Q: ( ( c > d ) > set_a ) > $o] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A3 @ ( collect_c_d_set_a @ P ) ) @ ( inf_in754637537901350525_set_a @ B4 @ ( collect_c_d_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_281_Int__greatest,axiom,
! [C4: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A3 )
=> ( ( ord_less_eq_set_a @ C4 @ B4 )
=> ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_282_Int__greatest,axiom,
! [C4: set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ C4 @ B4 )
=> ( ord_le5982164083705284911_set_a @ C4 @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_283_Int__absorb2,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_284_Int__absorb2,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_285_Int__absorb1,axiom,
! [B4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_286_Int__absorb1,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_287_Int__lower2,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_288_Int__lower2,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_289_Int__lower1,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_290_Int__lower1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_291_Int__mono,axiom,
! [A3: set_a,C4: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C4 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ C4 @ D ) ) ) ) ).
% Int_mono
thf(fact_292_Int__mono,axiom,
! [A3: set_c_d_set_a,C4: set_c_d_set_a,B4: set_c_d_set_a,D: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ C4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ D )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ ( inf_in754637537901350525_set_a @ C4 @ D ) ) ) ) ).
% Int_mono
thf(fact_293_finite__Int,axiom,
! [F3: set_c_d_set_a,G3: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ F3 )
| ( finite3330819693523053784_set_a @ G3 ) )
=> ( finite3330819693523053784_set_a @ ( inf_in754637537901350525_set_a @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_294_finite__imageI,axiom,
! [F3: set_c_d_set_a,H: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( finite3330819693523053784_set_a @ ( image_5710119992958135237_set_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_295_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_296_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C ) )
= ( ( ord_le8464990428230162895_set_a @ A @ B )
& ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_297_semilattice__inf__class_Oinf_Obounded__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( inf_in754637537901350525_set_a @ B @ C ) )
= ( ( ord_le5982164083705284911_set_a @ A @ B )
& ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.bounded_iff
thf(fact_298_semilattice__inf__class_Ole__inf__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_299_semilattice__inf__class_Ole__inf__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) )
= ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
& ( ord_le8464990428230162895_set_a @ X @ Z2 ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_300_semilattice__inf__class_Ole__inf__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
& ( ord_le5982164083705284911_set_a @ X @ Z2 ) ) ) ).
% semilattice_inf_class.le_inf_iff
thf(fact_301_ord__class_Omono__on__subset,axiom,
! [A3: set_set_a,F: set_a > set_a,B4: set_set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( monoto7172710143293369831_set_a @ B4 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_302_ord__class_Omono__on__subset,axiom,
! [A3: set_set_a,F: set_a > ( c > d ) > set_a,B4: set_set_a] :
( ( monoto2748056057003999288_set_a @ A3 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( monoto2748056057003999288_set_a @ B4 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_303_ord__class_Omono__on__subset,axiom,
! [A3: set_set_a,F: set_a > set_c_d_set_a,B4: set_set_a] :
( ( monoto7894950695950633880_set_a @ A3 @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ A3 )
=> ( monoto7894950695950633880_set_a @ B4 @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_304_ord__class_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,B4: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6316088450447394390_set_a @ B4 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_305_ord__class_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto2937423850181994535_set_a @ B4 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_306_ord__class_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6642458133393520519_set_a @ B4 @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_307_ord__class_Omono__on__subset,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_a,B4: set_set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le7272806397018272911_set_a @ B4 @ A3 )
=> ( monoto9091215303422693110_set_a @ B4 @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_308_ord__class_Omono__on__subset,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,B4: set_set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le7272806397018272911_set_a @ B4 @ A3 )
=> ( monoto5673664640695304391_set_a @ B4 @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_309_ord__class_Omono__on__subset,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,B4: set_set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le7272806397018272911_set_a @ B4 @ A3 )
=> ( monoto4733996707696316455_set_a @ B4 @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% ord_class.mono_on_subset
thf(fact_310_ord_Omono__on__subset,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a,B4: set_a] :
( ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( monotone_on_a_set_a @ B4 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_311_ord_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a,B4: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6316088450447394390_set_a @ B4 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_312_ord_Omono__on__subset,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > ( c > d ) > set_a,B4: set_a] :
( ( monoto2502030104860647832_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( monoto2502030104860647832_set_a @ B4 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_313_ord_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto2937423850181994535_set_a @ B4 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_314_ord_Omono__on__subset,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_c_d_set_a,B4: set_a] :
( ( monoto4999900198720154872_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( monoto4999900198720154872_set_a @ B4 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_315_ord_Omono__on__subset,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto6642458133393520519_set_a @ B4 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_316_all__finite__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A3 ) )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_317_all__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_c_d_set_a_a @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A3 ) )
=> ( P @ ( image_c_d_set_a_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_318_all__finite__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ! [B5: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ ( image_a_c_d_set_a @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A3 ) )
=> ( P @ ( image_a_c_d_set_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_319_all__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ! [B5: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ ( image_5710119992958135237_set_a @ F @ A3 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_c_d_set_a] :
( ( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A3 ) )
=> ( P @ ( image_5710119992958135237_set_a @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_320_ex__finite__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A3 )
& ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_321_ex__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ ( image_c_d_set_a_a @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A3 )
& ( P @ ( image_c_d_set_a_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_322_ex__finite__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ? [B5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ ( image_a_c_d_set_a @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ord_less_eq_set_a @ B5 @ A3 )
& ( P @ ( image_a_c_d_set_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_323_ex__finite__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ? [B5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B5 )
& ( ord_le5982164083705284911_set_a @ B5 @ A3 )
& ( P @ ( image_5710119992958135237_set_a @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_324_finite__subset__image,axiom,
! [B4: set_a,F: a > a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
& ( finite_finite_a @ C5 )
& ( B4
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_325_finite__subset__image,axiom,
! [B4: set_a,F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ? [C5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C5 @ A3 )
& ( finite3330819693523053784_set_a @ C5 )
& ( B4
= ( image_c_d_set_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_326_finite__subset__image,axiom,
! [B4: set_c_d_set_a,F: a > ( c > d ) > set_a,A3: set_a] :
( ( finite3330819693523053784_set_a @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
& ( finite_finite_a @ C5 )
& ( B4
= ( image_a_c_d_set_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_327_finite__subset__image,axiom,
! [B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ? [C5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C5 @ A3 )
& ( finite3330819693523053784_set_a @ C5 )
& ( B4
= ( image_5710119992958135237_set_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_328_semilattice__inf__class_Oinf_Oidem,axiom,
! [A: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A @ A )
= A ) ).
% semilattice_inf_class.inf.idem
thf(fact_329_semilattice__inf__class_Oinf__idem,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ X )
= X ) ).
% semilattice_inf_class.inf_idem
thf(fact_330_semilattice__inf__class_Oinf_Oleft__idem,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A @ ( inf_in754637537901350525_set_a @ A @ B ) )
= ( inf_in754637537901350525_set_a @ A @ B ) ) ).
% semilattice_inf_class.inf.left_idem
thf(fact_331_semilattice__inf__class_Oinf__left__idem,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ X @ Y2 ) )
= ( inf_in754637537901350525_set_a @ X @ Y2 ) ) ).
% semilattice_inf_class.inf_left_idem
thf(fact_332_semilattice__inf__class_Oinf_Oright__idem,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ B )
= ( inf_in754637537901350525_set_a @ A @ B ) ) ).
% semilattice_inf_class.inf.right_idem
thf(fact_333_semilattice__inf__class_Oinf__right__idem,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ Y2 )
= ( inf_in754637537901350525_set_a @ X @ Y2 ) ) ).
% semilattice_inf_class.inf_right_idem
thf(fact_334_lattice__class_Oinf__sup__aci_I4_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ X @ Y2 ) )
= ( inf_in754637537901350525_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_aci(4)
thf(fact_335_lattice__class_Oinf__sup__aci_I3_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( inf_in754637537901350525_set_a @ Y2 @ ( inf_in754637537901350525_set_a @ X @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(3)
thf(fact_336_lattice__class_Oinf__sup__aci_I2_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ Z2 )
= ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(2)
thf(fact_337_lattice__class_Oinf__sup__aci_I1_J,axiom,
( inf_in754637537901350525_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] : ( inf_in754637537901350525_set_a @ Y3 @ X2 ) ) ) ).
% lattice_class.inf_sup_aci(1)
thf(fact_338_semilattice__inf__class_Oinf_Oassoc,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ C )
= ( inf_in754637537901350525_set_a @ A @ ( inf_in754637537901350525_set_a @ B @ C ) ) ) ).
% semilattice_inf_class.inf.assoc
thf(fact_339_semilattice__inf__class_Oinf__assoc,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ Z2 )
= ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_inf_class.inf_assoc
thf(fact_340_semilattice__inf__class_Oinf_Ocommute,axiom,
( inf_in754637537901350525_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] : ( inf_in754637537901350525_set_a @ B2 @ A2 ) ) ) ).
% semilattice_inf_class.inf.commute
thf(fact_341_semilattice__inf__class_Oinf__commute,axiom,
( inf_in754637537901350525_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] : ( inf_in754637537901350525_set_a @ Y3 @ X2 ) ) ) ).
% semilattice_inf_class.inf_commute
thf(fact_342_semilattice__inf__class_Oinf_Oleft__commute,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ B @ ( inf_in754637537901350525_set_a @ A @ C ) )
= ( inf_in754637537901350525_set_a @ A @ ( inf_in754637537901350525_set_a @ B @ C ) ) ) ).
% semilattice_inf_class.inf.left_commute
thf(fact_343_semilattice__inf__class_Oinf__left__commute,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( inf_in754637537901350525_set_a @ Y2 @ ( inf_in754637537901350525_set_a @ X @ Z2 ) ) ) ).
% semilattice_inf_class.inf_left_commute
thf(fact_344_monotone__onD,axiom,
! [A3: set_c_d_set_a,Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ Orda @ Ordb @ F )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( ( member_c_d_set_a @ Y2 @ A3 )
=> ( ( Orda @ X @ Y2 )
=> ( Ordb @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_345_monotone__onI,axiom,
! [A3: set_c_d_set_a,Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( ( member_c_d_set_a @ Y4 @ A3 )
=> ( ( Orda @ X3 @ Y4 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y4 ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A3 @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_346_monotone__on__def,axiom,
( monoto2937423850181994535_set_a
= ( ^ [A5: set_c_d_set_a,Orda2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A5 )
=> ! [Y3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y3 @ A5 )
=> ( ( Orda2 @ X2 @ Y3 )
=> ( Ordb2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_347_order__class_Ofinite__has__maximal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal2
thf(fact_348_order__class_Ofinite__has__maximal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( ord_le8464990428230162895_set_a @ A @ X3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal2
thf(fact_349_order__class_Ofinite__has__maximal2,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ A @ A3 )
=> ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A3 )
& ( ord_le5982164083705284911_set_a @ A @ X3 )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal2
thf(fact_350_order__class_Ofinite__has__minimal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
thf(fact_351_order__class_Ofinite__has__minimal2,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ( ord_le8464990428230162895_set_a @ X3 @ A )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
thf(fact_352_order__class_Ofinite__has__minimal2,axiom,
! [A3: set_set_c_d_set_a,A: set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( member_set_c_d_set_a @ A @ A3 )
=> ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A3 )
& ( ord_le5982164083705284911_set_a @ X3 @ A )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal2
thf(fact_353_semilattice__inf__class_Oinf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI2
thf(fact_354_semilattice__inf__class_Oinf_OcoboundedI2,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ C )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI2
thf(fact_355_semilattice__inf__class_Oinf_OcoboundedI2,axiom,
! [B: set_c_d_set_a,C: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ C )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI2
thf(fact_356_semilattice__inf__class_Oinf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI1
thf(fact_357_semilattice__inf__class_Oinf_OcoboundedI1,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ C )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI1
thf(fact_358_semilattice__inf__class_Oinf_OcoboundedI1,axiom,
! [A: set_c_d_set_a,C: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ C )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ C ) ) ).
% semilattice_inf_class.inf.coboundedI1
thf(fact_359_semilattice__inf__class_Oinf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff2
thf(fact_360_semilattice__inf__class_Oinf_Oabsorb__iff2,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ A2 @ B2 )
= B2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff2
thf(fact_361_semilattice__inf__class_Oinf_Oabsorb__iff2,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A2 @ B2 )
= B2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff2
thf(fact_362_semilattice__inf__class_Oinf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff1
thf(fact_363_semilattice__inf__class_Oinf_Oabsorb__iff1,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff1
thf(fact_364_semilattice__inf__class_Oinf_Oabsorb__iff1,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_inf_class.inf.absorb_iff1
thf(fact_365_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_366_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_367_semilattice__inf__class_Oinf_Ocobounded2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ B ) ).
% semilattice_inf_class.inf.cobounded2
thf(fact_368_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_369_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_370_semilattice__inf__class_Oinf_Ocobounded1,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ A ) ).
% semilattice_inf_class.inf.cobounded1
thf(fact_371_semilattice__inf__class_Oinf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_inf_class.inf.order_iff
thf(fact_372_semilattice__inf__class_Oinf_Oorder__iff,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( A2
= ( inf_inf_c_d_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_inf_class.inf.order_iff
thf(fact_373_semilattice__inf__class_Oinf_Oorder__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] :
( A2
= ( inf_in754637537901350525_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_inf_class.inf.order_iff
thf(fact_374_semilattice__inf__class_Oinf__greatest,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y2 @ Z2 ) ) ) ) ).
% semilattice_inf_class.inf_greatest
thf(fact_375_semilattice__inf__class_Oinf__greatest,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ( ord_le8464990428230162895_set_a @ X @ Z2 )
=> ( ord_le8464990428230162895_set_a @ X @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) ) ) ) ).
% semilattice_inf_class.inf_greatest
thf(fact_376_semilattice__inf__class_Oinf__greatest,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ( ord_le5982164083705284911_set_a @ X @ Z2 )
=> ( ord_le5982164083705284911_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) ) ) ) ).
% semilattice_inf_class.inf_greatest
thf(fact_377_semilattice__inf__class_Oinf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% semilattice_inf_class.inf.boundedI
thf(fact_378_semilattice__inf__class_Oinf_OboundedI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( ord_le8464990428230162895_set_a @ A @ C )
=> ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C ) ) ) ) ).
% semilattice_inf_class.inf.boundedI
thf(fact_379_semilattice__inf__class_Oinf_OboundedI,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( ord_le5982164083705284911_set_a @ A @ C )
=> ( ord_le5982164083705284911_set_a @ A @ ( inf_in754637537901350525_set_a @ B @ C ) ) ) ) ).
% semilattice_inf_class.inf.boundedI
thf(fact_380_semilattice__inf__class_Oinf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.boundedE
thf(fact_381_semilattice__inf__class_Oinf_OboundedE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ ( inf_inf_c_d_set_a @ B @ C ) )
=> ~ ( ( ord_le8464990428230162895_set_a @ A @ B )
=> ~ ( ord_le8464990428230162895_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.boundedE
thf(fact_382_semilattice__inf__class_Oinf_OboundedE,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ ( inf_in754637537901350525_set_a @ B @ C ) )
=> ~ ( ( ord_le5982164083705284911_set_a @ A @ B )
=> ~ ( ord_le5982164083705284911_set_a @ A @ C ) ) ) ).
% semilattice_inf_class.inf.boundedE
thf(fact_383_semilattice__inf__class_Oinf__absorb2,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( inf_inf_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_inf_class.inf_absorb2
thf(fact_384_semilattice__inf__class_Oinf__absorb2,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y2 @ X )
=> ( ( inf_inf_c_d_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_inf_class.inf_absorb2
thf(fact_385_semilattice__inf__class_Oinf__absorb2,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y2 @ X )
=> ( ( inf_in754637537901350525_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_inf_class.inf_absorb2
thf(fact_386_semilattice__inf__class_Oinf__absorb1,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( inf_inf_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_inf_class.inf_absorb1
thf(fact_387_semilattice__inf__class_Oinf__absorb1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ( inf_inf_c_d_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_inf_class.inf_absorb1
thf(fact_388_semilattice__inf__class_Oinf__absorb1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ( inf_in754637537901350525_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_inf_class.inf_absorb1
thf(fact_389_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_390_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_391_semilattice__inf__class_Oinf_Oabsorb2,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( inf_in754637537901350525_set_a @ A @ B )
= B ) ) ).
% semilattice_inf_class.inf.absorb2
thf(fact_392_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_393_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_394_semilattice__inf__class_Oinf_Oabsorb1,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( inf_in754637537901350525_set_a @ A @ B )
= A ) ) ).
% semilattice_inf_class.inf.absorb1
thf(fact_395_semilattice__inf__class_Ole__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% semilattice_inf_class.le_iff_inf
thf(fact_396_semilattice__inf__class_Ole__iff__inf,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% semilattice_inf_class.le_iff_inf
thf(fact_397_semilattice__inf__class_Ole__iff__inf,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X2 @ Y3 )
= X2 ) ) ) ).
% semilattice_inf_class.le_iff_inf
thf(fact_398_semilattice__inf__class_Oinf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ( ord_less_eq_set_a @ X3 @ Z4 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_inf_class.inf_unique
thf(fact_399_semilattice__inf__class_Oinf__unique,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ( ord_le8464990428230162895_set_a @ X3 @ Z4 )
=> ( ord_le8464990428230162895_set_a @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_c_d_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_inf_class.inf_unique
thf(fact_400_semilattice__inf__class_Oinf__unique,axiom,
! [F: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a,Z4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ( ord_le5982164083705284911_set_a @ X3 @ Z4 )
=> ( ord_le5982164083705284911_set_a @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_in754637537901350525_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_inf_class.inf_unique
thf(fact_401_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_402_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_403_semilattice__inf__class_Oinf_OorderI,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( A
= ( inf_in754637537901350525_set_a @ A @ B ) )
=> ( ord_le5982164083705284911_set_a @ A @ B ) ) ).
% semilattice_inf_class.inf.orderI
thf(fact_404_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_405_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_406_semilattice__inf__class_Oinf_OorderE,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( A
= ( inf_in754637537901350525_set_a @ A @ B ) ) ) ).
% semilattice_inf_class.inf.orderE
thf(fact_407_semilattice__inf__class_Ole__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI2
thf(fact_408_semilattice__inf__class_Ole__infI2,axiom,
! [B: ( c > d ) > set_a,X: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ X )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI2
thf(fact_409_semilattice__inf__class_Ole__infI2,axiom,
! [B: set_c_d_set_a,X: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ X )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI2
thf(fact_410_semilattice__inf__class_Ole__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI1
thf(fact_411_semilattice__inf__class_Ole__infI1,axiom,
! [A: ( c > d ) > set_a,X: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ X )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI1
thf(fact_412_semilattice__inf__class_Ole__infI1,axiom,
! [A: set_c_d_set_a,X: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ X )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ X ) ) ).
% semilattice_inf_class.le_infI1
thf(fact_413_semilattice__inf__class_Oinf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% semilattice_inf_class.inf_mono
thf(fact_414_semilattice__inf__class_Oinf__mono,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ C )
=> ( ( ord_le8464990428230162895_set_a @ B @ D2 )
=> ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ A @ B ) @ ( inf_inf_c_d_set_a @ C @ D2 ) ) ) ) ).
% semilattice_inf_class.inf_mono
thf(fact_415_semilattice__inf__class_Oinf__mono,axiom,
! [A: set_c_d_set_a,C: set_c_d_set_a,B: set_c_d_set_a,D2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ C )
=> ( ( ord_le5982164083705284911_set_a @ B @ D2 )
=> ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ A @ B ) @ ( inf_in754637537901350525_set_a @ C @ D2 ) ) ) ) ).
% semilattice_inf_class.inf_mono
thf(fact_416_semilattice__inf__class_Ole__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% semilattice_inf_class.le_infI
thf(fact_417_semilattice__inf__class_Ole__infI,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ A )
=> ( ( ord_le8464990428230162895_set_a @ X @ B )
=> ( ord_le8464990428230162895_set_a @ X @ ( inf_inf_c_d_set_a @ A @ B ) ) ) ) ).
% semilattice_inf_class.le_infI
thf(fact_418_semilattice__inf__class_Ole__infI,axiom,
! [X: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ A )
=> ( ( ord_le5982164083705284911_set_a @ X @ B )
=> ( ord_le5982164083705284911_set_a @ X @ ( inf_in754637537901350525_set_a @ A @ B ) ) ) ) ).
% semilattice_inf_class.le_infI
thf(fact_419_semilattice__inf__class_Ole__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% semilattice_inf_class.le_infE
thf(fact_420_semilattice__inf__class_Ole__infE,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ ( inf_inf_c_d_set_a @ A @ B ) )
=> ~ ( ( ord_le8464990428230162895_set_a @ X @ A )
=> ~ ( ord_le8464990428230162895_set_a @ X @ B ) ) ) ).
% semilattice_inf_class.le_infE
thf(fact_421_semilattice__inf__class_Ole__infE,axiom,
! [X: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ ( inf_in754637537901350525_set_a @ A @ B ) )
=> ~ ( ( ord_le5982164083705284911_set_a @ X @ A )
=> ~ ( ord_le5982164083705284911_set_a @ X @ B ) ) ) ).
% semilattice_inf_class.le_infE
thf(fact_422_semilattice__inf__class_Oinf__le2,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% semilattice_inf_class.inf_le2
thf(fact_423_semilattice__inf__class_Oinf__le2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ Y2 ) ).
% semilattice_inf_class.inf_le2
thf(fact_424_semilattice__inf__class_Oinf__le2,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ Y2 ) ).
% semilattice_inf_class.inf_le2
thf(fact_425_semilattice__inf__class_Oinf__le1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% semilattice_inf_class.inf_le1
thf(fact_426_semilattice__inf__class_Oinf__le1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ X ) ).
% semilattice_inf_class.inf_le1
thf(fact_427_semilattice__inf__class_Oinf__le1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ X ) ).
% semilattice_inf_class.inf_le1
thf(fact_428_lattice__class_Oinf__sup__ord_I1_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ X ) ).
% lattice_class.inf_sup_ord(1)
thf(fact_429_lattice__class_Oinf__sup__ord_I1_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ X ) ).
% lattice_class.inf_sup_ord(1)
thf(fact_430_lattice__class_Oinf__sup__ord_I1_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ X ) ).
% lattice_class.inf_sup_ord(1)
thf(fact_431_lattice__class_Oinf__sup__ord_I2_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y2 ) @ Y2 ) ).
% lattice_class.inf_sup_ord(2)
thf(fact_432_lattice__class_Oinf__sup__ord_I2_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ Y2 ) ).
% lattice_class.inf_sup_ord(2)
thf(fact_433_lattice__class_Oinf__sup__ord_I2_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ Y2 ) ).
% lattice_class.inf_sup_ord(2)
thf(fact_434_all__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A3 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_435_all__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > a,A3: set_c_d_set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B5 @ A3 )
=> ( P @ ( image_c_d_set_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_436_all__subset__image,axiom,
! [F: a > ( c > d ) > set_a,A3: set_a,P: set_c_d_set_a > $o] :
( ( ! [B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B5 @ ( image_a_c_d_set_a @ F @ A3 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
=> ( P @ ( image_a_c_d_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_437_all__subset__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( ! [B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B5 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B5 @ A3 )
=> ( P @ ( image_5710119992958135237_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_438_finite__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( finite_finite_a @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_439_finite__subset,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( finite3330819693523053784_set_a @ B4 )
=> ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% finite_subset
thf(fact_440_infinite__super,axiom,
! [S4: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S4 @ T2 )
=> ( ~ ( finite_finite_a @ S4 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_441_infinite__super,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ S4 @ T2 )
=> ( ~ ( finite3330819693523053784_set_a @ S4 )
=> ~ ( finite3330819693523053784_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_442_rev__finite__subset,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_443_rev__finite__subset,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_444_ord__class_Omono__onI,axiom,
! [A3: set_set_a,F: set_a > set_a] :
( ! [R3: set_a,S: set_a] :
( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_445_ord__class_Omono__onI,axiom,
! [A3: set_set_a,F: set_a > ( c > d ) > set_a] :
( ! [R3: set_a,S: set_a] :
( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto2748056057003999288_set_a @ A3 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_446_ord__class_Omono__onI,axiom,
! [A3: set_set_a,F: set_a > set_c_d_set_a] :
( ! [R3: set_a,S: set_a] :
( ( member_set_a @ R3 @ A3 )
=> ( ( member_set_a @ S @ A3 )
=> ( ( ord_less_eq_set_a @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto7894950695950633880_set_a @ A3 @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_447_ord__class_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_448_ord__class_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_449_ord__class_Omono__onI,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6642458133393520519_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_450_ord__class_Omono__onI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_a] :
( ! [R3: set_c_d_set_a,S: set_c_d_set_a] :
( ( member_set_c_d_set_a @ R3 @ A3 )
=> ( ( member_set_c_d_set_a @ S @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto9091215303422693110_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_451_ord__class_Omono__onI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a] :
( ! [R3: set_c_d_set_a,S: set_c_d_set_a] :
( ( member_set_c_d_set_a @ R3 @ A3 )
=> ( ( member_set_c_d_set_a @ S @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto5673664640695304391_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_452_ord__class_Omono__onI,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ! [R3: set_c_d_set_a,S: set_c_d_set_a] :
( ( member_set_c_d_set_a @ R3 @ A3 )
=> ( ( member_set_c_d_set_a @ S @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto4733996707696316455_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% ord_class.mono_onI
thf(fact_453_ord__class_Omono__onD,axiom,
! [A3: set_set_a,F: set_a > set_a,R4: set_a,S3: set_a] :
( ( monoto7172710143293369831_set_a @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_454_ord__class_Omono__onD,axiom,
! [A3: set_set_a,F: set_a > ( c > d ) > set_a,R4: set_a,S3: set_a] :
( ( monoto2748056057003999288_set_a @ A3 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_455_ord__class_Omono__onD,axiom,
! [A3: set_set_a,F: set_a > set_c_d_set_a,R4: set_a,S3: set_a] :
( ( monoto7894950695950633880_set_a @ A3 @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_set_a @ R4 @ A3 )
=> ( ( member_set_a @ S3 @ A3 )
=> ( ( ord_less_eq_set_a @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_456_ord__class_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_457_ord__class_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_458_ord__class_Omono__onD,axiom,
! [A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > set_c_d_set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_459_ord__class_Omono__onD,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_a,R4: set_c_d_set_a,S3: set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_set_c_d_set_a @ R4 @ A3 )
=> ( ( member_set_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_460_ord__class_Omono__onD,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > ( c > d ) > set_a,R4: set_c_d_set_a,S3: set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_set_c_d_set_a @ R4 @ A3 )
=> ( ( member_set_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_461_ord__class_Omono__onD,axiom,
! [A3: set_set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a,R4: set_c_d_set_a,S3: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_set_c_d_set_a @ R4 @ A3 )
=> ( ( member_set_c_d_set_a @ S3 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord_class.mono_onD
thf(fact_462_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a] :
( ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
= ( ! [R5: a,S2: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_463_ord_Omono__on__def,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
= ( ! [R5: ( c > d ) > set_a,S2: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ R5 @ A3 )
& ( member_c_d_set_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_less_eq_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_464_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > ( c > d ) > set_a] :
( ( monoto2502030104860647832_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
= ( ! [R5: a,S2: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_465_ord_Omono__on__def,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
= ( ! [R5: ( c > d ) > set_a,S2: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ R5 @ A3 )
& ( member_c_d_set_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_le8464990428230162895_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_466_ord_Omono__on__def,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_c_d_set_a] :
( ( monoto4999900198720154872_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
= ( ! [R5: a,S2: a] :
( ( ( member_a @ R5 @ A3 )
& ( member_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_467_ord_Omono__on__def,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
= ( ! [R5: ( c > d ) > set_a,S2: ( c > d ) > set_a] :
( ( ( member_c_d_set_a @ R5 @ A3 )
& ( member_c_d_set_a @ S2 @ A3 )
& ( Less_eq2 @ R5 @ S2 ) )
=> ( ord_le5982164083705284911_set_a @ ( F @ R5 ) @ ( F @ S2 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_468_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a] :
( ! [R3: a,S: a] :
( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_469_ord_Omono__onI,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_less_eq_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6316088450447394390_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_470_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > ( c > d ) > set_a] :
( ! [R3: a,S: a] :
( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto2502030104860647832_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_471_ord_Omono__onI,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_le8464990428230162895_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto2937423850181994535_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_472_ord_Omono__onI,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_c_d_set_a] :
( ! [R3: a,S: a] :
( ( member_a @ R3 @ A3 )
=> ( ( member_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto4999900198720154872_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_473_ord_Omono__onI,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ! [R3: ( c > d ) > set_a,S: ( c > d ) > set_a] :
( ( member_c_d_set_a @ R3 @ A3 )
=> ( ( member_c_d_set_a @ S @ A3 )
=> ( ( Less_eq2 @ R3 @ S )
=> ( ord_le5982164083705284911_set_a @ ( F @ R3 ) @ ( F @ S ) ) ) ) )
=> ( monoto6642458133393520519_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F ) ) ).
% ord.mono_onI
thf(fact_474_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_a,R4: a,S3: a] :
( ( monotone_on_a_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_475_ord_Omono__onD,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ A3 @ Less_eq2 @ ord_less_eq_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_less_eq_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_476_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > ( c > d ) > set_a,R4: a,S3: a] :
( ( monoto2502030104860647832_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_477_ord_Omono__onD,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ Less_eq2 @ ord_le8464990428230162895_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_478_ord_Omono__onD,axiom,
! [A3: set_a,Less_eq2: a > a > $o,F: a > set_c_d_set_a,R4: a,S3: a] :
( ( monoto4999900198720154872_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_a @ R4 @ A3 )
=> ( ( member_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_479_ord_Omono__onD,axiom,
! [A3: set_c_d_set_a,Less_eq2: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > set_c_d_set_a,R4: ( c > d ) > set_a,S3: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ A3 @ Less_eq2 @ ord_le5982164083705284911_set_a @ F )
=> ( ( member_c_d_set_a @ R4 @ A3 )
=> ( ( member_c_d_set_a @ S3 @ A3 )
=> ( ( Less_eq2 @ R4 @ S3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ R4 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_480_monotone__on__subset,axiom,
! [A3: set_c_d_set_a,Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ A3 @ Orda @ Ordb @ F )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( monoto2937423850181994535_set_a @ B4 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_481_finite__surj,axiom,
! [A3: set_c_d_set_a,B4: set_a,F: ( ( c > d ) > set_a ) > a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ord_less_eq_set_a @ B4 @ ( image_c_d_set_a_a @ F @ A3 ) )
=> ( finite_finite_a @ B4 ) ) ) ).
% finite_surj
thf(fact_482_finite__surj,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ ( image_5710119992958135237_set_a @ F @ A3 ) )
=> ( finite3330819693523053784_set_a @ B4 ) ) ) ).
% finite_surj
thf(fact_483_local_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A3: set_c_d_set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( monoto6316088450447394390_set_a @ A3 @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F ) ) ).
% local.mono_imp_mono_on
thf(fact_484_local_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto2937423850181994535_set_a @ A3 @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% local.mono_imp_mono_on
thf(fact_485_local_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,A3: set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( monoto6642458133393520519_set_a @ A3 @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% local.mono_imp_mono_on
thf(fact_486_local_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F ) ) ).
% local.monoI
thf(fact_487_local_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% local.monoI
thf(fact_488_local_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% local.monoI
thf(fact_489_local_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoE
thf(fact_490_local_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoE
thf(fact_491_local_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoE
thf(fact_492_local_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoD
thf(fact_493_local_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoD
thf(fact_494_local_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% local.monoD
thf(fact_495_local_Ofinite__has__maximal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( smaller_interp_c_d_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_maximal
thf(fact_496_local_Ofinite__has__minimal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( smaller_interp_c_d_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% local.finite_has_minimal
thf(fact_497_preordering__bdd_Obdd_Ocong,axiom,
condit8154225043310684324_set_a = condit8154225043310684324_set_a ).
% preordering_bdd.bdd.cong
thf(fact_498_local_OSup__subset__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A3 ) @ ( sup_c_d_a @ B4 ) ) ) ).
% local.Sup_subset_mono
thf(fact_499_empty__interp__def,axiom,
( empty_interp_c_d_a
= ( ^ [S2: c > d] : bot_bot_set_a ) ) ).
% empty_interp_def
thf(fact_500_test__axiom__sup,axiom,
! [A3: set_c_d_set_a,Z2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ X3 @ Z2 ) )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A3 ) @ Z2 ) ) ).
% test_axiom_sup
thf(fact_501_local_OSup__upper2,axiom,
! [U: ( c > d ) > set_a,A3: set_c_d_set_a,V: ( c > d ) > set_a] :
( ( member_c_d_set_a @ U @ A3 )
=> ( ( smaller_interp_c_d_a @ V @ U )
=> ( smaller_interp_c_d_a @ V @ ( sup_c_d_a @ A3 ) ) ) ) ).
% local.Sup_upper2
thf(fact_502_local_OSup__upper,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( smaller_interp_c_d_a @ X @ ( sup_c_d_a @ A3 ) ) ) ).
% local.Sup_upper
thf(fact_503_local_OSup__mono,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ! [A4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A4 @ A3 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ B4 )
& ( smaller_interp_c_d_a @ A4 @ X4 ) ) )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a @ A3 ) @ ( sup_c_d_a @ B4 ) ) ) ).
% local.Sup_mono
thf(fact_504_local_OSup__le__iff,axiom,
! [A3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( sup_c_d_a @ A3 ) @ B )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ X2 @ B ) ) ) ) ).
% local.Sup_le_iff
thf(fact_505_local_OSup__eqI,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ! [Y4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y4 @ A3 )
=> ( smaller_interp_c_d_a @ Y4 @ X ) )
=> ( ! [Y4: ( c > d ) > set_a] :
( ! [Z5: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Z5 @ A3 )
=> ( smaller_interp_c_d_a @ Z5 @ Y4 ) )
=> ( smaller_interp_c_d_a @ X @ Y4 ) )
=> ( ( sup_c_d_a @ A3 )
= X ) ) ) ).
% local.Sup_eqI
thf(fact_506_local_OSUP__cong,axiom,
! [A3: set_a,B4: set_a,C4: a > ( c > d ) > set_a,D: a > ( c > d ) > set_a] :
( ( A3 = B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D @ X3 ) ) )
=> ( ( sup_c_d_a @ ( image_a_c_d_set_a @ C4 @ A3 ) )
= ( sup_c_d_a @ ( image_a_c_d_set_a @ D @ B4 ) ) ) ) ) ).
% local.SUP_cong
thf(fact_507_local_OSUP__cong,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: ( ( c > d ) > set_a ) > ( c > d ) > set_a,D: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( A3 = B4 )
=> ( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D @ X3 ) ) )
=> ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ C4 @ A3 ) )
= ( sup_c_d_a @ ( image_5710119992958135237_set_a @ D @ B4 ) ) ) ) ) ).
% local.SUP_cong
thf(fact_508_mono__same,axiom,
( monotonic_c_d_a
= ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a ) ) ).
% mono_same
thf(fact_509_local_OSUP__eq__const,axiom,
! [I: set_a,F: a > ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= X ) )
=> ( ( sup_c_d_a @ ( image_a_c_d_set_a @ F @ I ) )
= X ) ) ) ).
% local.SUP_eq_const
thf(fact_510_local_OSUP__eq__const,axiom,
! [I: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( I != bot_bo738396921950161403_set_a )
=> ( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ I )
=> ( ( F @ I2 )
= X ) )
=> ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ F @ I ) )
= X ) ) ) ).
% local.SUP_eq_const
thf(fact_511_local_OSUP__eq,axiom,
! [A3: set_a,B4: set_a,F: a > ( c > d ) > set_a,G2: a > ( c > d ) > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A3 )
=> ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( smaller_interp_c_d_a @ ( F @ I2 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( smaller_interp_c_d_a @ ( G2 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( sup_c_d_a @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( sup_c_d_a @ ( image_a_c_d_set_a @ G2 @ B4 ) ) ) ) ) ).
% local.SUP_eq
thf(fact_512_local_OSUP__eq,axiom,
! [A3: set_a,B4: set_c_d_set_a,F: a > ( c > d ) > set_a,G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A3 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ B4 )
& ( smaller_interp_c_d_a @ ( F @ I2 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( smaller_interp_c_d_a @ ( G2 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( sup_c_d_a @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( sup_c_d_a @ ( image_5710119992958135237_set_a @ G2 @ B4 ) ) ) ) ) ).
% local.SUP_eq
thf(fact_513_local_OSUP__eq,axiom,
! [A3: set_c_d_set_a,B4: set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G2: a > ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A3 )
=> ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( smaller_interp_c_d_a @ ( F @ I2 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A3 )
& ( smaller_interp_c_d_a @ ( G2 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( sup_c_d_a @ ( image_a_c_d_set_a @ G2 @ B4 ) ) ) ) ) ).
% local.SUP_eq
thf(fact_514_local_OSUP__eq,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A3 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ B4 )
& ( smaller_interp_c_d_a @ ( F @ I2 ) @ ( G2 @ X4 ) ) ) )
=> ( ! [J: ( c > d ) > set_a] :
( ( member_c_d_set_a @ J @ B4 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A3 )
& ( smaller_interp_c_d_a @ ( G2 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( sup_c_d_a @ ( image_5710119992958135237_set_a @ G2 @ B4 ) ) ) ) ) ).
% local.SUP_eq
thf(fact_515_local_Oless__eq__Sup,axiom,
! [A3: set_c_d_set_a,U: ( c > d ) > set_a] :
( ! [V2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ V2 @ A3 )
=> ( smaller_interp_c_d_a @ U @ V2 ) )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ U @ ( sup_c_d_a @ A3 ) ) ) ) ).
% local.less_eq_Sup
thf(fact_516_local_OSUP__eq__iff,axiom,
! [I: set_a,C: ( c > d ) > set_a,F: a > ( c > d ) > set_a] :
( ( I != bot_bot_set_a )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( smaller_interp_c_d_a @ C @ ( F @ I2 ) ) )
=> ( ( ( sup_c_d_a @ ( image_a_c_d_set_a @ F @ I ) )
= C )
= ( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% local.SUP_eq_iff
thf(fact_517_local_OSUP__eq__iff,axiom,
! [I: set_c_d_set_a,C: ( 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 )
=> ( smaller_interp_c_d_a @ C @ ( F @ I2 ) ) )
=> ( ( ( sup_c_d_a @ ( image_5710119992958135237_set_a @ F @ I ) )
= C )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% local.SUP_eq_iff
thf(fact_518_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_519_empty__iff,axiom,
! [C: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ C @ bot_bo738396921950161403_set_a ) ).
% empty_iff
thf(fact_520_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_521_all__not__in__conv,axiom,
! [A3: set_c_d_set_a] :
( ( ! [X2: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ X2 @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% all_not_in_conv
thf(fact_522_Collect__empty__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( ( collect_c_d_set_a @ P )
= bot_bo738396921950161403_set_a )
= ( ! [X2: ( c > d ) > set_a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_523_empty__Collect__eq,axiom,
! [P: ( ( c > d ) > set_a ) > $o] :
( ( bot_bo738396921950161403_set_a
= ( collect_c_d_set_a @ P ) )
= ( ! [X2: ( c > d ) > set_a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_524_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_525_UNIV__I,axiom,
! [X: ( c > d ) > set_a] : ( member_c_d_set_a @ X @ top_to4267977599310771935_set_a ) ).
% UNIV_I
thf(fact_526_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_527_empty__is__image,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% empty_is_image
thf(fact_528_image__is__empty,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( image_5710119992958135237_set_a @ F @ A3 )
= bot_bo738396921950161403_set_a )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% image_is_empty
thf(fact_529_finite__Plus__UNIV__iff,axiom,
( ( finite5989733633321134460_set_a @ top_to279427854467338187_set_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_530_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_531_empty__subsetI,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ bot_bo738396921950161403_set_a @ A3 ) ).
% empty_subsetI
thf(fact_532_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_533_subset__empty,axiom,
! [A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ bot_bo738396921950161403_set_a )
= ( A3 = bot_bo738396921950161403_set_a ) ) ).
% subset_empty
thf(fact_534_bounded__lattice__bot__class_Oinf__bot__right,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% bounded_lattice_bot_class.inf_bot_right
thf(fact_535_bounded__lattice__bot__class_Oinf__bot__left,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ X )
= bot_bo738396921950161403_set_a ) ).
% bounded_lattice_bot_class.inf_bot_left
thf(fact_536_bounded__semilattice__inf__top__class_Oinf__top_Oright__neutral,axiom,
! [A: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A @ top_to4267977599310771935_set_a )
= A ) ).
% bounded_semilattice_inf_top_class.inf_top.right_neutral
thf(fact_537_bounded__semilattice__inf__top__class_Oinf__top_Oneutr__eq__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( top_to4267977599310771935_set_a
= ( inf_in754637537901350525_set_a @ A @ B ) )
= ( ( A = top_to4267977599310771935_set_a )
& ( B = top_to4267977599310771935_set_a ) ) ) ).
% bounded_semilattice_inf_top_class.inf_top.neutr_eq_iff
thf(fact_538_bounded__semilattice__inf__top__class_Oinf__top_Oleft__neutral,axiom,
! [A: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ top_to4267977599310771935_set_a @ A )
= A ) ).
% bounded_semilattice_inf_top_class.inf_top.left_neutral
thf(fact_539_bounded__semilattice__inf__top__class_Oinf__top_Oeq__neutr__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A @ B )
= top_to4267977599310771935_set_a )
= ( ( A = top_to4267977599310771935_set_a )
& ( B = top_to4267977599310771935_set_a ) ) ) ).
% bounded_semilattice_inf_top_class.inf_top.eq_neutr_iff
thf(fact_540_bounded__semilattice__inf__top__class_Otop__eq__inf__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( top_to4267977599310771935_set_a
= ( inf_in754637537901350525_set_a @ X @ Y2 ) )
= ( ( X = top_to4267977599310771935_set_a )
& ( Y2 = top_to4267977599310771935_set_a ) ) ) ).
% bounded_semilattice_inf_top_class.top_eq_inf_iff
thf(fact_541_bounded__semilattice__inf__top__class_Oinf__eq__top__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ X @ Y2 )
= top_to4267977599310771935_set_a )
= ( ( X = top_to4267977599310771935_set_a )
& ( Y2 = top_to4267977599310771935_set_a ) ) ) ).
% bounded_semilattice_inf_top_class.inf_eq_top_iff
thf(fact_542_bounded__semilattice__inf__top__class_Oinf__top__right,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ top_to4267977599310771935_set_a )
= X ) ).
% bounded_semilattice_inf_top_class.inf_top_right
thf(fact_543_bounded__semilattice__inf__top__class_Oinf__top__left,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ top_to4267977599310771935_set_a @ X )
= X ) ).
% bounded_semilattice_inf_top_class.inf_top_left
thf(fact_544_Int__UNIV,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= top_to4267977599310771935_set_a )
= ( ( A3 = top_to4267977599310771935_set_a )
& ( B4 = top_to4267977599310771935_set_a ) ) ) ).
% Int_UNIV
thf(fact_545_local_Obdd__below__empty,axiom,
condit9007271454129256903_set_a @ smaller_interp_c_d_a @ bot_bo738396921950161403_set_a ).
% local.bdd_below_empty
thf(fact_546_local_Obdd__above__empty,axiom,
condit6926915774301931483_set_a @ smaller_interp_c_d_a @ bot_bo738396921950161403_set_a ).
% local.bdd_above_empty
thf(fact_547_local_Ogfp__upperbound,axiom,
! [X5: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X5 @ ( F @ X5 ) )
=> ( smaller_interp_c_d_a @ X5 @ ( comple4132920576971123013_set_a @ sup_c_d_a @ smaller_interp_c_d_a @ F ) ) ) ).
% local.gfp_upperbound
thf(fact_548_local_Ogfp__least,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X5: ( c > d ) > set_a] :
( ! [U2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ U2 @ ( F @ U2 ) )
=> ( smaller_interp_c_d_a @ U2 @ X5 ) )
=> ( smaller_interp_c_d_a @ ( comple4132920576971123013_set_a @ sup_c_d_a @ smaller_interp_c_d_a @ F ) @ X5 ) ) ).
% local.gfp_least
thf(fact_549_finite__Prod__UNIV,axiom,
( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( finite2397556900044337168_set_a @ top_to3895570120271872023_set_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_550_finite__prod,axiom,
( ( finite2397556900044337168_set_a @ top_to3895570120271872023_set_a )
= ( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
& ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ) ).
% finite_prod
thf(fact_551_Finite__Set_Ofinite__set,axiom,
( ( finite457288119118821432_set_a @ top_to5717711934741766719_set_a )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_552_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_553_emptyE,axiom,
! [A: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ).
% emptyE
thf(fact_554_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_555_equals0D,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( A3 = bot_bo738396921950161403_set_a )
=> ~ ( member_c_d_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_556_equals0I,axiom,
! [A3: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_557_equals0I,axiom,
! [A3: set_c_d_set_a] :
( ! [Y4: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ Y4 @ A3 )
=> ( A3 = bot_bo738396921950161403_set_a ) ) ).
% equals0I
thf(fact_558_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_559_UNIV__eq__I,axiom,
! [A3: set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ A3 )
=> ( top_to4267977599310771935_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_560_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_561_ex__in__conv,axiom,
! [A3: set_c_d_set_a] :
( ( ? [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ A3 ) )
= ( A3 != bot_bo738396921950161403_set_a ) ) ).
% ex_in_conv
thf(fact_562_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_563_UNIV__witness,axiom,
? [X3: ( c > d ) > set_a] : ( member_c_d_set_a @ X3 @ top_to4267977599310771935_set_a ) ).
% UNIV_witness
thf(fact_564_empty__not__UNIV,axiom,
bot_bo738396921950161403_set_a != top_to4267977599310771935_set_a ).
% empty_not_UNIV
thf(fact_565_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_566_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_567_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_568_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_569_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_570_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_571_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_572_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_573_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_574_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_575_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_576_order__top__class_Otop_Oextremum__uniqueI,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ top_to4267977599310771935_set_a @ A )
=> ( A = top_to4267977599310771935_set_a ) ) ).
% order_top_class.top.extremum_uniqueI
thf(fact_577_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_578_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_579_order__top__class_Otop_Oextremum__unique,axiom,
! [A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ top_to4267977599310771935_set_a @ A )
= ( A = top_to4267977599310771935_set_a ) ) ).
% order_top_class.top.extremum_unique
thf(fact_580_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_581_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_582_order__top__class_Otop__greatest,axiom,
! [A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ top_to4267977599310771935_set_a ) ).
% order_top_class.top_greatest
thf(fact_583_finite__fun__UNIVD2,axiom,
( ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( finite_finite_set_a @ top_top_set_set_a ) ) ).
% finite_fun_UNIVD2
thf(fact_584_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > a,X: ( c > d ) > set_a] : ( member_a @ ( F @ X ) @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_585_rangeI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a] : ( member_c_d_set_a @ ( F @ X ) @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) ) ).
% rangeI
thf(fact_586_range__eqI,axiom,
! [B: a,F: ( ( c > d ) > set_a ) > a,X: ( c > d ) > set_a] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_587_range__eqI,axiom,
! [B: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( B
= ( F @ X ) )
=> ( member_c_d_set_a @ B @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) ) ) ).
% range_eqI
thf(fact_588_surj__def,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
= ( ! [Y3: ( c > d ) > set_a] :
? [X2: ( c > d ) > set_a] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_589_surjI,axiom,
! [G2: ( ( c > d ) > set_a ) > ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_5710119992958135237_set_a @ G2 @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ) ).
% surjI
thf(fact_590_surjE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
=> ~ ! [X3: ( c > d ) > set_a] :
( Y2
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_591_surjD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a )
=> ? [X3: ( c > d ) > set_a] :
( Y2
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_592_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A4: a] :
~ ( member_a @ A4 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_593_ex__new__if__finite,axiom,
! [A3: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a )
=> ( ( finite3330819693523053784_set_a @ A3 )
=> ? [A4: ( c > d ) > set_a] :
~ ( member_c_d_set_a @ A4 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_594_infinite__imp__nonempty,axiom,
! [S4: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ S4 )
=> ( S4 != bot_bo738396921950161403_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_595_finite_OemptyI,axiom,
finite3330819693523053784_set_a @ bot_bo738396921950161403_set_a ).
% finite.emptyI
thf(fact_596_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_597_subset__UNIV,axiom,
! [A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ top_to4267977599310771935_set_a ) ).
% subset_UNIV
thf(fact_598_Int__UNIV__left,axiom,
! [B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ top_to4267977599310771935_set_a @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_599_Int__UNIV__right,axiom,
! [A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ top_to4267977599310771935_set_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_600_Int__emptyI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ~ ( member_a @ X3 @ B4 ) )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_601_Int__emptyI,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ~ ( member_c_d_set_a @ X3 @ B4 ) )
=> ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a ) ) ).
% Int_emptyI
thf(fact_602_disjoint__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ~ ( member_a @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_603_disjoint__iff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ~ ( member_c_d_set_a @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_604_Int__empty__left,axiom,
! [B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ B4 )
= bot_bo738396921950161403_set_a ) ).
% Int_empty_left
thf(fact_605_Int__empty__right,axiom,
! [A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% Int_empty_right
thf(fact_606_disjoint__iff__not__equal,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ! [Y3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_607_monotoneI,axiom,
! [Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( Orda @ X3 @ Y4 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_608_monotoneD,axiom,
! [Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y2 )
=> ( Ordb @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% monotoneD
thf(fact_609_monotone__on__empty,axiom,
! [Orda: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,Ordb: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] : ( monoto2937423850181994535_set_a @ bot_bo738396921950161403_set_a @ Orda @ Ordb @ F ) ).
% monotone_on_empty
thf(fact_610_antisymp__equality,axiom,
( antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a
@ ^ [Y: ( c > d ) > set_a,Z: ( c > d ) > set_a] : ( Y = Z ) ) ).
% antisymp_equality
thf(fact_611_antisympI,axiom,
! [R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ X3 )
=> ( X3 = Y4 ) ) )
=> ( antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a @ R ) ) ).
% antisympI
thf(fact_612_antisympD,axiom,
! [R: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a @ R )
=> ( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisympD
thf(fact_613_antisymp__less__eq,axiom,
! [R4: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,S3: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o] :
( ( ord_le1832228425591547726et_a_o @ R4 @ S3 )
=> ( ( antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a @ S3 )
=> ( antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a @ R4 ) ) ) ).
% antisymp_less_eq
thf(fact_614_order__class_Ofinite__has__minimal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal
thf(fact_615_order__class_Ofinite__has__minimal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal
thf(fact_616_order__class_Ofinite__has__minimal,axiom,
! [A3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A3 )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_minimal
thf(fact_617_order__class_Ofinite__has__maximal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal
thf(fact_618_order__class_Ofinite__has__maximal,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ? [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
& ! [Xa: ( c > d ) > set_a] :
( ( member_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le8464990428230162895_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal
thf(fact_619_order__class_Ofinite__has__maximal,axiom,
! [A3: set_set_c_d_set_a] :
( ( finite457288119118821432_set_a @ A3 )
=> ( ( A3 != bot_bo58555506362910043_set_a )
=> ? [X3: set_c_d_set_a] :
( ( member_set_c_d_set_a @ X3 @ A3 )
& ! [Xa: set_c_d_set_a] :
( ( member_set_c_d_set_a @ Xa @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% order_class.finite_has_maximal
thf(fact_620_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > a,B4: set_a,I3: ( c > d ) > set_a] :
( ( ord_less_eq_set_a @ ( image_c_d_set_a_a @ F @ top_to4267977599310771935_set_a ) @ B4 )
=> ( member_a @ ( F @ I3 ) @ B4 ) ) ).
% range_subsetD
thf(fact_621_range__subsetD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,B4: set_c_d_set_a,I3: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a ) @ B4 )
=> ( member_c_d_set_a @ ( F @ I3 ) @ B4 ) ) ).
% range_subsetD
thf(fact_622_order__class_Omono__imp__mono__on,axiom,
! [F: set_a > set_a,A3: 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 @ A3 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_623_order__class_Omono__imp__mono__on,axiom,
! [F: set_a > ( c > d ) > set_a,A3: set_set_a] :
( ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto2748056057003999288_set_a @ A3 @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_624_order__class_Omono__imp__mono__on,axiom,
! [F: set_a > set_c_d_set_a,A3: set_set_a] :
( ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( monoto7894950695950633880_set_a @ A3 @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_625_order__class_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A3: 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 @ A3 @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_626_order__class_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto2937423850181994535_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_627_order__class_Omono__imp__mono__on,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,A3: set_c_d_set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( monoto6642458133393520519_set_a @ A3 @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_628_order__class_Omono__imp__mono__on,axiom,
! [F: set_c_d_set_a > set_a,A3: set_set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( monoto9091215303422693110_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_629_order__class_Omono__imp__mono__on,axiom,
! [F: set_c_d_set_a > ( c > d ) > set_a,A3: set_set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( monoto5673664640695304391_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_630_order__class_Omono__imp__mono__on,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,A3: set_set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( monoto4733996707696316455_set_a @ A3 @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.mono_imp_mono_on
thf(fact_631_order__class_OmonoI,axiom,
! [F: set_a > set_a] :
( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.monoI
thf(fact_632_order__class_OmonoI,axiom,
! [F: set_a > ( c > d ) > set_a] :
( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto2748056057003999288_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.monoI
thf(fact_633_order__class_OmonoI,axiom,
! [F: set_a > set_c_d_set_a] :
( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.monoI
thf(fact_634_order__class_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.monoI
thf(fact_635_order__class_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.monoI
thf(fact_636_order__class_OmonoI,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.monoI
thf(fact_637_order__class_OmonoI,axiom,
! [F: set_c_d_set_a > set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F ) ) ).
% order_class.monoI
thf(fact_638_order__class_OmonoI,axiom,
! [F: set_c_d_set_a > ( c > d ) > set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto5673664640695304391_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F ) ) ).
% order_class.monoI
thf(fact_639_order__class_OmonoI,axiom,
! [F: set_c_d_set_a > set_c_d_set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X3 @ Y4 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F ) ) ).
% order_class.monoI
thf(fact_640_order__class_OmonoE,axiom,
! [F: set_a > set_a,X: set_a,Y2: 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 @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_641_order__class_OmonoE,axiom,
! [F: set_a > ( c > d ) > set_a,X: set_a,Y2: 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 @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_642_order__class_OmonoE,axiom,
! [F: set_a > set_c_d_set_a,X: set_a,Y2: set_a] :
( ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_643_order__class_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X: ( c > d ) > set_a,Y2: ( 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 @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_644_order__class_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_645_order__class_OmonoE,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_646_order__class_OmonoE,axiom,
! [F: set_c_d_set_a > set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_647_order__class_OmonoE,axiom,
! [F: set_c_d_set_a > ( c > d ) > set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_648_order__class_OmonoE,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoE
thf(fact_649_order__class_OmonoD,axiom,
! [F: set_a > set_a,X: set_a,Y2: 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 @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_650_order__class_OmonoD,axiom,
! [F: set_a > ( c > d ) > set_a,X: set_a,Y2: 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 @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_651_order__class_OmonoD,axiom,
! [F: set_a > set_c_d_set_a,X: set_a,Y2: set_a] :
( ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_652_order__class_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,X: ( c > d ) > set_a,Y2: ( 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 @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_653_order__class_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_654_order__class_OmonoD,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_655_order__class_OmonoD,axiom,
! [F: set_c_d_set_a > set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_656_order__class_OmonoD,axiom,
! [F: set_c_d_set_a > ( c > d ) > set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le8464990428230162895_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_657_order__class_OmonoD,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ord_le5982164083705284911_set_a @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% order_class.monoD
thf(fact_658_semilattice__inf__class_Omono__inf,axiom,
! [F: set_a > set_a,A3: set_a,B4: 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 @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_659_semilattice__inf__class_Omono__inf,axiom,
! [F: set_a > ( c > d ) > set_a,A3: set_a,B4: 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 @ A3 @ B4 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_660_semilattice__inf__class_Omono__inf,axiom,
! [F: set_a > set_c_d_set_a,A3: set_a,B4: set_a] :
( ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_661_semilattice__inf__class_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A3: ( c > d ) > set_a,B4: ( 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 @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_662_semilattice__inf__class_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: ( c > d ) > set_a,B4: ( 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 @ A3 @ B4 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_663_semilattice__inf__class_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,A3: ( c > d ) > set_a,B4: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_inf_c_d_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_664_semilattice__inf__class_Omono__inf,axiom,
! [F: set_c_d_set_a > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_665_semilattice__inf__class_Omono__inf,axiom,
! [F: set_c_d_set_a > ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto5673664640695304391_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ ( F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_666_semilattice__inf__class_Omono__inf,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% semilattice_inf_class.mono_inf
thf(fact_667_def__gfp__unfold,axiom,
! [A3: set_a,F: set_a > set_a] :
( ( A3
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( A3
= ( F @ A3 ) ) ) ) ).
% def_gfp_unfold
thf(fact_668_def__gfp__unfold,axiom,
! [A3: set_c_d_set_a,F: set_c_d_set_a > set_c_d_set_a] :
( ( A3
= ( comple5772108289334984589_set_a @ F ) )
=> ( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( A3
= ( F @ A3 ) ) ) ) ).
% def_gfp_unfold
thf(fact_669_def__gfp__unfold,axiom,
! [A3: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( A3
= ( comple4054414736020850733_set_a @ F ) )
=> ( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( A3
= ( F @ A3 ) ) ) ) ).
% def_gfp_unfold
thf(fact_670_gfp__fixpoint,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 )
=> ( ( F @ ( comple3341859861669737308_set_a @ F ) )
= ( comple3341859861669737308_set_a @ F ) ) ) ).
% gfp_fixpoint
thf(fact_671_gfp__fixpoint,axiom,
! [F: set_c_d_set_a > set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( F @ ( comple5772108289334984589_set_a @ F ) )
= ( comple5772108289334984589_set_a @ F ) ) ) ).
% gfp_fixpoint
thf(fact_672_gfp__fixpoint,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 )
=> ( ( F @ ( comple4054414736020850733_set_a @ F ) )
= ( comple4054414736020850733_set_a @ F ) ) ) ).
% gfp_fixpoint
thf(fact_673_gfp__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 )
=> ( ( comple3341859861669737308_set_a @ F )
= ( F @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% gfp_unfold
thf(fact_674_gfp__unfold,axiom,
! [F: set_c_d_set_a > set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( comple5772108289334984589_set_a @ F )
= ( F @ ( comple5772108289334984589_set_a @ F ) ) ) ) ).
% gfp_unfold
thf(fact_675_gfp__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 )
=> ( ( comple4054414736020850733_set_a @ F )
= ( F @ ( comple4054414736020850733_set_a @ F ) ) ) ) ).
% gfp_unfold
thf(fact_676_gfp__eqI,axiom,
! [F3: set_a > set_a,X: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F3 )
=> ( ( ( F3 @ X )
= X )
=> ( ! [Z4: set_a] :
( ( ( F3 @ Z4 )
= Z4 )
=> ( ord_less_eq_set_a @ Z4 @ X ) )
=> ( ( comple3341859861669737308_set_a @ F3 )
= X ) ) ) ) ).
% gfp_eqI
thf(fact_677_gfp__eqI,axiom,
! [F3: set_c_d_set_a > set_c_d_set_a,X: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F3 )
=> ( ( ( F3 @ X )
= X )
=> ( ! [Z4: set_c_d_set_a] :
( ( ( F3 @ Z4 )
= Z4 )
=> ( ord_le5982164083705284911_set_a @ Z4 @ X ) )
=> ( ( comple5772108289334984589_set_a @ F3 )
= X ) ) ) ) ).
% gfp_eqI
thf(fact_678_gfp__eqI,axiom,
! [F3: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F3 )
=> ( ( ( F3 @ X )
= X )
=> ( ! [Z4: ( c > d ) > set_a] :
( ( ( F3 @ Z4 )
= Z4 )
=> ( ord_le8464990428230162895_set_a @ Z4 @ X ) )
=> ( ( comple4054414736020850733_set_a @ F3 )
= X ) ) ) ) ).
% gfp_eqI
thf(fact_679_preorder_Obdd__below_Ocong,axiom,
condit9007271454129256903_set_a = condit9007271454129256903_set_a ).
% preorder.bdd_below.cong
thf(fact_680_preorder_Obdd__above_Ocong,axiom,
condit6926915774301931483_set_a = condit6926915774301931483_set_a ).
% preorder.bdd_above.cong
thf(fact_681_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ bot_bo738396921950161403_set_a @ X )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_682_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ bot_bo738396921950161403_set_a )
= bot_bo738396921950161403_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_683_finite__option__UNIV,axiom,
( ( finite1740182815655637662_set_a @ top_to1333438998097461157_set_a )
= ( finite3330819693523053784_set_a @ top_to4267977599310771935_set_a ) ) ).
% finite_option_UNIV
thf(fact_684_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_685_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_686_local_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > set_a,A3: ( c > d ) > set_a,B4: ( c > d ) > set_a] :
( ( monoto6316088450447394390_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_c_d_a2 @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% local.mono_inf
thf(fact_687_local_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: ( c > d ) > set_a,B4: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ ( F @ ( inf_c_d_a2 @ A3 @ B4 ) ) @ ( inf_inf_c_d_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% local.mono_inf
thf(fact_688_local_Omono__inf,axiom,
! [F: ( ( c > d ) > set_a ) > set_c_d_set_a,A3: ( c > d ) > set_a,B4: ( c > d ) > set_a] :
( ( monoto6642458133393520519_set_a @ top_to4267977599310771935_set_a @ smaller_interp_c_d_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_c_d_a2 @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% local.mono_inf
thf(fact_689_GFP__preserves__set__closure__property__aux,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,S4: a > a > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( set_cl2807270042661212426_a_c_d @ S4 @ full_interp_c_d_a )
=> ( ! [Delta3: ( c > d ) > set_a] :
( ( set_cl2807270042661212426_a_c_d @ S4 @ Delta3 )
=> ( set_cl2807270042661212426_a_c_d @ S4 @ ( F @ Delta3 ) ) )
=> ( set_cl2807270042661212426_a_c_d @ S4 @ ( comple4054414736020850733_set_a @ F ) ) ) ) ) ).
% GFP_preserves_set_closure_property_aux
thf(fact_690_local_Obot__least,axiom,
! [A: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ empty_interp_c_d_a @ A ) ).
% local.bot_least
thf(fact_691_local_Obot__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ empty_interp_c_d_a )
= ( A = empty_interp_c_d_a ) ) ).
% local.bot_unique
thf(fact_692_local_Ole__bot,axiom,
! [A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ empty_interp_c_d_a )
=> ( A = empty_interp_c_d_a ) ) ).
% local.le_bot
thf(fact_693_smaller__empty,axiom,
! [X: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ empty_interp_c_d_a @ X ) ).
% smaller_empty
thf(fact_694_inf__def,axiom,
( inf_c_d_a2
= ( ^ [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a,S2: c > d] : ( inf_inf_set_a @ ( Delta5 @ S2 ) @ ( Delta6 @ S2 ) ) ) ) ).
% inf_def
thf(fact_695_local_Oinf_Oassoc,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ ( inf_c_d_a2 @ A @ B ) @ C )
= ( inf_c_d_a2 @ A @ ( inf_c_d_a2 @ B @ C ) ) ) ).
% local.inf.assoc
thf(fact_696_local_Oinf_Ocommute,axiom,
( inf_c_d_a2
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] : ( inf_c_d_a2 @ B2 @ A2 ) ) ) ).
% local.inf.commute
thf(fact_697_local_Oinf_Oleft__commute,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ B @ ( inf_c_d_a2 @ A @ C ) )
= ( inf_c_d_a2 @ A @ ( inf_c_d_a2 @ B @ C ) ) ) ).
% local.inf.left_commute
thf(fact_698_local_Oinf__assoc,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ ( inf_c_d_a2 @ X @ Y2 ) @ Z2 )
= ( inf_c_d_a2 @ X @ ( inf_c_d_a2 @ Y2 @ Z2 ) ) ) ).
% local.inf_assoc
thf(fact_699_local_Oinf__commute,axiom,
( inf_c_d_a2
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] : ( inf_c_d_a2 @ Y3 @ X2 ) ) ) ).
% local.inf_commute
thf(fact_700_local_Oinf__left__commute,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X @ ( inf_c_d_a2 @ Y2 @ Z2 ) )
= ( inf_c_d_a2 @ Y2 @ ( inf_c_d_a2 @ X @ Z2 ) ) ) ).
% local.inf_left_commute
thf(fact_701_full__interp__def,axiom,
( full_interp_c_d_a
= ( ^ [S2: c > d] : top_top_set_a ) ) ).
% full_interp_def
thf(fact_702_set__closure__prop__empty__all_I1_J,axiom,
! [S4: a > a > set_a] : ( set_cl2807270042661212426_a_c_d @ S4 @ empty_interp_c_d_a ) ).
% set_closure_prop_empty_all(1)
thf(fact_703_local_Oinf_Oabsorb1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( inf_c_d_a2 @ A @ B )
= A ) ) ).
% local.inf.absorb1
thf(fact_704_local_Oinf_Oabsorb2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( inf_c_d_a2 @ A @ B )
= B ) ) ).
% local.inf.absorb2
thf(fact_705_local_Oinf_Oabsorb__iff1,axiom,
( smaller_interp_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ A2 @ B2 )
= A2 ) ) ) ).
% local.inf.absorb_iff1
thf(fact_706_local_Oinf_Oabsorb__iff2,axiom,
( smaller_interp_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ A2 @ B2 )
= B2 ) ) ) ).
% local.inf.absorb_iff2
thf(fact_707_local_Oinf_OboundedE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ ( inf_c_d_a2 @ B @ C ) )
=> ~ ( ( smaller_interp_c_d_a @ A @ B )
=> ~ ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% local.inf.boundedE
thf(fact_708_local_Oinf_OboundedI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ A @ C )
=> ( smaller_interp_c_d_a @ A @ ( inf_c_d_a2 @ B @ C ) ) ) ) ).
% local.inf.boundedI
thf(fact_709_local_Oinf_Ocobounded1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ A ) ).
% local.inf.cobounded1
thf(fact_710_local_Oinf_Ocobounded2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ B ) ).
% local.inf.cobounded2
thf(fact_711_local_Oinf_OcoboundedI1,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ C )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ C ) ) ).
% local.inf.coboundedI1
thf(fact_712_local_Oinf_OcoboundedI2,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ C ) ) ).
% local.inf.coboundedI2
thf(fact_713_local_Oinf_OorderE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( A
= ( inf_c_d_a2 @ A @ B ) ) ) ).
% local.inf.orderE
thf(fact_714_local_Oinf_OorderI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A
= ( inf_c_d_a2 @ A @ B ) )
=> ( smaller_interp_c_d_a @ A @ B ) ) ).
% local.inf.orderI
thf(fact_715_local_Oinf_Oorder__iff,axiom,
( smaller_interp_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( A2
= ( inf_c_d_a2 @ A2 @ B2 ) ) ) ) ).
% local.inf.order_iff
thf(fact_716_local_Oinf__absorb1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( inf_c_d_a2 @ X @ Y2 )
= X ) ) ).
% local.inf_absorb1
thf(fact_717_local_Oinf__absorb2,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y2 @ X )
=> ( ( inf_c_d_a2 @ X @ Y2 )
= Y2 ) ) ).
% local.inf_absorb2
thf(fact_718_local_Oinf__greatest,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( smaller_interp_c_d_a @ X @ Z2 )
=> ( smaller_interp_c_d_a @ X @ ( inf_c_d_a2 @ Y2 @ Z2 ) ) ) ) ).
% local.inf_greatest
thf(fact_719_local_Oinf__le1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ X @ Y2 ) @ X ) ).
% local.inf_le1
thf(fact_720_local_Oinf__le2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ X @ Y2 ) @ Y2 ) ).
% local.inf_le2
thf(fact_721_local_Oinf__mono,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ C )
=> ( ( smaller_interp_c_d_a @ B @ D2 )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ ( inf_c_d_a2 @ C @ D2 ) ) ) ) ).
% local.inf_mono
thf(fact_722_local_Oinf__unique,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X3 @ Y4 )
=> ( ( smaller_interp_c_d_a @ X3 @ Z4 )
=> ( smaller_interp_c_d_a @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_c_d_a2 @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% local.inf_unique
thf(fact_723_local_Ole__iff__inf,axiom,
( smaller_interp_c_d_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X2 @ Y3 )
= X2 ) ) ) ).
% local.le_iff_inf
thf(fact_724_local_Ole__infE,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ ( inf_c_d_a2 @ A @ B ) )
=> ~ ( ( smaller_interp_c_d_a @ X @ A )
=> ~ ( smaller_interp_c_d_a @ X @ B ) ) ) ).
% local.le_infE
thf(fact_725_local_Ole__infI,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ A )
=> ( ( smaller_interp_c_d_a @ X @ B )
=> ( smaller_interp_c_d_a @ X @ ( inf_c_d_a2 @ A @ B ) ) ) ) ).
% local.le_infI
thf(fact_726_local_Ole__infI1,axiom,
! [A: ( c > d ) > set_a,X: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ X )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ X ) ) ).
% local.le_infI1
thf(fact_727_local_Ole__infI2,axiom,
! [B: ( c > d ) > set_a,X: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ X )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a2 @ A @ B ) @ X ) ) ).
% local.le_infI2
thf(fact_728_smaller__full,axiom,
! [X: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ X @ full_interp_c_d_a ) ).
% smaller_full
thf(fact_729_local_Otop__greatest,axiom,
! [A: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ A @ full_interp_c_d_a ) ).
% local.top_greatest
thf(fact_730_local_Otop__le,axiom,
! [A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ full_interp_c_d_a @ A )
=> ( A = full_interp_c_d_a ) ) ).
% local.top_le
thf(fact_731_local_Otop__unique,axiom,
! [A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ full_interp_c_d_a @ A )
= ( A = full_interp_c_d_a ) ) ).
% local.top_unique
thf(fact_732_set__closure__prop__empty__all_I2_J,axiom,
! [S4: a > a > set_a] : ( set_cl2807270042661212426_a_c_d @ S4 @ full_interp_c_d_a ) ).
% set_closure_prop_empty_all(2)
thf(fact_733_local_OatLeast__eq__UNIV__iff,axiom,
! [X: ( c > d ) > set_a] :
( ( ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ X )
= top_to4267977599310771935_set_a )
= ( X = empty_interp_c_d_a ) ) ).
% local.atLeast_eq_UNIV_iff
thf(fact_734_local_OSup__inter__less__eq,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( smaller_interp_c_d_a @ ( sup_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_c_d_a2 @ ( sup_c_d_a @ A3 ) @ ( sup_c_d_a @ B4 ) ) ) ).
% local.Sup_inter_less_eq
thf(fact_735_local_OatMost__eq__UNIV__iff,axiom,
! [X: ( c > d ) > set_a] :
( ( ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ X )
= top_to4267977599310771935_set_a )
= ( X = full_interp_c_d_a ) ) ).
% local.atMost_eq_UNIV_iff
thf(fact_736_local_Oinf_Oidem,axiom,
! [A: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ A @ A )
= A ) ).
% local.inf.idem
thf(fact_737_local_Oinf_Oleft__idem,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ A @ ( inf_c_d_a2 @ A @ B ) )
= ( inf_c_d_a2 @ A @ B ) ) ).
% local.inf.left_idem
thf(fact_738_local_Oinf_Oright__idem,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ ( inf_c_d_a2 @ A @ B ) @ B )
= ( inf_c_d_a2 @ A @ B ) ) ).
% local.inf.right_idem
thf(fact_739_local_Oinf__bot__left,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ empty_interp_c_d_a @ X )
= empty_interp_c_d_a ) ).
% local.inf_bot_left
thf(fact_740_local_Oinf__bot__right,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X @ empty_interp_c_d_a )
= empty_interp_c_d_a ) ).
% local.inf_bot_right
thf(fact_741_local_Oinf__idem,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_742_local_Oinf__left__idem,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X @ ( inf_c_d_a2 @ X @ Y2 ) )
= ( inf_c_d_a2 @ X @ Y2 ) ) ).
% local.inf_left_idem
thf(fact_743_local_Oinf__right__idem,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ ( inf_c_d_a2 @ X @ Y2 ) @ Y2 )
= ( inf_c_d_a2 @ X @ Y2 ) ) ).
% local.inf_right_idem
thf(fact_744_local_OSup__bot__conv_I2_J,axiom,
! [A3: set_c_d_set_a] :
( ( empty_interp_c_d_a
= ( sup_c_d_a @ A3 ) )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( X2 = empty_interp_c_d_a ) ) ) ) ).
% local.Sup_bot_conv(2)
thf(fact_745_local_OSup__bot__conv_I1_J,axiom,
! [A3: set_c_d_set_a] :
( ( ( sup_c_d_a @ A3 )
= empty_interp_c_d_a )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( X2 = empty_interp_c_d_a ) ) ) ) ).
% local.Sup_bot_conv(1)
thf(fact_746_local_Oinf_Obounded__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ ( inf_c_d_a2 @ B @ C ) )
= ( ( smaller_interp_c_d_a @ A @ B )
& ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% local.inf.bounded_iff
thf(fact_747_local_Ole__inf__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ ( inf_c_d_a2 @ Y2 @ Z2 ) )
= ( ( smaller_interp_c_d_a @ X @ Y2 )
& ( smaller_interp_c_d_a @ X @ Z2 ) ) ) ).
% local.le_inf_iff
thf(fact_748_local_Oinf__top_Oeq__neutr__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( inf_c_d_a2 @ A @ B )
= full_interp_c_d_a )
= ( ( A = full_interp_c_d_a )
& ( B = full_interp_c_d_a ) ) ) ).
% local.inf_top.eq_neutr_iff
thf(fact_749_local_Oinf__top_Oleft__neutral,axiom,
! [A: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ full_interp_c_d_a @ A )
= A ) ).
% local.inf_top.left_neutral
thf(fact_750_local_Oinf__top_Oneutr__eq__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( full_interp_c_d_a
= ( inf_c_d_a2 @ A @ B ) )
= ( ( A = full_interp_c_d_a )
& ( B = full_interp_c_d_a ) ) ) ).
% local.inf_top.neutr_eq_iff
thf(fact_751_local_Oinf__top_Oright__neutral,axiom,
! [A: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ A @ full_interp_c_d_a )
= A ) ).
% local.inf_top.right_neutral
thf(fact_752_local_Oinf__eq__top__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ( inf_c_d_a2 @ X @ Y2 )
= full_interp_c_d_a )
= ( ( X = full_interp_c_d_a )
& ( Y2 = full_interp_c_d_a ) ) ) ).
% local.inf_eq_top_iff
thf(fact_753_local_Oinf__top__left,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ full_interp_c_d_a @ X )
= X ) ).
% local.inf_top_left
thf(fact_754_local_Oinf__top__right,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a2 @ X @ full_interp_c_d_a )
= X ) ).
% local.inf_top_right
thf(fact_755_local_Otop__eq__inf__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( full_interp_c_d_a
= ( inf_c_d_a2 @ X @ Y2 ) )
= ( ( X = full_interp_c_d_a )
& ( Y2 = full_interp_c_d_a ) ) ) ).
% local.top_eq_inf_iff
thf(fact_756_sup__empty,axiom,
( ( sup_c_d_a @ bot_bo738396921950161403_set_a )
= empty_interp_c_d_a ) ).
% sup_empty
thf(fact_757_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_758_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_759_local_Omax__bot,axiom,
! [X: ( c > d ) > set_a] :
( ( max_c_d_set_a @ smaller_interp_c_d_a @ empty_interp_c_d_a @ X )
= X ) ).
% local.max_bot
thf(fact_760_local_Omax__bot2,axiom,
! [X: ( c > d ) > set_a] :
( ( max_c_d_set_a @ smaller_interp_c_d_a @ X @ empty_interp_c_d_a )
= X ) ).
% local.max_bot2
thf(fact_761_local_Omin__bot,axiom,
! [X: ( c > d ) > set_a] :
( ( min_c_d_set_a @ smaller_interp_c_d_a @ empty_interp_c_d_a @ X )
= empty_interp_c_d_a ) ).
% local.min_bot
thf(fact_762_local_Omin__bot2,axiom,
! [X: ( c > d ) > set_a] :
( ( min_c_d_set_a @ smaller_interp_c_d_a @ X @ empty_interp_c_d_a )
= empty_interp_c_d_a ) ).
% local.min_bot2
thf(fact_763_local_OSup__UNIV,axiom,
( ( sup_c_d_a @ top_to4267977599310771935_set_a )
= full_interp_c_d_a ) ).
% local.Sup_UNIV
thf(fact_764_local_OSup__atMost,axiom,
! [Y2: ( c > d ) > set_a] :
( ( sup_c_d_a @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ Y2 ) )
= Y2 ) ).
% local.Sup_atMost
thf(fact_765_local_Obdd__below__Ici,axiom,
! [A: ( c > d ) > set_a] : ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ A ) ) ).
% local.bdd_below_Ici
thf(fact_766_local_Obdd__above__Iic,axiom,
! [B: ( c > d ) > set_a] : ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ B ) ) ).
% local.bdd_above_Iic
thf(fact_767_local_Omax__top,axiom,
! [X: ( c > d ) > set_a] :
( ( max_c_d_set_a @ smaller_interp_c_d_a @ full_interp_c_d_a @ X )
= full_interp_c_d_a ) ).
% local.max_top
thf(fact_768_local_Omax__top2,axiom,
! [X: ( c > d ) > set_a] :
( ( max_c_d_set_a @ smaller_interp_c_d_a @ X @ full_interp_c_d_a )
= full_interp_c_d_a ) ).
% local.max_top2
thf(fact_769_local_Omin__top,axiom,
! [X: ( c > d ) > set_a] :
( ( min_c_d_set_a @ smaller_interp_c_d_a @ full_interp_c_d_a @ X )
= X ) ).
% local.min_top
thf(fact_770_local_Omin__top2,axiom,
! [X: ( c > d ) > set_a] :
( ( min_c_d_set_a @ smaller_interp_c_d_a @ X @ full_interp_c_d_a )
= X ) ).
% local.min_top2
thf(fact_771_local_OSup__atLeast,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_c_d_a @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ X ) )
= full_interp_c_d_a ) ).
% local.Sup_atLeast
thf(fact_772_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_773_top__empty__eq,axiom,
( top_top_c_d_set_a_o
= ( ^ [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ top_to4267977599310771935_set_a ) ) ) ).
% top_empty_eq
thf(fact_774_top__set__def,axiom,
( top_to4267977599310771935_set_a
= ( collect_c_d_set_a @ top_top_c_d_set_a_o ) ) ).
% top_set_def
thf(fact_775_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_776_bot__empty__eq,axiom,
( bot_bot_c_d_set_a_o
= ( ^ [X2: ( c > d ) > set_a] : ( member_c_d_set_a @ X2 @ bot_bo738396921950161403_set_a ) ) ) ).
% bot_empty_eq
thf(fact_777_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_778_complete__lattice_Ogfp_Ocong,axiom,
comple4132920576971123013_set_a = comple4132920576971123013_set_a ).
% complete_lattice.gfp.cong
thf(fact_779_antisym__bot,axiom,
antisy1518167394357443548_set_a @ top_to4267977599310771935_set_a @ bot_bo919924463001950746et_a_o ).
% antisym_bot
thf(fact_780_mono__Int,axiom,
! [F: set_a > set_a,A3: set_a,B4: 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 @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% mono_Int
thf(fact_781_mono__Int,axiom,
! [F: set_a > set_c_d_set_a,A3: set_a,B4: set_a] :
( ( monoto7894950695950633880_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_inf_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% mono_Int
thf(fact_782_mono__Int,axiom,
! [F: set_c_d_set_a > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto9091215303422693110_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% mono_Int
thf(fact_783_mono__Int,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ord_le5982164083705284911_set_a @ ( F @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) @ ( inf_in754637537901350525_set_a @ ( F @ A3 ) @ ( F @ B4 ) ) ) ) ).
% mono_Int
thf(fact_784_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_c_d_set_a,K: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( A3
= ( inf_in754637537901350525_set_a @ K @ A ) )
=> ( ( inf_in754637537901350525_set_a @ A3 @ B )
= ( inf_in754637537901350525_set_a @ K @ ( inf_in754637537901350525_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_785_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_c_d_set_a,K: set_c_d_set_a,B: set_c_d_set_a,A: set_c_d_set_a] :
( ( B4
= ( inf_in754637537901350525_set_a @ K @ B ) )
=> ( ( inf_in754637537901350525_set_a @ A @ B4 )
= ( inf_in754637537901350525_set_a @ K @ ( inf_in754637537901350525_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_786_boolean__algebra_Oconj__one__right,axiom,
! [X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ top_to4267977599310771935_set_a )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_787_LFP__preserves__set__closure__property__aux,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,S4: a > a > set_a] :
( ( monotonic_c_d_a @ F )
=> ( ( set_cl2807270042661212426_a_c_d @ S4 @ empty_interp_c_d_a )
=> ( ! [Delta3: ( c > d ) > set_a] :
( ( set_cl2807270042661212426_a_c_d @ S4 @ Delta3 )
=> ( set_cl2807270042661212426_a_c_d @ S4 @ ( F @ Delta3 ) ) )
=> ( set_cl2807270042661212426_a_c_d @ S4 @ ( comple2361085228800170300_set_a @ F ) ) ) ) ) ).
% LFP_preserves_set_closure_property_aux
thf(fact_788_local_OatLeastAtMost__def,axiom,
! [L: ( c > d ) > set_a,U: ( c > d ) > set_a] :
( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L @ U )
= ( inf_in754637537901350525_set_a @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ L ) @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ U ) ) ) ).
% local.atLeastAtMost_def
thf(fact_789_local_OInf__fin_Osubset__imp,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( finite3330819693523053784_set_a @ B4 )
=> ( smaller_interp_c_d_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ B4 ) @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ) ).
% local.Inf_fin.subset_imp
thf(fact_790_local_Oinf__top_Osemilattice__neutr__axioms,axiom,
semila7616582506879544593_set_a @ inf_c_d_a2 @ full_interp_c_d_a ).
% local.inf_top.semilattice_neutr_axioms
thf(fact_791_local_OInf__fin_Osubset,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( B4 != bot_bo738396921950161403_set_a )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( ( inf_c_d_a2 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ B4 ) @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) )
= ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ) ).
% local.Inf_fin.subset
thf(fact_792_local_Oinf__top_Omonoid__axioms,axiom,
monoid_c_d_set_a @ inf_c_d_a2 @ full_interp_c_d_a ).
% local.inf_top.monoid_axioms
thf(fact_793_local_OInf__fin_Oin__idem,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( ( inf_c_d_a2 @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) )
= ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ).
% local.Inf_fin.in_idem
thf(fact_794_local_OInf__fin_OcoboundedI,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ( smaller_interp_c_d_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) @ A ) ) ) ).
% local.Inf_fin.coboundedI
thf(fact_795_local_OInf__fin_Ohom__commute,axiom,
! [H: ( ( c > d ) > set_a ) > ( c > d ) > set_a,N: set_c_d_set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] :
( ( H @ ( inf_c_d_a2 @ X3 @ Y4 ) )
= ( inf_c_d_a2 @ ( H @ X3 ) @ ( H @ Y4 ) ) )
=> ( ( 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_796_local_OInf__fin_OboundedE,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) )
=> ! [A6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A6 @ A3 )
=> ( smaller_interp_c_d_a @ X @ A6 ) ) ) ) ) ).
% local.Inf_fin.boundedE
thf(fact_797_local_OInf__fin_OboundedI,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A4 @ A3 )
=> ( smaller_interp_c_d_a @ X @ A4 ) )
=> ( smaller_interp_c_d_a @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ) ).
% local.Inf_fin.boundedI
thf(fact_798_local_OInf__fin_Obounded__iff,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ X @ X2 ) ) ) ) ) ) ).
% local.Inf_fin.bounded_iff
thf(fact_799_local_OatLeastAtMost__eq__UNIV__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ X @ Y2 )
= top_to4267977599310771935_set_a )
= ( ( X = empty_interp_c_d_a )
& ( Y2 = full_interp_c_d_a ) ) ) ).
% local.atLeastAtMost_eq_UNIV_iff
thf(fact_800_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_801_local_OIcc__eq__Icc,axiom,
! [L: ( c > d ) > set_a,H: ( c > d ) > set_a,L2: ( c > d ) > set_a,H2: ( c > d ) > set_a] :
( ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L @ H )
= ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( smaller_interp_c_d_a @ L @ H )
& ~ ( smaller_interp_c_d_a @ L2 @ H2 ) ) ) ) ).
% local.Icc_eq_Icc
thf(fact_802_local_OSup__atLeastAtMost,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( sup_c_d_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ X @ Y2 ) )
= Y2 ) ) ).
% local.Sup_atLeastAtMost
thf(fact_803_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_804_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_805_local_Obdd__below__Icc,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) ) ).
% local.bdd_below_Icc
thf(fact_806_local_Obdd__above__Icc,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) ) ).
% local.bdd_above_Icc
thf(fact_807_local_OatLeastatMost__subset__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ C @ D2 ) )
= ( ~ ( smaller_interp_c_d_a @ A @ B )
| ( ( smaller_interp_c_d_a @ C @ A )
& ( smaller_interp_c_d_a @ B @ D2 ) ) ) ) ).
% local.atLeastatMost_subset_iff
thf(fact_808_local_OIcc__subset__Ici__iff,axiom,
! [L: ( c > d ) > set_a,H: ( c > d ) > set_a,L2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L @ H ) @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ L2 ) )
= ( ~ ( smaller_interp_c_d_a @ L @ H )
| ( smaller_interp_c_d_a @ L2 @ L ) ) ) ).
% local.Icc_subset_Ici_iff
thf(fact_809_local_OIcc__subset__Iic__iff,axiom,
! [L: ( c > d ) > set_a,H: ( c > d ) > set_a,H2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ L @ H ) @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ H2 ) )
= ( ~ ( smaller_interp_c_d_a @ L @ H )
| ( smaller_interp_c_d_a @ H @ H2 ) ) ) ).
% local.Icc_subset_Iic_iff
thf(fact_810_bounded__semilattice__inf__top__class_Oinf__top_Omonoid__axioms,axiom,
monoid_set_c_d_set_a @ inf_in754637537901350525_set_a @ top_to4267977599310771935_set_a ).
% bounded_semilattice_inf_top_class.inf_top.monoid_axioms
thf(fact_811_bounded__semilattice__inf__top__class_Oinf__top_Osemilattice__neutr__axioms,axiom,
semila3717735699007493233_set_a @ inf_in754637537901350525_set_a @ top_to4267977599310771935_set_a ).
% bounded_semilattice_inf_top_class.inf_top.semilattice_neutr_axioms
thf(fact_812_ord_OatLeastAtMost__def,axiom,
( set_at2224545791267470424_set_a
= ( ^ [Less_eq: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o,L3: ( c > d ) > set_a,U3: ( c > d ) > set_a] : ( inf_in754637537901350525_set_a @ ( set_at4358065015900363374_set_a @ Less_eq @ L3 ) @ ( set_atMost_c_d_set_a @ Less_eq @ U3 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_813_fixp__lowerbound,axiom,
! [F: set_a > set_a,Z2: 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 @ Z2 ) @ Z2 )
=> ( ord_less_eq_set_a @ ( comple6813827801316615403_set_a @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_814_fixp__lowerbound,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,Z2: ( 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 @ Z2 ) @ Z2 )
=> ( ord_le8464990428230162895_set_a @ ( comple2361085228800170300_set_a @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_815_fixp__lowerbound,axiom,
! [F: set_c_d_set_a > set_c_d_set_a,Z2: set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( ord_le5982164083705284911_set_a @ ( F @ Z2 ) @ Z2 )
=> ( ord_le5982164083705284911_set_a @ ( comple7316045241340859548_set_a @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_816_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_817_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_818_fixp__unfold,axiom,
! [F: set_c_d_set_a > set_c_d_set_a] :
( ( monoto4733996707696316455_set_a @ top_to5717711934741766719_set_a @ ord_le5982164083705284911_set_a @ ord_le5982164083705284911_set_a @ F )
=> ( ( comple7316045241340859548_set_a @ F )
= ( F @ ( comple7316045241340859548_set_a @ F ) ) ) ) ).
% fixp_unfold
thf(fact_819_local_OInf__fin_Oclosed,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( member_c_d_set_a @ ( inf_c_d_a2 @ X3 @ Y4 ) @ ( insert_c_d_set_a @ X3 @ ( insert_c_d_set_a @ Y4 @ bot_bo738396921950161403_set_a ) ) )
=> ( member_c_d_set_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) @ A3 ) ) ) ) ).
% local.Inf_fin.closed
thf(fact_820_local_OInf__fin_Oinsert__not__elem,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ~ ( member_c_d_set_a @ X @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X @ A3 ) )
= ( inf_c_d_a2 @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ) ) ).
% local.Inf_fin.insert_not_elem
thf(fact_821_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_822_insert__absorb2,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ X @ ( insert_c_d_set_a @ X @ A3 ) )
= ( insert_c_d_set_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_823_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_824_insert__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_c_d_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_825_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_826_insertCI,axiom,
! [A: ( c > d ) > set_a,B4: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ~ ( member_c_d_set_a @ A @ B4 )
=> ( A = B ) )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_827_insert__image,axiom,
! [X: a,A3: set_a,F: a > ( c > d ) > set_a] :
( ( member_a @ X @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X ) @ ( image_a_c_d_set_a @ F @ A3 ) )
= ( image_a_c_d_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_828_insert__image,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( ( insert_c_d_set_a @ ( F @ X ) @ ( image_5710119992958135237_set_a @ F @ A3 ) )
= ( image_5710119992958135237_set_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_829_image__insert,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( image_5710119992958135237_set_a @ F @ ( insert_c_d_set_a @ A @ B4 ) )
= ( insert_c_d_set_a @ ( F @ A ) @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_830_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_831_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_832_finite__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ ( insert_c_d_set_a @ A @ A3 ) )
= ( finite3330819693523053784_set_a @ A3 ) ) ).
% finite_insert
thf(fact_833_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_834_insert__subset,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ X @ A3 ) @ B4 )
= ( ( member_c_d_set_a @ X @ B4 )
& ( ord_le5982164083705284911_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_835_Int__insert__right__if1,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_836_Int__insert__right__if1,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_837_Int__insert__right__if0,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_838_Int__insert__right__if0,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_839_insert__inter__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ ( insert_c_d_set_a @ A @ B4 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_840_Int__insert__left__if1,axiom,
! [A: a,C4: set_a,B4: set_a] :
( ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C4 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_841_Int__insert__left__if1,axiom,
! [A: ( c > d ) > set_a,C4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ C4 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B4 ) @ C4 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_842_Int__insert__left__if0,axiom,
! [A: a,C4: set_a,B4: set_a] :
( ~ ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C4 )
= ( inf_inf_set_a @ B4 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_843_Int__insert__left__if0,axiom,
! [A: ( c > d ) > set_a,C4: set_c_d_set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ C4 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B4 ) @ C4 )
= ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_844_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_845_singleton__insert__inj__eq,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a )
= ( insert_c_d_set_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_846_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_847_singleton__insert__inj__eq_H,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ( insert_c_d_set_a @ A @ A3 )
= ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) )
= ( ( A = B )
& ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ bot_bo738396921950161403_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_848_insert__disjoint_I1_J,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B4 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_849_insert__disjoint_I1_J,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ B4 )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B4 )
& ( ( inf_in754637537901350525_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_850_insert__disjoint_I2_J,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A3 ) @ B4 ) )
= ( ~ ( member_a @ A @ B4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_851_insert__disjoint_I2_J,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ A3 ) @ B4 ) )
= ( ~ ( member_c_d_set_a @ A @ B4 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_852_disjoint__insert_I1_J,axiom,
! [B4: set_a,A: a,A3: set_a] :
( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ B4 @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_853_disjoint__insert_I1_J,axiom,
! [B4: set_c_d_set_a,A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( ( inf_in754637537901350525_set_a @ B4 @ ( insert_c_d_set_a @ A @ A3 ) )
= bot_bo738396921950161403_set_a )
= ( ~ ( member_c_d_set_a @ A @ B4 )
& ( ( inf_in754637537901350525_set_a @ B4 @ A3 )
= bot_bo738396921950161403_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_854_disjoint__insert_I2_J,axiom,
! [A3: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_855_disjoint__insert_I2_J,axiom,
! [A3: set_c_d_set_a,B: ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ B @ B4 ) ) )
= ( ~ ( member_c_d_set_a @ B @ A3 )
& ( bot_bo738396921950161403_set_a
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_856_local_Obdd__below__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( insert_c_d_set_a @ A @ A3 ) )
= ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_below_insert
thf(fact_857_local_Obdd__above__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( insert_c_d_set_a @ A @ A3 ) )
= ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 ) ) ).
% local.bdd_above_insert
thf(fact_858_local_OatLeastAtMost__singleton__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B )
= ( insert_c_d_set_a @ C @ bot_bo738396921950161403_set_a ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% local.atLeastAtMost_singleton_iff
thf(fact_859_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_860_local_OInf__fin_Osingleton,axiom,
! [X: ( c > d ) > set_a] :
( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
= X ) ).
% local.Inf_fin.singleton
thf(fact_861_local_OInf__fin_Oinsert,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X @ A3 ) )
= ( inf_c_d_a2 @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) ) ) ) ) ).
% local.Inf_fin.insert
thf(fact_862_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_863_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_864_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_865_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_866_doubleton__eq__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( 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 @ C @ ( insert_c_d_set_a @ D2 @ bot_bo738396921950161403_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_867_insert__not__empty,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ A3 )
!= bot_bo738396921950161403_set_a ) ).
% insert_not_empty
thf(fact_868_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_869_insert__UNIV,axiom,
! [X: ( c > d ) > set_a] :
( ( insert_c_d_set_a @ X @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ).
% insert_UNIV
thf(fact_870_finite_OinsertI,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( finite3330819693523053784_set_a @ ( insert_c_d_set_a @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_871_subset__insertI2,axiom,
! [A3: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_872_subset__insertI2,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_873_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_874_subset__insertI,axiom,
! [B4: set_c_d_set_a,A: ( c > d ) > set_a] : ( ord_le5982164083705284911_set_a @ B4 @ ( insert_c_d_set_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_875_subset__insert,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_876_subset__insert,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X @ B4 ) )
= ( ord_le5982164083705284911_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_877_insert__mono,axiom,
! [C4: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_878_insert__mono,axiom,
! [C4: set_c_d_set_a,D: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ D )
=> ( ord_le5982164083705284911_set_a @ ( insert_c_d_set_a @ A @ C4 ) @ ( insert_c_d_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_879_Int__insert__left,axiom,
! [A: a,C4: set_a,B4: set_a] :
( ( ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C4 )
= ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C4 ) ) ) )
& ( ~ ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C4 )
= ( inf_inf_set_a @ B4 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_880_Int__insert__left,axiom,
! [A: ( c > d ) > set_a,C4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ C4 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B4 ) @ C4 )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ C4 )
=> ( ( inf_in754637537901350525_set_a @ ( insert_c_d_set_a @ A @ B4 ) @ C4 )
= ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_881_Int__insert__right,axiom,
! [A: a,A3: set_a,B4: set_a] :
( ( ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A3 @ B4 ) ) ) )
& ( ~ ( member_a @ A @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_882_Int__insert__right,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) )
= ( insert_c_d_set_a @ A @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) )
& ( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ( inf_in754637537901350525_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_883_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_884_mk__disjoint__insert,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ? [B6: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ A @ B6 ) )
& ~ ( member_c_d_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_885_insert__commute,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ X @ ( insert_c_d_set_a @ Y2 @ A3 ) )
= ( insert_c_d_set_a @ Y2 @ ( insert_c_d_set_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_886_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C6: set_a] :
( ( A3
= ( insert_a @ B @ C6 ) )
& ~ ( member_a @ B @ C6 )
& ( B4
= ( insert_a @ A @ C6 ) )
& ~ ( member_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_887_insert__eq__iff,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a,B: ( c > d ) > set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ A @ A3 )
=> ( ~ ( member_c_d_set_a @ B @ B4 )
=> ( ( ( insert_c_d_set_a @ A @ A3 )
= ( insert_c_d_set_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C6: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ B @ C6 ) )
& ~ ( member_c_d_set_a @ B @ C6 )
& ( B4
= ( insert_c_d_set_a @ A @ C6 ) )
& ~ ( member_c_d_set_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_888_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_889_insert__absorb,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ A3 )
=> ( ( insert_c_d_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_890_insert__ident,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_891_insert__ident,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X @ A3 )
=> ( ~ ( member_c_d_set_a @ X @ B4 )
=> ( ( ( insert_c_d_set_a @ X @ A3 )
= ( insert_c_d_set_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_892_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_893_Set_Oset__insert,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ~ ! [B6: set_c_d_set_a] :
( ( A3
= ( insert_c_d_set_a @ X @ B6 ) )
=> ( member_c_d_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_894_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_895_insertI2,axiom,
! [A: ( c > d ) > set_a,B4: set_c_d_set_a,B: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A @ B4 )
=> ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_896_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_897_insertI1,axiom,
! [A: ( c > d ) > set_a,B4: set_c_d_set_a] : ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ A @ B4 ) ) ).
% insertI1
thf(fact_898_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_899_insertE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ A @ ( insert_c_d_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_c_d_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_900_finite_Ocases,axiom,
! [A: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A )
=> ( ( A != bot_bo738396921950161403_set_a )
=> ~ ! [A7: set_c_d_set_a] :
( ? [A4: ( c > d ) > set_a] :
( A
= ( insert_c_d_set_a @ A4 @ A7 ) )
=> ~ ( finite3330819693523053784_set_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_901_finite_Osimps,axiom,
( finite3330819693523053784_set_a
= ( ^ [A2: set_c_d_set_a] :
( ( A2 = bot_bo738396921950161403_set_a )
| ? [A5: set_c_d_set_a,B2: ( c > d ) > set_a] :
( ( A2
= ( insert_c_d_set_a @ B2 @ A5 ) )
& ( finite3330819693523053784_set_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_902_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_903_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 )
=> ( ! [X3: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ~ ( member_c_d_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_904_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_905_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 )
=> ( ! [X3: ( c > d ) > set_a] : ( P @ ( insert_c_d_set_a @ X3 @ bot_bo738396921950161403_set_a ) )
=> ( ! [X3: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( F4 != bot_bo738396921950161403_set_a )
=> ( ~ ( member_c_d_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_906_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_907_infinite__finite__induct,axiom,
! [P: set_c_d_set_a > $o,A3: set_c_d_set_a] :
( ! [A7: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [X3: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ~ ( member_c_d_set_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_908_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_909_subset__singleton__iff,axiom,
! [X5: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ X5 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
= ( ( X5 = bot_bo738396921950161403_set_a )
| ( X5
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_910_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_911_subset__singletonD,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
=> ( ( A3 = bot_bo738396921950161403_set_a )
| ( A3
= ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_912_range__eq__singletonD,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( ( image_5710119992958135237_set_a @ F @ top_to4267977599310771935_set_a )
= ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) )
=> ( ( F @ X )
= A ) ) ).
% range_eq_singletonD
thf(fact_913_finite__subset__induct,axiom,
! [F3: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A3 )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_914_finite__subset__induct,axiom,
! [F3: set_c_d_set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A4: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( member_c_d_set_a @ A4 @ A3 )
=> ( ~ ( member_c_d_set_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_915_finite__subset__induct_H,axiom,
! [F3: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A3 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_916_finite__subset__induct_H,axiom,
! [F3: set_c_d_set_a,A3: set_c_d_set_a,P: set_c_d_set_a > $o] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( ord_le5982164083705284911_set_a @ F3 @ A3 )
=> ( ( P @ bot_bo738396921950161403_set_a )
=> ( ! [A4: ( c > d ) > set_a,F4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F4 )
=> ( ( member_c_d_set_a @ A4 @ A3 )
=> ( ( ord_le5982164083705284911_set_a @ F4 @ A3 )
=> ( ~ ( member_c_d_set_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_c_d_set_a @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_917_local_OInf__fin_Oremove,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ X @ A3 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 )
= X ) )
& ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 )
= ( inf_c_d_a2 @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ) ).
% local.Inf_fin.remove
thf(fact_918_local_OInf__fin_Oinsert__remove,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X @ A3 ) )
= X ) )
& ( ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) )
!= bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( insert_c_d_set_a @ X @ A3 ) )
= ( inf_c_d_a2 @ X @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ bot_bo738396921950161403_set_a ) ) ) ) ) ) ) ) ).
% local.Inf_fin.insert_remove
thf(fact_919_local_OInf__fin_Ounion,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( finite3330819693523053784_set_a @ B4 )
=> ( ( B4 != bot_bo738396921950161403_set_a )
=> ( ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( inf_c_d_a2 @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ B4 ) ) ) ) ) ) ) ).
% local.Inf_fin.union
thf(fact_920_semilattice__sup__class_Osup_Oidem,axiom,
! [A: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A @ A )
= A ) ).
% semilattice_sup_class.sup.idem
thf(fact_921_semilattice__sup__class_Osup_Oidem,axiom,
! [A: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ A @ A )
= A ) ).
% semilattice_sup_class.sup.idem
thf(fact_922_semilattice__sup__class_Osup__idem,axiom,
! [X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ X )
= X ) ).
% semilattice_sup_class.sup_idem
thf(fact_923_semilattice__sup__class_Osup__idem,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ X )
= X ) ).
% semilattice_sup_class.sup_idem
thf(fact_924_semilattice__sup__class_Osup_Oleft__idem,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A @ ( sup_su3175602471750379875_set_a @ A @ B ) )
= ( sup_su3175602471750379875_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.left_idem
thf(fact_925_semilattice__sup__class_Osup_Oleft__idem,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ A @ ( sup_sup_c_d_set_a @ A @ B ) )
= ( sup_sup_c_d_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.left_idem
thf(fact_926_semilattice__sup__class_Osup__left__idem,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) )
= ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_left_idem
thf(fact_927_semilattice__sup__class_Osup__left__idem,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ X @ Y2 ) )
= ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_left_idem
thf(fact_928_semilattice__sup__class_Osup_Oright__idem,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ A @ B ) @ B )
= ( sup_su3175602471750379875_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.right_idem
thf(fact_929_semilattice__sup__class_Osup_Oright__idem,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( sup_sup_c_d_set_a @ A @ B ) @ B )
= ( sup_sup_c_d_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.right_idem
thf(fact_930_sup__apply,axiom,
( sup_sup_c_d_set_a
= ( ^ [F2: ( c > d ) > set_a,G: ( c > d ) > set_a,X2: c > d] : ( sup_sup_set_a @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_apply
thf(fact_931_UnCI,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_932_UnCI,axiom,
! [C: ( c > d ) > set_a,B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ~ ( member_c_d_set_a @ C @ B4 )
=> ( member_c_d_set_a @ C @ A3 ) )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_933_Un__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_934_Un__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( ( member_c_d_set_a @ C @ A3 )
| ( member_c_d_set_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_935_DiffI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_936_DiffI,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( ~ ( member_c_d_set_a @ C @ B4 )
=> ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_937_Diff__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_938_Diff__iff,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
= ( ( member_c_d_set_a @ C @ A3 )
& ~ ( member_c_d_set_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_939_Diff__idemp,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) @ B4 )
= ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ).
% Diff_idemp
thf(fact_940_semilattice__sup__class_Ole__sup__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_sup_class.le_sup_iff
thf(fact_941_semilattice__sup__class_Ole__sup__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_le8464990428230162895_set_a @ X @ Z2 )
& ( ord_le8464990428230162895_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_sup_class.le_sup_iff
thf(fact_942_semilattice__sup__class_Ole__sup__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_le5982164083705284911_set_a @ X @ Z2 )
& ( ord_le5982164083705284911_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_sup_class.le_sup_iff
thf(fact_943_semilattice__sup__class_Osup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.bounded_iff
thf(fact_944_semilattice__sup__class_Osup_Obounded__iff,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ B @ C ) @ A )
= ( ( ord_le8464990428230162895_set_a @ B @ A )
& ( ord_le8464990428230162895_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.bounded_iff
thf(fact_945_semilattice__sup__class_Osup_Obounded__iff,axiom,
! [B: set_c_d_set_a,C: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ B @ C ) @ A )
= ( ( ord_le5982164083705284911_set_a @ B @ A )
& ( ord_le5982164083705284911_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.bounded_iff
thf(fact_946_bounded__semilattice__sup__bot__class_Osup__bot_Oright__neutral,axiom,
! [A: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ A @ bot_bot_c_d_set_a )
= A ) ).
% bounded_semilattice_sup_bot_class.sup_bot.right_neutral
thf(fact_947_bounded__semilattice__sup__bot__class_Osup__bot_Oright__neutral,axiom,
! [A: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A @ bot_bo738396921950161403_set_a )
= A ) ).
% bounded_semilattice_sup_bot_class.sup_bot.right_neutral
thf(fact_948_bounded__semilattice__sup__bot__class_Osup__bot_Oneutr__eq__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( bot_bot_c_d_set_a
= ( sup_sup_c_d_set_a @ A @ B ) )
= ( ( A = bot_bot_c_d_set_a )
& ( B = bot_bot_c_d_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_bot.neutr_eq_iff
thf(fact_949_bounded__semilattice__sup__bot__class_Osup__bot_Oneutr__eq__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( sup_su3175602471750379875_set_a @ A @ B ) )
= ( ( A = bot_bo738396921950161403_set_a )
& ( B = bot_bo738396921950161403_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_bot.neutr_eq_iff
thf(fact_950_bounded__semilattice__sup__bot__class_Osup__bot_Oleft__neutral,axiom,
! [A: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ bot_bot_c_d_set_a @ A )
= A ) ).
% bounded_semilattice_sup_bot_class.sup_bot.left_neutral
thf(fact_951_bounded__semilattice__sup__bot__class_Osup__bot_Oleft__neutral,axiom,
! [A: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ bot_bo738396921950161403_set_a @ A )
= A ) ).
% bounded_semilattice_sup_bot_class.sup_bot.left_neutral
thf(fact_952_bounded__semilattice__sup__bot__class_Osup__bot_Oeq__neutr__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ( sup_sup_c_d_set_a @ A @ B )
= bot_bot_c_d_set_a )
= ( ( A = bot_bot_c_d_set_a )
& ( B = bot_bot_c_d_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_bot.eq_neutr_iff
thf(fact_953_bounded__semilattice__sup__bot__class_Osup__bot_Oeq__neutr__iff,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ( sup_su3175602471750379875_set_a @ A @ B )
= bot_bo738396921950161403_set_a )
= ( ( A = bot_bo738396921950161403_set_a )
& ( B = bot_bo738396921950161403_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_bot.eq_neutr_iff
thf(fact_954_bounded__semilattice__sup__bot__class_Osup__eq__bot__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ( sup_sup_c_d_set_a @ X @ Y2 )
= bot_bot_c_d_set_a )
= ( ( X = bot_bot_c_d_set_a )
& ( Y2 = bot_bot_c_d_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_eq_bot_iff
thf(fact_955_bounded__semilattice__sup__bot__class_Osup__eq__bot__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ( sup_su3175602471750379875_set_a @ X @ Y2 )
= bot_bo738396921950161403_set_a )
= ( ( X = bot_bo738396921950161403_set_a )
& ( Y2 = bot_bo738396921950161403_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.sup_eq_bot_iff
thf(fact_956_bounded__semilattice__sup__bot__class_Obot__eq__sup__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( bot_bot_c_d_set_a
= ( sup_sup_c_d_set_a @ X @ Y2 ) )
= ( ( X = bot_bot_c_d_set_a )
& ( Y2 = bot_bot_c_d_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.bot_eq_sup_iff
thf(fact_957_bounded__semilattice__sup__bot__class_Obot__eq__sup__iff,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( bot_bo738396921950161403_set_a
= ( sup_su3175602471750379875_set_a @ X @ Y2 ) )
= ( ( X = bot_bo738396921950161403_set_a )
& ( Y2 = bot_bo738396921950161403_set_a ) ) ) ).
% bounded_semilattice_sup_bot_class.bot_eq_sup_iff
thf(fact_958_bounded__semilattice__sup__bot__class_Osup__bot__right,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ bot_bot_c_d_set_a )
= X ) ).
% bounded_semilattice_sup_bot_class.sup_bot_right
thf(fact_959_bounded__semilattice__sup__bot__class_Osup__bot__right,axiom,
! [X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ bot_bo738396921950161403_set_a )
= X ) ).
% bounded_semilattice_sup_bot_class.sup_bot_right
thf(fact_960_bounded__semilattice__sup__bot__class_Osup__bot__left,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ bot_bot_c_d_set_a @ X )
= X ) ).
% bounded_semilattice_sup_bot_class.sup_bot_left
thf(fact_961_bounded__semilattice__sup__bot__class_Osup__bot__left,axiom,
! [X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ bot_bo738396921950161403_set_a @ X )
= X ) ).
% bounded_semilattice_sup_bot_class.sup_bot_left
thf(fact_962_bounded__lattice__top__class_Osup__top__right,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ top_top_c_d_set_a )
= top_top_c_d_set_a ) ).
% bounded_lattice_top_class.sup_top_right
thf(fact_963_bounded__lattice__top__class_Osup__top__right,axiom,
! [X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ).
% bounded_lattice_top_class.sup_top_right
thf(fact_964_bounded__lattice__top__class_Osup__top__left,axiom,
! [X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ top_top_c_d_set_a @ X )
= top_top_c_d_set_a ) ).
% bounded_lattice_top_class.sup_top_left
thf(fact_965_bounded__lattice__top__class_Osup__top__left,axiom,
! [X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ top_to4267977599310771935_set_a @ X )
= top_to4267977599310771935_set_a ) ).
% bounded_lattice_top_class.sup_top_left
thf(fact_966_lattice__class_Oinf__sup__absorb,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) )
= X ) ).
% lattice_class.inf_sup_absorb
thf(fact_967_lattice__class_Oinf__sup__absorb,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ X @ Y2 ) )
= X ) ).
% lattice_class.inf_sup_absorb
thf(fact_968_lattice__class_Osup__inf__absorb,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( inf_in754637537901350525_set_a @ X @ Y2 ) )
= X ) ).
% lattice_class.sup_inf_absorb
thf(fact_969_lattice__class_Osup__inf__absorb,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( inf_inf_c_d_set_a @ X @ Y2 ) )
= X ) ).
% lattice_class.sup_inf_absorb
thf(fact_970_Un__empty,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( sup_su3175602471750379875_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a )
= ( ( A3 = bot_bo738396921950161403_set_a )
& ( B4 = bot_bo738396921950161403_set_a ) ) ) ).
% Un_empty
thf(fact_971_Diff__empty,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ bot_bo738396921950161403_set_a )
= A3 ) ).
% Diff_empty
thf(fact_972_empty__Diff,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ bot_bo738396921950161403_set_a @ A3 )
= bot_bo738396921950161403_set_a ) ).
% empty_Diff
thf(fact_973_Diff__cancel,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ A3 )
= bot_bo738396921950161403_set_a ) ).
% Diff_cancel
thf(fact_974_finite__Un,axiom,
! [F3: set_c_d_set_a,G3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ ( sup_su3175602471750379875_set_a @ F3 @ G3 ) )
= ( ( finite3330819693523053784_set_a @ F3 )
& ( finite3330819693523053784_set_a @ G3 ) ) ) ).
% finite_Un
thf(fact_975_finite__Diff2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ B4 )
=> ( ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
= ( finite3330819693523053784_set_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_976_finite__Diff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_977_Un__subset__iff,axiom,
! [A3: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C4 )
= ( ( ord_less_eq_set_a @ A3 @ C4 )
& ( ord_less_eq_set_a @ B4 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_978_Un__subset__iff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ C4 )
= ( ( ord_le5982164083705284911_set_a @ A3 @ C4 )
& ( ord_le5982164083705284911_set_a @ B4 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_979_Un__insert__left,axiom,
! [A: ( c > d ) > set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( insert_c_d_set_a @ A @ B4 ) @ C4 )
= ( insert_c_d_set_a @ A @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_980_Un__insert__right,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) )
= ( insert_c_d_set_a @ A @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_981_Diff__insert0,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_982_Diff__insert0,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ~ ( member_c_d_set_a @ X @ A3 )
=> ( ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ X @ B4 ) )
= ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_983_insert__Diff1,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_984_insert__Diff1,axiom,
! [X: ( c > d ) > set_a,B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X @ B4 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X @ A3 ) @ B4 )
= ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_985_Int__Un__eq_I4_J,axiom,
! [T2: set_c_d_set_a,S4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ T2 @ ( inf_in754637537901350525_set_a @ S4 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_986_Int__Un__eq_I3_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ S4 @ ( inf_in754637537901350525_set_a @ S4 @ T2 ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_987_Int__Un__eq_I2_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ S4 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_988_Int__Un__eq_I1_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ S4 @ T2 ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_989_Un__Int__eq_I4_J,axiom,
! [T2: set_c_d_set_a,S4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ T2 @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_990_Un__Int__eq_I3_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ S4 @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_991_Un__Int__eq_I2_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_992_Un__Int__eq_I1_J,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_993_Un__Diff__cancel,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B4 @ A3 ) )
= ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_994_Un__Diff__cancel2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( minus_1665977719694084726_set_a @ B4 @ A3 ) @ A3 )
= ( sup_su3175602471750379875_set_a @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_995_Diff__UNIV,axiom,
! [A3: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ top_to4267977599310771935_set_a )
= bot_bo738396921950161403_set_a ) ).
% Diff_UNIV
thf(fact_996_Diff__eq__empty__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_997_Diff__eq__empty__iff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ( minus_1665977719694084726_set_a @ A3 @ B4 )
= bot_bo738396921950161403_set_a )
= ( ord_le5982164083705284911_set_a @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_998_insert__Diff__single,axiom,
! [A: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( insert_c_d_set_a @ A @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ bot_bo738396921950161403_set_a ) ) )
= ( insert_c_d_set_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_999_finite__Diff__insert,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a,B4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A3 @ ( insert_c_d_set_a @ A @ B4 ) ) )
= ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1000_Diff__disjoint,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B4 @ A3 ) )
= bot_bo738396921950161403_set_a ) ).
% Diff_disjoint
thf(fact_1001_local_Obdd__below__Un,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ A3 )
& ( condit9007271454129256903_set_a @ smaller_interp_c_d_a @ B4 ) ) ) ).
% local.bdd_below_Un
thf(fact_1002_local_Obdd__above__Un,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ A3 )
& ( condit6926915774301931483_set_a @ smaller_interp_c_d_a @ B4 ) ) ) ).
% local.bdd_above_Un
thf(fact_1003_image__Un,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( image_5710119992958135237_set_a @ F @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( sup_su3175602471750379875_set_a @ ( image_5710119992958135237_set_a @ F @ A3 ) @ ( image_5710119992958135237_set_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_1004_insert__Diff__if,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1005_insert__Diff__if,axiom,
! [X: ( c > d ) > set_a,B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ( member_c_d_set_a @ X @ B4 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X @ A3 ) @ B4 )
= ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) )
& ( ~ ( member_c_d_set_a @ X @ B4 )
=> ( ( minus_1665977719694084726_set_a @ ( insert_c_d_set_a @ X @ A3 ) @ B4 )
= ( insert_c_d_set_a @ X @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1006_Diff__subset__conv,axiom,
! [A3: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ C4 )
= ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_1007_Diff__subset__conv,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) @ C4 )
= ( ord_le5982164083705284911_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_1008_Diff__partition,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1009_Diff__partition,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( sup_su3175602471750379875_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1010_lattice__class_Oinf__sup__aci_I8_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) )
= ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_aci(8)
thf(fact_1011_lattice__class_Oinf__sup__aci_I8_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ X @ Y2 ) )
= ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_aci(8)
thf(fact_1012_lattice__class_Oinf__sup__aci_I7_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) )
= ( sup_su3175602471750379875_set_a @ Y2 @ ( sup_su3175602471750379875_set_a @ X @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(7)
thf(fact_1013_lattice__class_Oinf__sup__aci_I7_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) )
= ( sup_sup_c_d_set_a @ Y2 @ ( sup_sup_c_d_set_a @ X @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(7)
thf(fact_1014_lattice__class_Oinf__sup__aci_I6_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ Z2 )
= ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(6)
thf(fact_1015_lattice__class_Oinf__sup__aci_I6_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ Z2 )
= ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) ) ) ).
% lattice_class.inf_sup_aci(6)
thf(fact_1016_lattice__class_Oinf__sup__aci_I5_J,axiom,
( sup_su3175602471750379875_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] : ( sup_su3175602471750379875_set_a @ Y3 @ X2 ) ) ) ).
% lattice_class.inf_sup_aci(5)
thf(fact_1017_lattice__class_Oinf__sup__aci_I5_J,axiom,
( sup_sup_c_d_set_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] : ( sup_sup_c_d_set_a @ Y3 @ X2 ) ) ) ).
% lattice_class.inf_sup_aci(5)
thf(fact_1018_semilattice__sup__class_Osup_Oassoc,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,C: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ A @ B ) @ C )
= ( sup_su3175602471750379875_set_a @ A @ ( sup_su3175602471750379875_set_a @ B @ C ) ) ) ).
% semilattice_sup_class.sup.assoc
thf(fact_1019_semilattice__sup__class_Osup_Oassoc,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( sup_sup_c_d_set_a @ A @ B ) @ C )
= ( sup_sup_c_d_set_a @ A @ ( sup_sup_c_d_set_a @ B @ C ) ) ) ).
% semilattice_sup_class.sup.assoc
thf(fact_1020_semilattice__sup__class_Osup__assoc,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ Z2 )
= ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_sup_class.sup_assoc
thf(fact_1021_semilattice__sup__class_Osup__assoc,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ Z2 )
= ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) ) ) ).
% semilattice_sup_class.sup_assoc
thf(fact_1022_semilattice__sup__class_Osup_Ocommute,axiom,
( sup_su3175602471750379875_set_a
= ( ^ [A2: set_c_d_set_a,B2: set_c_d_set_a] : ( sup_su3175602471750379875_set_a @ B2 @ A2 ) ) ) ).
% semilattice_sup_class.sup.commute
thf(fact_1023_semilattice__sup__class_Osup_Ocommute,axiom,
( sup_sup_c_d_set_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] : ( sup_sup_c_d_set_a @ B2 @ A2 ) ) ) ).
% semilattice_sup_class.sup.commute
thf(fact_1024_semilattice__sup__class_Osup__commute,axiom,
( sup_su3175602471750379875_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] : ( sup_su3175602471750379875_set_a @ Y3 @ X2 ) ) ) ).
% semilattice_sup_class.sup_commute
thf(fact_1025_semilattice__sup__class_Osup__commute,axiom,
( sup_sup_c_d_set_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] : ( sup_sup_c_d_set_a @ Y3 @ X2 ) ) ) ).
% semilattice_sup_class.sup_commute
thf(fact_1026_semilattice__sup__class_Osup_Oleft__commute,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ B @ ( sup_su3175602471750379875_set_a @ A @ C ) )
= ( sup_su3175602471750379875_set_a @ A @ ( sup_su3175602471750379875_set_a @ B @ C ) ) ) ).
% semilattice_sup_class.sup.left_commute
thf(fact_1027_semilattice__sup__class_Osup_Oleft__commute,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ B @ ( sup_sup_c_d_set_a @ A @ C ) )
= ( sup_sup_c_d_set_a @ A @ ( sup_sup_c_d_set_a @ B @ C ) ) ) ).
% semilattice_sup_class.sup.left_commute
thf(fact_1028_semilattice__sup__class_Osup__left__commute,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) )
= ( sup_su3175602471750379875_set_a @ Y2 @ ( sup_su3175602471750379875_set_a @ X @ Z2 ) ) ) ).
% semilattice_sup_class.sup_left_commute
thf(fact_1029_semilattice__sup__class_Osup__left__commute,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) )
= ( sup_sup_c_d_set_a @ Y2 @ ( sup_sup_c_d_set_a @ X @ Z2 ) ) ) ).
% semilattice_sup_class.sup_left_commute
thf(fact_1030_sup__fun__def,axiom,
( sup_sup_c_d_set_a
= ( ^ [F2: ( c > d ) > set_a,G: ( c > d ) > set_a,X2: c > d] : ( sup_sup_set_a @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_fun_def
thf(fact_1031_UnE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_1032_UnE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
=> ( ~ ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_1033_UnI1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1034_UnI1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_1035_UnI2,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1036_UnI2,axiom,
! [C: ( c > d ) > set_a,B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ B4 )
=> ( member_c_d_set_a @ C @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_1037_DiffE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1038_DiffE,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
=> ~ ( ( member_c_d_set_a @ C @ A3 )
=> ( member_c_d_set_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1039_DiffD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1040_DiffD1,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
=> ( member_c_d_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1041_DiffD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_1042_DiffD2,axiom,
! [C: ( c > d ) > set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( member_c_d_set_a @ C @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
=> ~ ( member_c_d_set_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_1043_bex__Un,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1044_Un__Diff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ C4 )
= ( sup_su3175602471750379875_set_a @ ( minus_1665977719694084726_set_a @ A3 @ C4 ) @ ( minus_1665977719694084726_set_a @ B4 @ C4 ) ) ) ).
% Un_Diff
thf(fact_1045_ball__Un,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,P: ( ( c > d ) > set_a ) > $o] :
( ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1046_Un__assoc,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ C4 )
= ( sup_su3175602471750379875_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) ) ) ).
% Un_assoc
thf(fact_1047_Un__absorb,axiom,
! [A3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1048_Un__commute,axiom,
( sup_su3175602471750379875_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] : ( sup_su3175602471750379875_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_1049_Un__left__absorb,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
= ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_1050_Un__left__commute,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) )
= ( sup_su3175602471750379875_set_a @ B4 @ ( sup_su3175602471750379875_set_a @ A3 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_1051_Diff__Int__distrib2,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) @ C4 )
= ( minus_1665977719694084726_set_a @ ( inf_in754637537901350525_set_a @ A3 @ C4 ) @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1052_Diff__Int__distrib,axiom,
! [C4: set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ C4 @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
= ( minus_1665977719694084726_set_a @ ( inf_in754637537901350525_set_a @ C4 @ A3 ) @ ( inf_in754637537901350525_set_a @ C4 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1053_Diff__Diff__Int,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ A3 @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) )
= ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1054_Diff__Int2,axiom,
! [A3: set_c_d_set_a,C4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ ( inf_in754637537901350525_set_a @ A3 @ C4 ) @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) )
= ( minus_1665977719694084726_set_a @ ( inf_in754637537901350525_set_a @ A3 @ C4 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1055_Int__Diff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( minus_1665977719694084726_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ C4 )
= ( inf_in754637537901350525_set_a @ A3 @ ( minus_1665977719694084726_set_a @ B4 @ C4 ) ) ) ).
% Int_Diff
thf(fact_1056_Un__Int__distrib2,axiom,
! [B4: set_c_d_set_a,C4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) @ A3 )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ B4 @ A3 ) @ ( sup_su3175602471750379875_set_a @ C4 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1057_Int__Un__distrib2,axiom,
! [B4: set_c_d_set_a,C4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) @ A3 )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ B4 @ A3 ) @ ( inf_in754637537901350525_set_a @ C4 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1058_Un__Int__distrib,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ ( sup_su3175602471750379875_set_a @ A3 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_1059_Int__Un__distrib,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ ( inf_in754637537901350525_set_a @ A3 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_1060_Un__Int__crazy,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) @ ( inf_in754637537901350525_set_a @ B4 @ C4 ) ) @ ( inf_in754637537901350525_set_a @ C4 @ A3 ) )
= ( inf_in754637537901350525_set_a @ ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ ( sup_su3175602471750379875_set_a @ B4 @ C4 ) ) @ ( sup_su3175602471750379875_set_a @ C4 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1061_Diff__mono,axiom,
! [A3: set_a,C4: set_a,D: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ C4 )
=> ( ( ord_less_eq_set_a @ D @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ C4 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1062_Diff__mono,axiom,
! [A3: set_c_d_set_a,C4: set_c_d_set_a,D: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ C4 )
=> ( ( ord_le5982164083705284911_set_a @ D @ B4 )
=> ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) @ ( minus_1665977719694084726_set_a @ C4 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1063_Diff__subset,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_1064_Diff__subset,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ ( minus_1665977719694084726_set_a @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_1065_double__diff,axiom,
! [A3: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C4 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C4 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1066_double__diff,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a,C4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ C4 )
=> ( ( minus_1665977719694084726_set_a @ B4 @ ( minus_1665977719694084726_set_a @ C4 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1067_Diff__infinite__finite,axiom,
! [T2: set_c_d_set_a,S4: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ T2 )
=> ( ~ ( finite3330819693523053784_set_a @ S4 )
=> ~ ( finite3330819693523053784_set_a @ ( minus_1665977719694084726_set_a @ S4 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1068_Un__mono,axiom,
! [A3: set_a,C4: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A3 @ C4 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ ( sup_sup_set_a @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_1069_Un__mono,axiom,
! [A3: set_c_d_set_a,C4: set_c_d_set_a,B4: set_c_d_set_a,D: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ C4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ D )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ ( sup_su3175602471750379875_set_a @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_1070_Un__least,axiom,
! [A3: set_a,C4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ C4 )
=> ( ( ord_less_eq_set_a @ B4 @ C4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C4 ) ) ) ).
% Un_least
thf(fact_1071_Un__least,axiom,
! [A3: set_c_d_set_a,C4: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ C4 )
=> ( ( ord_le5982164083705284911_set_a @ B4 @ C4 )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) @ C4 ) ) ) ).
% Un_least
thf(fact_1072_Un__upper1,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_1073_Un__upper1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A3 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_1074_Un__upper2,axiom,
! [B4: set_a,A3: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_1075_Un__upper2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ B4 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_1076_Un__absorb1,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( sup_sup_set_a @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1077_Un__absorb1,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( sup_su3175602471750379875_set_a @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1078_Un__absorb2,axiom,
! [B4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1079_Un__absorb2,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( ( sup_su3175602471750379875_set_a @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1080_subset__UnE,axiom,
! [C4: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C4 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A3 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B4 )
=> ( C4
!= ( sup_sup_set_a @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1081_subset__UnE,axiom,
! [C4: set_c_d_set_a,A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C4 @ ( sup_su3175602471750379875_set_a @ A3 @ B4 ) )
=> ~ ! [A8: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A8 @ A3 )
=> ! [B7: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B7 @ B4 )
=> ( C4
!= ( sup_su3175602471750379875_set_a @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1082_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1083_subset__Un__eq,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [A5: set_c_d_set_a,B5: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1084_infinite__Un,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ( ~ ( finite3330819693523053784_set_a @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) ) )
= ( ~ ( finite3330819693523053784_set_a @ S4 )
| ~ ( finite3330819693523053784_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_1085_Un__infinite,axiom,
! [S4: set_c_d_set_a,T2: set_c_d_set_a] :
( ~ ( finite3330819693523053784_set_a @ S4 )
=> ~ ( finite3330819693523053784_set_a @ ( sup_su3175602471750379875_set_a @ S4 @ T2 ) ) ) ).
% Un_infinite
thf(fact_1086_finite__UnI,axiom,
! [F3: set_c_d_set_a,G3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ F3 )
=> ( ( finite3330819693523053784_set_a @ G3 )
=> ( finite3330819693523053784_set_a @ ( sup_su3175602471750379875_set_a @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_1087_Un__UNIV__right,axiom,
! [A3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ top_to4267977599310771935_set_a )
= top_to4267977599310771935_set_a ) ).
% Un_UNIV_right
thf(fact_1088_Un__UNIV__left,axiom,
! [B4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ top_to4267977599310771935_set_a @ B4 )
= top_to4267977599310771935_set_a ) ).
% Un_UNIV_left
thf(fact_1089_Un__empty__right,axiom,
! [A3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A3 @ bot_bo738396921950161403_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_1090_Un__empty__left,axiom,
! [B4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ bot_bo738396921950161403_set_a @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1091_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_c_d_set_a,Z2: set_c_d_set_a,X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) @ X )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ Y2 @ X ) @ ( sup_su3175602471750379875_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1092_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) @ X )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ Y2 @ X ) @ ( sup_sup_c_d_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1093_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_c_d_set_a,Z2: set_c_d_set_a,X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) @ X )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ Y2 @ X ) @ ( inf_in754637537901350525_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1094_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) @ X )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ Y2 @ X ) @ ( inf_inf_c_d_set_a @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1095_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ ( sup_su3175602471750379875_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1096_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ ( sup_sup_c_d_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1097_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ ( inf_in754637537901350525_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1098_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ ( inf_inf_c_d_set_a @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1099_sup__inf__distrib2,axiom,
! [Y2: set_c_d_set_a,Z2: set_c_d_set_a,X: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) @ X )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ Y2 @ X ) @ ( sup_su3175602471750379875_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1100_sup__inf__distrib2,axiom,
! [Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) @ X )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ Y2 @ X ) @ ( sup_sup_c_d_set_a @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1101_sup__inf__distrib1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ ( sup_su3175602471750379875_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1102_sup__inf__distrib1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ ( sup_sup_c_d_set_a @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1103_inf__sup__distrib2,axiom,
! [Y2: set_c_d_set_a,Z2: set_c_d_set_a,X: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) @ X )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ Y2 @ X ) @ ( inf_in754637537901350525_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1104_inf__sup__distrib2,axiom,
! [Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) @ X )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ Y2 @ X ) @ ( inf_inf_c_d_set_a @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1105_inf__sup__distrib1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ ( inf_in754637537901350525_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1106_inf__sup__distrib1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ ( inf_inf_c_d_set_a @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1107_lattice__class_Odistrib__imp2,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a,Z4: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X3 @ ( inf_in754637537901350525_set_a @ Y4 @ Z4 ) )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ X3 @ Y4 ) @ ( sup_su3175602471750379875_set_a @ X3 @ Z4 ) ) )
=> ( ( inf_in754637537901350525_set_a @ X @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ X @ Y2 ) @ ( inf_in754637537901350525_set_a @ X @ Z2 ) ) ) ) ).
% lattice_class.distrib_imp2
thf(fact_1108_lattice__class_Odistrib__imp2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X3 @ ( inf_inf_c_d_set_a @ Y4 @ Z4 ) )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ X3 @ Y4 ) @ ( sup_sup_c_d_set_a @ X3 @ Z4 ) ) )
=> ( ( inf_inf_c_d_set_a @ X @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ X @ Y2 ) @ ( inf_inf_c_d_set_a @ X @ Z2 ) ) ) ) ).
% lattice_class.distrib_imp2
thf(fact_1109_lattice__class_Odistrib__imp1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a,Z2: set_c_d_set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a,Z4: set_c_d_set_a] :
( ( inf_in754637537901350525_set_a @ X3 @ ( sup_su3175602471750379875_set_a @ Y4 @ Z4 ) )
= ( sup_su3175602471750379875_set_a @ ( inf_in754637537901350525_set_a @ X3 @ Y4 ) @ ( inf_in754637537901350525_set_a @ X3 @ Z4 ) ) )
=> ( ( sup_su3175602471750379875_set_a @ X @ ( inf_in754637537901350525_set_a @ Y2 @ Z2 ) )
= ( inf_in754637537901350525_set_a @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) @ ( sup_su3175602471750379875_set_a @ X @ Z2 ) ) ) ) ).
% lattice_class.distrib_imp1
thf(fact_1110_lattice__class_Odistrib__imp1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( inf_inf_c_d_set_a @ X3 @ ( sup_sup_c_d_set_a @ Y4 @ Z4 ) )
= ( sup_sup_c_d_set_a @ ( inf_inf_c_d_set_a @ X3 @ Y4 ) @ ( inf_inf_c_d_set_a @ X3 @ Z4 ) ) )
=> ( ( sup_sup_c_d_set_a @ X @ ( inf_inf_c_d_set_a @ Y2 @ Z2 ) )
= ( inf_inf_c_d_set_a @ ( sup_sup_c_d_set_a @ X @ Y2 ) @ ( sup_sup_c_d_set_a @ X @ Z2 ) ) ) ) ).
% lattice_class.distrib_imp1
thf(fact_1111_lattice__class_Oinf__sup__ord_I4_J,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(4)
thf(fact_1112_lattice__class_Oinf__sup__ord_I4_J,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ Y2 @ ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(4)
thf(fact_1113_lattice__class_Oinf__sup__ord_I4_J,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ Y2 @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(4)
thf(fact_1114_lattice__class_Oinf__sup__ord_I3_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(3)
thf(fact_1115_lattice__class_Oinf__sup__ord_I3_J,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X @ ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(3)
thf(fact_1116_lattice__class_Oinf__sup__ord_I3_J,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% lattice_class.inf_sup_ord(3)
thf(fact_1117_semilattice__sup__class_Ole__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% semilattice_sup_class.le_supE
thf(fact_1118_semilattice__sup__class_Ole__supE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ A @ B ) @ X )
=> ~ ( ( ord_le8464990428230162895_set_a @ A @ X )
=> ~ ( ord_le8464990428230162895_set_a @ B @ X ) ) ) ).
% semilattice_sup_class.le_supE
thf(fact_1119_semilattice__sup__class_Ole__supE,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a,X: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A @ B ) @ X )
=> ~ ( ( ord_le5982164083705284911_set_a @ A @ X )
=> ~ ( ord_le5982164083705284911_set_a @ B @ X ) ) ) ).
% semilattice_sup_class.le_supE
thf(fact_1120_semilattice__sup__class_Ole__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% semilattice_sup_class.le_supI
thf(fact_1121_semilattice__sup__class_Ole__supI,axiom,
! [A: ( c > d ) > set_a,X: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ X )
=> ( ( ord_le8464990428230162895_set_a @ B @ X )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ A @ B ) @ X ) ) ) ).
% semilattice_sup_class.le_supI
thf(fact_1122_semilattice__sup__class_Ole__supI,axiom,
! [A: set_c_d_set_a,X: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ X )
=> ( ( ord_le5982164083705284911_set_a @ B @ X )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A @ B ) @ X ) ) ) ).
% semilattice_sup_class.le_supI
thf(fact_1123_semilattice__sup__class_Osup__ge1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge1
thf(fact_1124_semilattice__sup__class_Osup__ge1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X @ ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge1
thf(fact_1125_semilattice__sup__class_Osup__ge1,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge1
thf(fact_1126_semilattice__sup__class_Osup__ge2,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge2
thf(fact_1127_semilattice__sup__class_Osup__ge2,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ Y2 @ ( sup_sup_c_d_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge2
thf(fact_1128_semilattice__sup__class_Osup__ge2,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ Y2 @ ( sup_su3175602471750379875_set_a @ X @ Y2 ) ) ).
% semilattice_sup_class.sup_ge2
thf(fact_1129_semilattice__sup__class_Ole__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI1
thf(fact_1130_semilattice__sup__class_Ole__supI1,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ A )
=> ( ord_le8464990428230162895_set_a @ X @ ( sup_sup_c_d_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI1
thf(fact_1131_semilattice__sup__class_Ole__supI1,axiom,
! [X: set_c_d_set_a,A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ A )
=> ( ord_le5982164083705284911_set_a @ X @ ( sup_su3175602471750379875_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI1
thf(fact_1132_semilattice__sup__class_Ole__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI2
thf(fact_1133_semilattice__sup__class_Ole__supI2,axiom,
! [X: ( c > d ) > set_a,B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ B )
=> ( ord_le8464990428230162895_set_a @ X @ ( sup_sup_c_d_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI2
thf(fact_1134_semilattice__sup__class_Ole__supI2,axiom,
! [X: set_c_d_set_a,B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ B )
=> ( ord_le5982164083705284911_set_a @ X @ ( sup_su3175602471750379875_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.le_supI2
thf(fact_1135_semilattice__sup__class_Osup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% semilattice_sup_class.sup.mono
thf(fact_1136_semilattice__sup__class_Osup_Omono,axiom,
! [C: ( c > d ) > set_a,A: ( c > d ) > set_a,D2: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ C @ A )
=> ( ( ord_le8464990428230162895_set_a @ D2 @ B )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ C @ D2 ) @ ( sup_sup_c_d_set_a @ A @ B ) ) ) ) ).
% semilattice_sup_class.sup.mono
thf(fact_1137_semilattice__sup__class_Osup_Omono,axiom,
! [C: set_c_d_set_a,A: set_c_d_set_a,D2: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ C @ A )
=> ( ( ord_le5982164083705284911_set_a @ D2 @ B )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ C @ D2 ) @ ( sup_su3175602471750379875_set_a @ A @ B ) ) ) ) ).
% semilattice_sup_class.sup.mono
thf(fact_1138_semilattice__sup__class_Osup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% semilattice_sup_class.sup_mono
thf(fact_1139_semilattice__sup__class_Osup__mono,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ C )
=> ( ( ord_le8464990428230162895_set_a @ B @ D2 )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ A @ B ) @ ( sup_sup_c_d_set_a @ C @ D2 ) ) ) ) ).
% semilattice_sup_class.sup_mono
thf(fact_1140_semilattice__sup__class_Osup__mono,axiom,
! [A: set_c_d_set_a,C: set_c_d_set_a,B: set_c_d_set_a,D2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ C )
=> ( ( ord_le5982164083705284911_set_a @ B @ D2 )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ A @ B ) @ ( sup_su3175602471750379875_set_a @ C @ D2 ) ) ) ) ).
% semilattice_sup_class.sup_mono
thf(fact_1141_semilattice__sup__class_Osup__least,axiom,
! [Y2: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% semilattice_sup_class.sup_least
thf(fact_1142_semilattice__sup__class_Osup__least,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y2 @ X )
=> ( ( ord_le8464990428230162895_set_a @ Z2 @ X )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% semilattice_sup_class.sup_least
thf(fact_1143_semilattice__sup__class_Osup__least,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a,Z2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y2 @ X )
=> ( ( ord_le5982164083705284911_set_a @ Z2 @ X )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% semilattice_sup_class.sup_least
thf(fact_1144_semilattice__sup__class_Ole__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% semilattice_sup_class.le_iff_sup
thf(fact_1145_semilattice__sup__class_Ole__iff__sup,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( sup_sup_c_d_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% semilattice_sup_class.le_iff_sup
thf(fact_1146_semilattice__sup__class_Ole__iff__sup,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [X2: set_c_d_set_a,Y3: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ X2 @ Y3 )
= Y3 ) ) ) ).
% semilattice_sup_class.le_iff_sup
thf(fact_1147_semilattice__sup__class_Osup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.sup.orderE
thf(fact_1148_semilattice__sup__class_Osup_OorderE,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( A
= ( sup_sup_c_d_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.sup.orderE
thf(fact_1149_semilattice__sup__class_Osup_OorderE,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( A
= ( sup_su3175602471750379875_set_a @ A @ B ) ) ) ).
% semilattice_sup_class.sup.orderE
thf(fact_1150_semilattice__sup__class_Osup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% semilattice_sup_class.sup.orderI
thf(fact_1151_semilattice__sup__class_Osup_OorderI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A
= ( sup_sup_c_d_set_a @ A @ B ) )
=> ( ord_le8464990428230162895_set_a @ B @ A ) ) ).
% semilattice_sup_class.sup.orderI
thf(fact_1152_semilattice__sup__class_Osup_OorderI,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( A
= ( sup_su3175602471750379875_set_a @ A @ B ) )
=> ( ord_le5982164083705284911_set_a @ B @ A ) ) ).
% semilattice_sup_class.sup.orderI
thf(fact_1153_semilattice__sup__class_Osup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_a,Y4: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X3 )
=> ( ( ord_less_eq_set_a @ Z4 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_sup_class.sup_unique
thf(fact_1154_semilattice__sup__class_Osup__unique,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y4 @ X3 )
=> ( ( ord_le8464990428230162895_set_a @ Z4 @ X3 )
=> ( ord_le8464990428230162895_set_a @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_c_d_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_sup_class.sup_unique
thf(fact_1155_semilattice__sup__class_Osup__unique,axiom,
! [F: set_c_d_set_a > set_c_d_set_a > set_c_d_set_a,X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: set_c_d_set_a,Y4: set_c_d_set_a,Z4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y4 @ X3 )
=> ( ( ord_le5982164083705284911_set_a @ Z4 @ X3 )
=> ( ord_le5982164083705284911_set_a @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su3175602471750379875_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% semilattice_sup_class.sup_unique
thf(fact_1156_semilattice__sup__class_Osup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% semilattice_sup_class.sup.absorb1
thf(fact_1157_semilattice__sup__class_Osup_Oabsorb1,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( sup_sup_c_d_set_a @ A @ B )
= A ) ) ).
% semilattice_sup_class.sup.absorb1
thf(fact_1158_semilattice__sup__class_Osup_Oabsorb1,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( sup_su3175602471750379875_set_a @ A @ B )
= A ) ) ).
% semilattice_sup_class.sup.absorb1
thf(fact_1159_semilattice__sup__class_Osup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% semilattice_sup_class.sup.absorb2
thf(fact_1160_semilattice__sup__class_Osup_Oabsorb2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ A @ B )
=> ( ( sup_sup_c_d_set_a @ A @ B )
= B ) ) ).
% semilattice_sup_class.sup.absorb2
thf(fact_1161_semilattice__sup__class_Osup_Oabsorb2,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A @ B )
=> ( ( sup_su3175602471750379875_set_a @ A @ B )
= B ) ) ).
% semilattice_sup_class.sup.absorb2
thf(fact_1162_semilattice__sup__class_Osup__absorb1,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( sup_sup_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_sup_class.sup_absorb1
thf(fact_1163_semilattice__sup__class_Osup__absorb1,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ Y2 @ X )
=> ( ( sup_sup_c_d_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_sup_class.sup_absorb1
thf(fact_1164_semilattice__sup__class_Osup__absorb1,axiom,
! [Y2: set_c_d_set_a,X: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ Y2 @ X )
=> ( ( sup_su3175602471750379875_set_a @ X @ Y2 )
= X ) ) ).
% semilattice_sup_class.sup_absorb1
thf(fact_1165_semilattice__sup__class_Osup__absorb2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( sup_sup_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_sup_class.sup_absorb2
thf(fact_1166_semilattice__sup__class_Osup__absorb2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X @ Y2 )
=> ( ( sup_sup_c_d_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_sup_class.sup_absorb2
thf(fact_1167_semilattice__sup__class_Osup__absorb2,axiom,
! [X: set_c_d_set_a,Y2: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ X @ Y2 )
=> ( ( sup_su3175602471750379875_set_a @ X @ Y2 )
= Y2 ) ) ).
% semilattice_sup_class.sup_absorb2
thf(fact_1168_semilattice__sup__class_Osup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.boundedE
thf(fact_1169_semilattice__sup__class_Osup_OboundedE,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ B @ C ) @ A )
=> ~ ( ( ord_le8464990428230162895_set_a @ B @ A )
=> ~ ( ord_le8464990428230162895_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.boundedE
thf(fact_1170_semilattice__sup__class_Osup_OboundedE,axiom,
! [B: set_c_d_set_a,C: set_c_d_set_a,A: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ B @ C ) @ A )
=> ~ ( ( ord_le5982164083705284911_set_a @ B @ A )
=> ~ ( ord_le5982164083705284911_set_a @ C @ A ) ) ) ).
% semilattice_sup_class.sup.boundedE
thf(fact_1171_semilattice__sup__class_Osup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% semilattice_sup_class.sup.boundedI
thf(fact_1172_semilattice__sup__class_Osup_OboundedI,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ B @ A )
=> ( ( ord_le8464990428230162895_set_a @ C @ A )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ B @ C ) @ A ) ) ) ).
% semilattice_sup_class.sup.boundedI
thf(fact_1173_semilattice__sup__class_Osup_OboundedI,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a,C: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B @ A )
=> ( ( ord_le5982164083705284911_set_a @ C @ A )
=> ( ord_le5982164083705284911_set_a @ ( sup_su3175602471750379875_set_a @ B @ C ) @ A ) ) ) ).
% semilattice_sup_class.sup.boundedI
thf(fact_1174_semilattice__sup__class_Osup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A2: set_a] :
( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_sup_class.sup.order_iff
thf(fact_1175_semilattice__sup__class_Osup_Oorder__iff,axiom,
( ord_le8464990428230162895_set_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( A2
= ( sup_sup_c_d_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_sup_class.sup.order_iff
thf(fact_1176_semilattice__sup__class_Osup_Oorder__iff,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( A2
= ( sup_su3175602471750379875_set_a @ A2 @ B2 ) ) ) ) ).
% semilattice_sup_class.sup.order_iff
thf(fact_1177_semilattice__sup__class_Osup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded1
thf(fact_1178_semilattice__sup__class_Osup_Ocobounded1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ A @ ( sup_sup_c_d_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded1
thf(fact_1179_semilattice__sup__class_Osup_Ocobounded1,axiom,
! [A: set_c_d_set_a,B: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ A @ ( sup_su3175602471750379875_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded1
thf(fact_1180_semilattice__sup__class_Osup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded2
thf(fact_1181_semilattice__sup__class_Osup_Ocobounded2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] : ( ord_le8464990428230162895_set_a @ B @ ( sup_sup_c_d_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded2
thf(fact_1182_semilattice__sup__class_Osup_Ocobounded2,axiom,
! [B: set_c_d_set_a,A: set_c_d_set_a] : ( ord_le5982164083705284911_set_a @ B @ ( sup_su3175602471750379875_set_a @ A @ B ) ) ).
% semilattice_sup_class.sup.cobounded2
thf(fact_1183_semilattice__sup__class_Osup_Oabsorb__iff1,axiom,
( ord_le5982164083705284911_set_a
= ( ^ [B2: set_c_d_set_a,A2: set_c_d_set_a] :
( ( sup_su3175602471750379875_set_a @ A2 @ B2 )
= A2 ) ) ) ).
% semilattice_sup_class.sup.absorb_iff1
thf(fact_1184_coinduct,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X5: ( c > d ) > set_a] :
( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X5 @ ( F @ ( sup_sup_c_d_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) ) )
=> ( ord_le8464990428230162895_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) ) ) ).
% coinduct
thf(fact_1185_def__coinduct,axiom,
! [A3: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,X5: ( c > d ) > set_a] :
( ( A3
= ( comple4054414736020850733_set_a @ F ) )
=> ( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ( ord_le8464990428230162895_set_a @ X5 @ ( F @ ( sup_sup_c_d_set_a @ X5 @ A3 ) ) )
=> ( ord_le8464990428230162895_set_a @ X5 @ A3 ) ) ) ) ).
% def_coinduct
thf(fact_1186_coinduct__lemma,axiom,
! [X5: ( c > d ) > set_a,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( ord_le8464990428230162895_set_a @ X5 @ ( F @ ( sup_sup_c_d_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) ) )
=> ( ( monoto2937423850181994535_set_a @ top_to4267977599310771935_set_a @ ord_le8464990428230162895_set_a @ ord_le8464990428230162895_set_a @ F )
=> ( ord_le8464990428230162895_set_a @ ( sup_sup_c_d_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) @ ( F @ ( sup_sup_c_d_set_a @ X5 @ ( comple4054414736020850733_set_a @ F ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_1187_local_Osup_Oabsorb1,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( sup_c_d_a2 @ A @ B )
= A ) ) ).
% local.sup.absorb1
thf(fact_1188_local_Osup_Oabsorb2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( sup_c_d_a2 @ A @ B )
= B ) ) ).
% local.sup.absorb2
thf(fact_1189_local_Osup_Oabsorb__iff1,axiom,
( smaller_interp_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( sup_c_d_a2 @ A2 @ B2 )
= A2 ) ) ) ).
% local.sup.absorb_iff1
thf(fact_1190_local_Osup_Oabsorb__iff2,axiom,
( smaller_interp_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( sup_c_d_a2 @ A2 @ B2 )
= B2 ) ) ) ).
% local.sup.absorb_iff2
thf(fact_1191_local_Osup_OboundedE,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ B @ C ) @ A )
=> ~ ( ( smaller_interp_c_d_a @ B @ A )
=> ~ ( smaller_interp_c_d_a @ C @ A ) ) ) ).
% local.sup.boundedE
thf(fact_1192_local_Osup_OboundedI,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( smaller_interp_c_d_a @ C @ A )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ B @ C ) @ A ) ) ) ).
% local.sup.boundedI
thf(fact_1193_local_Osup_Ocobounded1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ A @ ( sup_c_d_a2 @ A @ B ) ) ).
% local.sup.cobounded1
thf(fact_1194_local_Osup_Ocobounded2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ B @ ( sup_c_d_a2 @ A @ B ) ) ).
% local.sup.cobounded2
thf(fact_1195_local_Osup_OcoboundedI1,axiom,
! [C: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ C @ A )
=> ( smaller_interp_c_d_a @ C @ ( sup_c_d_a2 @ A @ B ) ) ) ).
% local.sup.coboundedI1
thf(fact_1196_local_Osup_OcoboundedI2,axiom,
! [C: ( c > d ) > set_a,B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ C @ B )
=> ( smaller_interp_c_d_a @ C @ ( sup_c_d_a2 @ A @ B ) ) ) ).
% local.sup.coboundedI2
thf(fact_1197_local_Osup_Omono,axiom,
! [C: ( c > d ) > set_a,A: ( c > d ) > set_a,D2: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ C @ A )
=> ( ( smaller_interp_c_d_a @ D2 @ B )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ C @ D2 ) @ ( sup_c_d_a2 @ A @ B ) ) ) ) ).
% local.sup.mono
thf(fact_1198_local_Osup_OorderE,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( A
= ( sup_c_d_a2 @ A @ B ) ) ) ).
% local.sup.orderE
thf(fact_1199_local_Osup_OorderI,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A
= ( sup_c_d_a2 @ A @ B ) )
=> ( smaller_interp_c_d_a @ B @ A ) ) ).
% local.sup.orderI
thf(fact_1200_local_Osup_Oorder__iff,axiom,
( smaller_interp_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( A2
= ( sup_c_d_a2 @ A2 @ B2 ) ) ) ) ).
% local.sup.order_iff
thf(fact_1201_local_Ole__iff__sup,axiom,
( smaller_interp_c_d_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( sup_c_d_a2 @ X2 @ Y3 )
= Y3 ) ) ) ).
% local.le_iff_sup
thf(fact_1202_local_Ole__supE,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ A @ B ) @ X )
=> ~ ( ( smaller_interp_c_d_a @ A @ X )
=> ~ ( smaller_interp_c_d_a @ B @ X ) ) ) ).
% local.le_supE
thf(fact_1203_local_Ole__supI,axiom,
! [A: ( c > d ) > set_a,X: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ X )
=> ( ( smaller_interp_c_d_a @ B @ X )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ A @ B ) @ X ) ) ) ).
% local.le_supI
thf(fact_1204_local_Ole__supI1,axiom,
! [X: ( c > d ) > set_a,A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ A )
=> ( smaller_interp_c_d_a @ X @ ( sup_c_d_a2 @ A @ B ) ) ) ).
% local.le_supI1
thf(fact_1205_local_Ole__supI2,axiom,
! [X: ( c > d ) > set_a,B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ B )
=> ( smaller_interp_c_d_a @ X @ ( sup_c_d_a2 @ A @ B ) ) ) ).
% local.le_supI2
thf(fact_1206_local_Osup__absorb1,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y2 @ X )
=> ( ( sup_c_d_a2 @ X @ Y2 )
= X ) ) ).
% local.sup_absorb1
thf(fact_1207_local_Osup__absorb2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( sup_c_d_a2 @ X @ Y2 )
= Y2 ) ) ).
% local.sup_absorb2
thf(fact_1208_local_Osup__ge1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ X @ ( sup_c_d_a2 @ X @ Y2 ) ) ).
% local.sup_ge1
thf(fact_1209_local_Osup__ge2,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ Y2 @ ( sup_c_d_a2 @ X @ Y2 ) ) ).
% local.sup_ge2
thf(fact_1210_local_Osup__least,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y2 @ X )
=> ( ( smaller_interp_c_d_a @ Z2 @ X )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ Y2 @ Z2 ) @ X ) ) ) ).
% local.sup_least
thf(fact_1211_local_Osup__mono,axiom,
! [A: ( c > d ) > set_a,C: ( c > d ) > set_a,B: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ C )
=> ( ( smaller_interp_c_d_a @ B @ D2 )
=> ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ A @ B ) @ ( sup_c_d_a2 @ C @ D2 ) ) ) ) ).
% local.sup_mono
thf(fact_1212_local_Osup__unique,axiom,
! [F: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > ( c > d ) > set_a,X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: ( c > d ) > set_a,Y4: ( c > d ) > set_a,Z4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y4 @ X3 )
=> ( ( smaller_interp_c_d_a @ Z4 @ X3 )
=> ( smaller_interp_c_d_a @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_c_d_a2 @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% local.sup_unique
thf(fact_1213_local_Odistrib__sup__le,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ X @ ( inf_c_d_a2 @ Y2 @ Z2 ) ) @ ( inf_c_d_a2 @ ( sup_c_d_a2 @ X @ Y2 ) @ ( sup_c_d_a2 @ X @ Z2 ) ) ) ).
% local.distrib_sup_le
thf(fact_1214_local_Odistrib__inf__le,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ ( inf_c_d_a2 @ X @ Y2 ) @ ( inf_c_d_a2 @ X @ Z2 ) ) @ ( inf_c_d_a2 @ X @ ( sup_c_d_a2 @ Y2 @ Z2 ) ) ) ).
% local.distrib_inf_le
thf(fact_1215_local_Ole__sup__iff,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ X @ Y2 ) @ Z2 )
= ( ( smaller_interp_c_d_a @ X @ Z2 )
& ( smaller_interp_c_d_a @ Y2 @ Z2 ) ) ) ).
% local.le_sup_iff
thf(fact_1216_local_Osup_Obounded__iff,axiom,
! [B: ( c > d ) > set_a,C: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ B @ C ) @ A )
= ( ( smaller_interp_c_d_a @ B @ A )
& ( smaller_interp_c_d_a @ C @ A ) ) ) ).
% local.sup.bounded_iff
thf(fact_1217_local_OInf__fin__le__Sup__fin,axiom,
! [A3: set_c_d_set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ ( lattic1898000229760699588_set_a @ inf_c_d_a2 @ A3 ) @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) ) ) ) ).
% local.Inf_fin_le_Sup_fin
thf(fact_1218_local_OSup__fin_OcoboundedI,axiom,
! [A3: set_c_d_set_a,A: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( member_c_d_set_a @ A @ A3 )
=> ( smaller_interp_c_d_a @ A @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) ) ) ) ).
% local.Sup_fin.coboundedI
thf(fact_1219_local_OSup__fin_OboundedE,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) @ X )
=> ! [A6: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A6 @ A3 )
=> ( smaller_interp_c_d_a @ A6 @ X ) ) ) ) ) ).
% local.Sup_fin.boundedE
thf(fact_1220_local_OSup__fin_OboundedI,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ! [A4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ A4 @ A3 )
=> ( smaller_interp_c_d_a @ A4 @ X ) )
=> ( smaller_interp_c_d_a @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) @ X ) ) ) ) ).
% local.Sup_fin.boundedI
thf(fact_1221_local_OSup__fin_Obounded__iff,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ( finite3330819693523053784_set_a @ A3 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( smaller_interp_c_d_a @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) @ X )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ X2 @ X ) ) ) ) ) ) ).
% local.Sup_fin.bounded_iff
thf(fact_1222_local_OSup__fin_Osubset__imp,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ A3 @ B4 )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( ( finite3330819693523053784_set_a @ B4 )
=> ( smaller_interp_c_d_a @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ A3 ) @ ( lattic5849929604656016644_set_a @ sup_c_d_a2 @ B4 ) ) ) ) ) ).
% local.Sup_fin.subset_imp
thf(fact_1223_local_OInf__eqI,axiom,
! [A3: set_c_d_set_a,X: ( c > d ) > set_a] :
( ! [I2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I2 @ A3 )
=> ( smaller_interp_c_d_a @ X @ I2 ) )
=> ( ! [Y4: ( c > d ) > set_a] :
( ! [I4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ I4 @ A3 )
=> ( smaller_interp_c_d_a @ Y4 @ I4 ) )
=> ( smaller_interp_c_d_a @ Y4 @ X ) )
=> ( ( inf_c_d_a @ A3 )
= X ) ) ) ).
% local.Inf_eqI
thf(fact_1224_local_OInf__lower,axiom,
! [X: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( member_c_d_set_a @ X @ A3 )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ X ) ) ).
% local.Inf_lower
thf(fact_1225_local_OInf__lower2,axiom,
! [U: ( c > d ) > set_a,A3: set_c_d_set_a,V: ( c > d ) > set_a] :
( ( member_c_d_set_a @ U @ A3 )
=> ( ( smaller_interp_c_d_a @ U @ V )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ V ) ) ) ).
% local.Inf_lower2
thf(fact_1226_local_OInf__mono,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ! [B3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ B3 @ B4 )
=> ? [X4: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X4 @ A3 )
& ( smaller_interp_c_d_a @ X4 @ B3 ) ) )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ ( inf_c_d_a @ B4 ) ) ) ).
% local.Inf_mono
thf(fact_1227_local_Ole__Inf__iff,axiom,
! [B: ( c > d ) > set_a,A3: set_c_d_set_a] :
( ( smaller_interp_c_d_a @ B @ ( inf_c_d_a @ A3 ) )
= ( ! [X2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X2 @ A3 )
=> ( smaller_interp_c_d_a @ B @ X2 ) ) ) ) ).
% local.le_Inf_iff
thf(fact_1228_test__axiom__inf,axiom,
! [A3: set_c_d_set_a,Z2: ( c > d ) > set_a] :
( ! [X3: ( c > d ) > set_a] :
( ( member_c_d_set_a @ X3 @ A3 )
=> ( smaller_interp_c_d_a @ Z2 @ X3 ) )
=> ( smaller_interp_c_d_a @ Z2 @ ( inf_c_d_a @ A3 ) ) ) ).
% test_axiom_inf
thf(fact_1229_local_OInf__less__eq,axiom,
! [A3: set_c_d_set_a,U: ( c > d ) > set_a] :
( ! [V2: ( c > d ) > set_a] :
( ( member_c_d_set_a @ V2 @ A3 )
=> ( smaller_interp_c_d_a @ V2 @ U ) )
=> ( ( A3 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ U ) ) ) ).
% local.Inf_less_eq
thf(fact_1230_local_OInf__superset__mono,axiom,
! [B4: set_c_d_set_a,A3: set_c_d_set_a] :
( ( ord_le5982164083705284911_set_a @ B4 @ A3 )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ ( inf_c_d_a @ B4 ) ) ) ).
% local.Inf_superset_mono
thf(fact_1231_local_OInf__le__Sup,axiom,
! [A3: set_c_d_set_a] :
( ( A3 != bot_bo738396921950161403_set_a )
=> ( smaller_interp_c_d_a @ ( inf_c_d_a @ A3 ) @ ( sup_c_d_a @ A3 ) ) ) ).
% local.Inf_le_Sup
thf(fact_1232_local_Oless__eq__Inf__inter,axiom,
! [A3: set_c_d_set_a,B4: set_c_d_set_a] : ( smaller_interp_c_d_a @ ( sup_c_d_a2 @ ( inf_c_d_a @ A3 ) @ ( inf_c_d_a @ B4 ) ) @ ( inf_c_d_a @ ( inf_in754637537901350525_set_a @ A3 @ B4 ) ) ) ).
% local.less_eq_Inf_inter
thf(fact_1233_local_OInf__atLeastAtMost,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( inf_c_d_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ X @ Y2 ) )
= X ) ) ).
% local.Inf_atLeastAtMost
thf(fact_1234_local_Olfp__greatest,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ! [U2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( F @ U2 ) @ U2 )
=> ( smaller_interp_c_d_a @ A3 @ U2 ) )
=> ( smaller_interp_c_d_a @ A3 @ ( comple5961674822413889664_set_a @ inf_c_d_a @ smaller_interp_c_d_a @ F ) ) ) ).
% local.lfp_greatest
thf(fact_1235_local_Olfp__lowerbound,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a,A3: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ ( F @ A3 ) @ A3 )
=> ( smaller_interp_c_d_a @ ( comple5961674822413889664_set_a @ inf_c_d_a @ smaller_interp_c_d_a @ F ) @ A3 ) ) ).
% local.lfp_lowerbound
thf(fact_1236_local_OInf__atLeast,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a @ ( set_at4358065015900363374_set_a @ smaller_interp_c_d_a @ X ) )
= X ) ).
% local.Inf_atLeast
thf(fact_1237_local_OInf__atMost,axiom,
! [X: ( c > d ) > set_a] :
( ( inf_c_d_a @ ( set_atMost_c_d_set_a @ smaller_interp_c_d_a @ X ) )
= empty_interp_c_d_a ) ).
% local.Inf_atMost
thf(fact_1238_local_Onless__le,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( ~ ( less_c_d_a @ A @ B ) )
= ( ~ ( smaller_interp_c_d_a @ A @ B )
| ( A = B ) ) ) ).
% local.nless_le
thf(fact_1239_local_Oneq__le__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A != B )
=> ( ( smaller_interp_c_d_a @ A @ B )
=> ( less_c_d_a @ A @ B ) ) ) ).
% local.neq_le_trans
thf(fact_1240_local_Oless__le__trans,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( less_c_d_a @ X @ Y2 )
=> ( ( smaller_interp_c_d_a @ Y2 @ Z2 )
=> ( less_c_d_a @ X @ Z2 ) ) ) ).
% local.less_le_trans
thf(fact_1241_local_Oless__le__not__le,axiom,
( less_c_d_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X2 @ Y3 )
& ~ ( smaller_interp_c_d_a @ Y3 @ X2 ) ) ) ) ).
% local.less_le_not_le
thf(fact_1242_local_Oless__le,axiom,
( less_c_d_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% local.less_le
thf(fact_1243_local_Oless__imp__le,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( less_c_d_a @ X @ Y2 )
=> ( smaller_interp_c_d_a @ X @ Y2 ) ) ).
% local.less_imp_le
thf(fact_1244_less__def,axiom,
( less_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% less_def
thf(fact_1245_local_Ole__neq__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( A != B )
=> ( less_c_d_a @ A @ B ) ) ) ).
% local.le_neq_trans
thf(fact_1246_local_Ole__less__trans,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a,Z2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( less_c_d_a @ Y2 @ Z2 )
=> ( less_c_d_a @ X @ Z2 ) ) ) ).
% local.le_less_trans
thf(fact_1247_local_Ole__less,axiom,
( smaller_interp_c_d_a
= ( ^ [X2: ( c > d ) > set_a,Y3: ( c > d ) > set_a] :
( ( less_c_d_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% local.le_less
thf(fact_1248_local_Ole__imp__less__or__eq,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( less_c_d_a @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% local.le_imp_less_or_eq
thf(fact_1249_local_OleD,axiom,
! [Y2: ( c > d ) > set_a,X: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Y2 @ X )
=> ~ ( less_c_d_a @ X @ Y2 ) ) ).
% local.leD
thf(fact_1250_local_Oantisym__conv2,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ X @ Y2 )
=> ( ( ~ ( less_c_d_a @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% local.antisym_conv2
thf(fact_1251_local_Oantisym__conv1,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ~ ( less_c_d_a @ X @ Y2 )
=> ( ( smaller_interp_c_d_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% local.antisym_conv1
thf(fact_1252_local_Oorder_Ostrict__trans2,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( less_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( less_c_d_a @ A @ C ) ) ) ).
% local.order.strict_trans2
thf(fact_1253_local_Oorder_Ostrict__trans1,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( less_c_d_a @ B @ C )
=> ( less_c_d_a @ A @ C ) ) ) ).
% local.order.strict_trans1
thf(fact_1254_local_Oorder_Ostrict__implies__order,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( less_c_d_a @ A @ B )
=> ( smaller_interp_c_d_a @ A @ B ) ) ).
% local.order.strict_implies_order
thf(fact_1255_local_Oorder_Ostrict__iff__not,axiom,
( less_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A2 @ B2 )
& ~ ( smaller_interp_c_d_a @ B2 @ A2 ) ) ) ) ).
% local.order.strict_iff_not
thf(fact_1256_local_Oorder_Oorder__iff__strict,axiom,
( smaller_interp_c_d_a
= ( ^ [A2: ( c > d ) > set_a,B2: ( c > d ) > set_a] :
( ( less_c_d_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% local.order.order_iff_strict
thf(fact_1257_local_Odual__order_Ostrict__trans2,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( less_c_d_a @ B @ A )
=> ( ( smaller_interp_c_d_a @ C @ B )
=> ( less_c_d_a @ C @ A ) ) ) ).
% local.dual_order.strict_trans2
thf(fact_1258_local_Odual__order_Ostrict__trans1,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B @ A )
=> ( ( less_c_d_a @ C @ B )
=> ( less_c_d_a @ C @ A ) ) ) ).
% local.dual_order.strict_trans1
thf(fact_1259_local_Odual__order_Ostrict__implies__order,axiom,
! [B: ( c > d ) > set_a,A: ( c > d ) > set_a] :
( ( less_c_d_a @ B @ A )
=> ( smaller_interp_c_d_a @ B @ A ) ) ).
% local.dual_order.strict_implies_order
thf(fact_1260_local_Odual__order_Ostrict__iff__order,axiom,
( less_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% local.dual_order.strict_iff_order
thf(fact_1261_local_Odual__order_Ostrict__iff__not,axiom,
( less_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ B2 @ A2 )
& ~ ( smaller_interp_c_d_a @ A2 @ B2 ) ) ) ) ).
% local.dual_order.strict_iff_not
thf(fact_1262_local_Odual__order_Oorder__iff__strict,axiom,
( smaller_interp_c_d_a
= ( ^ [B2: ( c > d ) > set_a,A2: ( c > d ) > set_a] :
( ( less_c_d_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% local.dual_order.order_iff_strict
thf(fact_1263_local_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a] :
( ( A != B )
=> ( ( smaller_interp_c_d_a @ B @ A )
=> ( less_c_d_a @ B @ A ) ) ) ).
% local.dual_order.not_eq_order_implies_strict
thf(fact_1264_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_1265_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_1266_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_1267_local_OatLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a,D2: ( c > d ) > set_a] :
( ( ord_le5982164083705284911_set_a @ ( set_at2224545791267470424_set_a @ smaller_interp_c_d_a @ A @ B ) @ ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ C @ D2 ) )
= ( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ C @ A )
& ( less_c_d_a @ B @ D2 ) ) ) ) ).
% local.atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1268_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_1269_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_1270_local_OInf__atLeastLessThan,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( less_c_d_a @ X @ Y2 )
=> ( ( inf_c_d_a @ ( set_at2139306834251651636_set_a @ smaller_interp_c_d_a @ less_c_d_a @ X @ Y2 ) )
= X ) ) ).
% local.Inf_atLeastLessThan
thf(fact_1271_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_1272_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_1273_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_1274_local_OSup__greaterThanAtMost,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( less_c_d_a @ X @ Y2 )
=> ( ( sup_c_d_a @ ( set_gr4053032598485390707_set_a @ smaller_interp_c_d_a @ less_c_d_a @ X @ Y2 ) )
= Y2 ) ) ).
% local.Sup_greaterThanAtMost
thf(fact_1275_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_1276_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_1277_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
% Helper facts (3)
thf(help_If_3_1_If_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_T,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( if_c_d_set_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_T,axiom,
! [X: ( c > d ) > set_a,Y2: ( c > d ) > set_a] :
( ( if_c_d_set_a @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( comple4054414736020850733_set_a @ f )
= ( gFP_c_d_a @ f ) ) ).
%------------------------------------------------------------------------------