TPTP Problem File: SLH0674^1.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 : FO_Theory_Rewriting/0068_Ground_MCtxt/prob_00774_031983__18518322_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1679 ( 413 unt; 404 typ; 0 def)
% Number of atoms : 3898 (1084 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12793 ( 438 ~; 37 |; 219 &;10245 @)
% ( 0 <=>;1854 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 46 ( 45 usr)
% Number of type conns : 1591 (1591 >; 0 *; 0 +; 0 <<)
% Number of symbols : 360 ( 359 usr; 39 con; 0-4 aty)
% Number of variables : 3662 ( 91 ^;3413 !; 158 ?;3662 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:01:30.544
%------------------------------------------------------------------------------
% Could-be-implicit typings (45)
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% Explicit typings (359)
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
inj_on5187228193288102844_a_nat: ( produc8175205341597284509ctxt_a > nat ) > set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
inj_on4839580843290501553ctxt_a: ( produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ) > set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Nat__Onat,type,
inj_on_rat_nat: ( rat > nat ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
inj_on8318268736959861138ctxt_a: ( rat > produc8175205341597284509ctxt_a ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Rat__Orat,type,
inj_on_rat_rat: ( rat > rat ) > set_rat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
inj_on1096178645466186887et_rat: ( set_nat > set_rat ) > set_set_nat > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Rat__Orat,type,
monotone_on_nat_rat: set_nat > ( nat > nat > $o ) > ( rat > rat > $o ) > ( nat > rat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
monoto2101069531372151366_a_nat: set_Pr8705490787052945405ctxt_a > ( produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o ) > ( nat > nat > $o ) > ( produc8175205341597284509ctxt_a > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Nat__Onat,type,
monotone_on_rat_nat: set_rat > ( rat > rat > $o ) > ( nat > nat > $o ) > ( rat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Rat__Orat,type,
monotone_on_rat_rat: set_rat > ( rat > rat > $o ) > ( rat > rat > $o ) > ( rat > rat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
monoto1748750089227133045et_nat: set_set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
monoto6151637465316607655ctxt_a: set_se6366015629438403421ctxt_a > ( set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > $o ) > ( set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > $o ) > ( set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
monoto6994633993926870149et_rat: set_set_rat > ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > ( set_rat > set_rat ) > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
the_in1598212552454528472ctxt_a: set_nat > ( nat > produc8175205341597284509ctxt_a ) > produc8175205341597284509ctxt_a > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Rat__Orat,type,
the_inv_into_nat_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
the_in7535783866242650106_a_nat: set_Pr8705490787052945405ctxt_a > ( produc8175205341597284509ctxt_a > nat ) > nat > produc8175205341597284509ctxt_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
the_in7972105333848070131ctxt_a: set_Pr8705490787052945405ctxt_a > ( produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ) > produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Nat__Onat,type,
the_inv_into_rat_nat: set_rat > ( rat > nat ) > nat > rat ).
thf(sy_c_Ground__MCtxt_Ocomp__gmctxt_001tf__a,type,
ground_comp_gmctxt_a: set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001tf__a,type,
ground_GMFun_a: a > list_Ground_gmctxt_a > ground_gmctxt_a ).
thf(sy_c_Ground__MCtxt_Ogmctxt_OGMHole_001tf__a,type,
ground_GMHole_a: ground_gmctxt_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
minus_minus_rat: rat > rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
minus_3912115830687649252ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
minus_3029729196815224238ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr490164551564945991ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
minus_9022085185153369176at_rat: set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
minus_minus_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
uminus5163801920556549748ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Rat__Orat_J,type,
uminus2201863774496077783et_rat: set_rat > set_rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Nat__Onat,type,
hilber5277034221543178913on_nat: ( nat > nat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Rat__Orat,type,
hilber4641904161456683177on_rat: ( rat > rat ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
hilber1942506781820365823ctxt_a: set_nat > ( nat > produc8175205341597284509ctxt_a ) > produc8175205341597284509ctxt_a > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Rat__Orat,type,
hilber2998747136712319222at_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
hilber7880078095608487457_a_nat: set_Pr8705490787052945405ctxt_a > ( produc8175205341597284509ctxt_a > nat ) > nat > produc8175205341597284509ctxt_a ).
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hilber1117725407648346380ctxt_a: set_Pr8705490787052945405ctxt_a > ( produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ) > produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Nat__Onat,type,
hilber3317322552863949046at_nat: set_rat > ( rat > nat ) > nat > rat ).
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hilber1787746602425469943ctxt_a: set_rat > ( rat > produc8175205341597284509ctxt_a ) > produc8175205341597284509ctxt_a > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Rat__Orat,type,
hilber2682192492777453310at_rat: set_rat > ( rat > rat ) > rat > rat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Nat__Onat_J,type,
comple1596078789208929544et_nat: ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
comple5045702773340321787ctxt_a: ( set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ) > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Rat__Orat_J,type,
comple7311222267670308624et_rat: ( set_rat > set_rat ) > set_rat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
sup_sup_rat: rat > rat > rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
sup_su721339912568147456ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a > set_Ground_gmctxt_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
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sup_su1394892571904902097ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
sup_su4770203945505871429at_rat: set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
sup_sup_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
compow_rat_rat: nat > ( rat > rat ) > rat > rat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
compow8708494347934031032et_nat: nat > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Rat__Orat_J_Mt__Set__Oset_It__Rat__Orat_J_J,type,
compow1859248436694711496et_rat: nat > ( set_rat > set_rat ) > set_rat > set_rat ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
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thf(sy_c_Order__Relation_Olinear__order__on_001t__Nat__Onat,type,
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order_3248224809147042642ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr490164551564945991ctxt_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
bot_bo542495261630389981at_rat: set_Pr8928021450653196913at_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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ord_le254475249434039465ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_062_It__Rat__Orat_M_Eo_J_J,type,
ord_le4717968354871517046_rat_o: ( rat > rat > $o ) > ( rat > rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
vimage172744354138440144ctxt_a: ( nat > produc8175205341597284509ctxt_a ) > set_Pr8705490787052945405ctxt_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Rat__Orat,type,
vimage_nat_rat: ( nat > rat ) > set_rat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Nat__Onat,type,
vimage6110315667926561778_a_nat: ( produc8175205341597284509ctxt_a > nat ) > set_nat > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
vimage1911517659246870011ctxt_a: ( produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a ) > set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Set_Ovimage_001t__Rat__Orat_001t__Nat__Onat,type,
vimage_rat_nat: ( rat > nat ) > set_nat > set_rat ).
thf(sy_c_Set_Ovimage_001t__Rat__Orat_001t__Rat__Orat,type,
vimage_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
set_or4029947393144176647an_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or458868116921152288et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_set_rat ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
transi8399219972279719055ctxt_a: set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Nat__Onat,type,
transi3139058337892597825ic_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Transitive__Closure_Oacyclic_001t__Rat__Orat,type,
transi2503928277806102089ic_rat: set_Pr8928021450653196913at_rat > $o ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
transi2468631686036622171ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
transi2118924707453815072ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Rat__Orat,type,
transi2270211269848806677cl_rat: set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
transi3225167780237802813ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
transi6952278968286904510ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Rat__Orat,type,
transi5628869978870870775cl_rat: set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
lex_pr7880170277141576941ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a > set_Pr490164551564945991ctxt_a ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Rat__Orat,type,
lex_prod_nat_rat: set_Pr1261947904930325089at_nat > set_Pr8928021450653196913at_rat > set_Pr8235500881516561735at_rat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Rat__Orat_001t__Nat__Onat,type,
lex_prod_rat_nat: set_Pr8928021450653196913at_rat > set_Pr1261947904930325089at_nat > set_Pr5829696820385088839at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Rat__Orat_001t__Rat__Orat,type,
lex_prod_rat_rat: set_Pr8928021450653196913at_rat > set_Pr8928021450653196913at_rat > set_Pr5371460266479843143at_rat ).
thf(sy_c_Wellfounded_Omax__ext_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
max_ex5522983650612646819ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr2420497028500194409ctxt_a ).
thf(sy_c_Wellfounded_Omax__ext_001t__Nat__Onat,type,
max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
max_ex7231900002760029912ctxt_a: set_Pr490164551564945991ctxt_a > set_Pr671546610523601479ctxt_a ).
thf(sy_c_Wellfounded_Omeasure_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
measur2076251321048797014ctxt_a: ( ground_gmctxt_a > nat ) > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
mlex_p7507609278379928015ctxt_a: ( ground_gmctxt_a > nat ) > set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a ).
thf(sy_c_Wellfounded_Owf_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
wf_Ground_gmctxt_a: set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
wf_Pro4728511896666056454ctxt_a: set_Pr490164551564945991ctxt_a > $o ).
thf(sy_c_Wellfounded_Owf_001t__Rat__Orat,type,
wf_rat: set_Pr8928021450653196913at_rat > $o ).
thf(sy_c_Wfrec_Osame__fst_001t__Ground____MCtxt__Ogmctxt_Itf__a_J_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
same_f3606491088509793834ctxt_a: ( ground_gmctxt_a > $o ) > ( ground_gmctxt_a > set_Pr8705490787052945405ctxt_a ) > set_Pr490164551564945991ctxt_a ).
thf(sy_c_member_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
member3306386364193653941ctxt_a: ground_gmctxt_a > set_Ground_gmctxt_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
member461722496487281478ctxt_a: produc8175205341597284509ctxt_a > set_Pr8705490787052945405ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
member4022424729989878928ctxt_a: produc5976605159854024551ctxt_a > set_Pr490164551564945991ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J,type,
member4615739369455165370at_rat: product_prod_rat_rat > set_Pr8928021450653196913at_rat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_Mt__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
member5333943563032178098ctxt_a: produc301344754507701001ctxt_a > set_Pr2420497028500194409ctxt_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J_J,type,
member2401363563938284688ctxt_a: produc5919946334080622439ctxt_a > set_Pr671546610523601479ctxt_a > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____MCtxt__Ogmctxt_Itf__a_J_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
member6521329796374918822ctxt_a: set_Pr8705490787052945405ctxt_a > set_se6366015629438403421ctxt_a > $o ).
thf(sy_v_C,type,
c: ground_gmctxt_a ).
thf(sy_v_D,type,
d: ground_gmctxt_a ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ c @ d ) @ ground_comp_gmctxt_a ).
% assms
thf(fact_1_comp__gmctxt__refl,axiom,
! [C: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ C ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt_refl
thf(fact_2_prod_Oinject,axiom,
! [X1: set_Ground_gmctxt_a,X2: set_Ground_gmctxt_a,Y1: set_Ground_gmctxt_a,Y2: set_Ground_gmctxt_a] :
( ( ( produc3704910457525973569ctxt_a @ X1 @ X2 )
= ( produc3704910457525973569ctxt_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_3_prod_Oinject,axiom,
! [X1: set_Pr8705490787052945405ctxt_a,X2: set_Pr8705490787052945405ctxt_a,Y1: set_Pr8705490787052945405ctxt_a,Y2: set_Pr8705490787052945405ctxt_a] :
( ( ( produc795262113445460951ctxt_a @ X1 @ X2 )
= ( produc795262113445460951ctxt_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_4_prod_Oinject,axiom,
! [X1: set_nat,X2: set_nat,Y1: set_nat,Y2: set_nat] :
( ( ( produc4532415448927165861et_nat @ X1 @ X2 )
= ( produc4532415448927165861et_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_prod_Oinject,axiom,
! [X1: rat,X2: rat,Y1: rat,Y2: rat] :
( ( ( product_Pair_rat_rat @ X1 @ X2 )
= ( product_Pair_rat_rat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_6_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_7_prod_Oinject,axiom,
! [X1: ground_gmctxt_a,X2: ground_gmctxt_a,Y1: ground_gmctxt_a,Y2: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ X1 @ X2 )
= ( produc6347872285926497493ctxt_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
! [A: set_Ground_gmctxt_a,B: set_Ground_gmctxt_a,A2: set_Ground_gmctxt_a,B2: set_Ground_gmctxt_a] :
( ( ( produc3704910457525973569ctxt_a @ A @ B )
= ( produc3704910457525973569ctxt_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
! [A: set_Pr8705490787052945405ctxt_a,B: set_Pr8705490787052945405ctxt_a,A2: set_Pr8705490787052945405ctxt_a,B2: set_Pr8705490787052945405ctxt_a] :
( ( ( produc795262113445460951ctxt_a @ A @ B )
= ( produc795262113445460951ctxt_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: set_nat,B: set_nat,A2: set_nat,B2: set_nat] :
( ( ( produc4532415448927165861et_nat @ A @ B )
= ( produc4532415448927165861et_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: rat,B: rat,A2: rat,B2: rat] :
( ( ( product_Pair_rat_rat @ A @ B )
= ( product_Pair_rat_rat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ A @ B )
= ( produc6347872285926497493ctxt_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [Y: produc301344754507701001ctxt_a] :
~ ! [A3: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] :
( Y
!= ( produc3704910457525973569ctxt_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_15_old_Oprod_Oexhaust,axiom,
! [Y: produc5919946334080622439ctxt_a] :
~ ! [A3: set_Pr8705490787052945405ctxt_a,B3: set_Pr8705490787052945405ctxt_a] :
( Y
!= ( produc795262113445460951ctxt_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_16_old_Oprod_Oexhaust,axiom,
! [Y: produc7819656566062154093et_nat] :
~ ! [A3: set_nat,B3: set_nat] :
( Y
!= ( produc4532415448927165861et_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_17_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_rat_rat] :
~ ! [A3: rat,B3: rat] :
( Y
!= ( product_Pair_rat_rat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_18_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_19_old_Oprod_Oexhaust,axiom,
! [Y: produc8175205341597284509ctxt_a] :
~ ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( Y
!= ( produc6347872285926497493ctxt_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_20_surj__pair,axiom,
! [P: produc301344754507701001ctxt_a] :
? [X: set_Ground_gmctxt_a,Y3: set_Ground_gmctxt_a] :
( P
= ( produc3704910457525973569ctxt_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_21_surj__pair,axiom,
! [P: produc5919946334080622439ctxt_a] :
? [X: set_Pr8705490787052945405ctxt_a,Y3: set_Pr8705490787052945405ctxt_a] :
( P
= ( produc795262113445460951ctxt_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_22_surj__pair,axiom,
! [P: produc7819656566062154093et_nat] :
? [X: set_nat,Y3: set_nat] :
( P
= ( produc4532415448927165861et_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_23_surj__pair,axiom,
! [P: product_prod_rat_rat] :
? [X: rat,Y3: rat] :
( P
= ( product_Pair_rat_rat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_24_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_25_surj__pair,axiom,
! [P: produc8175205341597284509ctxt_a] :
? [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( P
= ( produc6347872285926497493ctxt_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_26_prod__cases,axiom,
! [P2: produc301344754507701001ctxt_a > $o,P: produc301344754507701001ctxt_a] :
( ! [A3: set_Ground_gmctxt_a,B3: set_Ground_gmctxt_a] : ( P2 @ ( produc3704910457525973569ctxt_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_27_prod__cases,axiom,
! [P2: produc5919946334080622439ctxt_a > $o,P: produc5919946334080622439ctxt_a] :
( ! [A3: set_Pr8705490787052945405ctxt_a,B3: set_Pr8705490787052945405ctxt_a] : ( P2 @ ( produc795262113445460951ctxt_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_28_prod__cases,axiom,
! [P2: produc7819656566062154093et_nat > $o,P: produc7819656566062154093et_nat] :
( ! [A3: set_nat,B3: set_nat] : ( P2 @ ( produc4532415448927165861et_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_29_prod__cases,axiom,
! [P2: product_prod_rat_rat > $o,P: product_prod_rat_rat] :
( ! [A3: rat,B3: rat] : ( P2 @ ( product_Pair_rat_rat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_30_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_31_prod__cases,axiom,
! [P2: produc8175205341597284509ctxt_a > $o,P: produc8175205341597284509ctxt_a] :
( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] : ( P2 @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_32_Pair__inject,axiom,
! [A: set_Ground_gmctxt_a,B: set_Ground_gmctxt_a,A2: set_Ground_gmctxt_a,B2: set_Ground_gmctxt_a] :
( ( ( produc3704910457525973569ctxt_a @ A @ B )
= ( produc3704910457525973569ctxt_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_33_Pair__inject,axiom,
! [A: set_Pr8705490787052945405ctxt_a,B: set_Pr8705490787052945405ctxt_a,A2: set_Pr8705490787052945405ctxt_a,B2: set_Pr8705490787052945405ctxt_a] :
( ( ( produc795262113445460951ctxt_a @ A @ B )
= ( produc795262113445460951ctxt_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_34_Pair__inject,axiom,
! [A: set_nat,B: set_nat,A2: set_nat,B2: set_nat] :
( ( ( produc4532415448927165861et_nat @ A @ B )
= ( produc4532415448927165861et_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_35_Pair__inject,axiom,
! [A: rat,B: rat,A2: rat,B2: rat] :
( ( ( product_Pair_rat_rat @ A @ B )
= ( product_Pair_rat_rat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_36_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_37_Pair__inject,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A2: ground_gmctxt_a,B2: ground_gmctxt_a] :
( ( ( produc6347872285926497493ctxt_a @ A @ B )
= ( produc6347872285926497493ctxt_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_ssubst__Pair__rhs,axiom,
! [R: set_Ground_gmctxt_a,S: set_Ground_gmctxt_a,R2: set_Pr2420497028500194409ctxt_a,S2: set_Ground_gmctxt_a] :
( ( member5333943563032178098ctxt_a @ ( produc3704910457525973569ctxt_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member5333943563032178098ctxt_a @ ( produc3704910457525973569ctxt_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_39_ssubst__Pair__rhs,axiom,
! [R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a,R2: set_Pr671546610523601479ctxt_a,S2: set_Pr8705490787052945405ctxt_a] :
( ( member2401363563938284688ctxt_a @ ( produc795262113445460951ctxt_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member2401363563938284688ctxt_a @ ( produc795262113445460951ctxt_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_40_ssubst__Pair__rhs,axiom,
! [R: set_nat,S: set_nat,R2: set_Pr5488025237498180813et_nat,S2: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_41_ssubst__Pair__rhs,axiom,
! [R: rat,S: rat,R2: set_Pr8928021450653196913at_rat,S2: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_42_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_43_ssubst__Pair__rhs,axiom,
! [R: ground_gmctxt_a,S: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,S2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_44_curry__conv,axiom,
( produc1032961633260156623xt_a_o
= ( ^ [F: produc8175205341597284509ctxt_a > $o,A4: ground_gmctxt_a,B4: ground_gmctxt_a] : ( F @ ( produc6347872285926497493ctxt_a @ A4 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_45_curryI,axiom,
! [F2: produc301344754507701001ctxt_a > $o,A: set_Ground_gmctxt_a,B: set_Ground_gmctxt_a] :
( ( F2 @ ( produc3704910457525973569ctxt_a @ A @ B ) )
=> ( produc3700491911096464995xt_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_46_curryI,axiom,
! [F2: produc5919946334080622439ctxt_a > $o,A: set_Pr8705490787052945405ctxt_a,B: set_Pr8705490787052945405ctxt_a] :
( ( F2 @ ( produc795262113445460951ctxt_a @ A @ B ) )
=> ( produc2410358161934032845xt_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_47_curryI,axiom,
! [F2: produc7819656566062154093et_nat > $o,A: set_nat,B: set_nat] :
( ( F2 @ ( produc4532415448927165861et_nat @ A @ B ) )
=> ( produc8481458855017059839_nat_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_48_curryI,axiom,
! [F2: product_prod_rat_rat > $o,A: rat,B: rat] :
( ( F2 @ ( product_Pair_rat_rat @ A @ B ) )
=> ( produc9054346771070301275_rat_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_49_curryI,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( F2 @ ( product_Pair_nat_nat @ A @ B ) )
=> ( produc1310100445399344235_nat_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_50_curryI,axiom,
! [F2: produc8175205341597284509ctxt_a > $o,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( F2 @ ( produc6347872285926497493ctxt_a @ A @ B ) )
=> ( produc1032961633260156623xt_a_o @ F2 @ A @ B ) ) ).
% curryI
thf(fact_51_comp__gmctxt_OGMHole2,axiom,
! [C: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C @ ground_GMHole_a ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt.GMHole2
thf(fact_52_comp__gmctxt_OGMHole1,axiom,
! [D: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ground_GMHole_a @ D ) @ ground_comp_gmctxt_a ) ).
% comp_gmctxt.GMHole1
thf(fact_53_swap__simp,axiom,
! [X3: set_Ground_gmctxt_a,Y: set_Ground_gmctxt_a] :
( ( produc2833711947310153697ctxt_a @ ( produc3704910457525973569ctxt_a @ X3 @ Y ) )
= ( produc3704910457525973569ctxt_a @ Y @ X3 ) ) ).
% swap_simp
thf(fact_54_swap__simp,axiom,
! [X3: set_Pr8705490787052945405ctxt_a,Y: set_Pr8705490787052945405ctxt_a] :
( ( produc4292740997436692855ctxt_a @ ( produc795262113445460951ctxt_a @ X3 @ Y ) )
= ( produc795262113445460951ctxt_a @ Y @ X3 ) ) ).
% swap_simp
thf(fact_55_swap__simp,axiom,
! [X3: set_nat,Y: set_nat] :
( ( produc6238068474642164549et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y ) )
= ( produc4532415448927165861et_nat @ Y @ X3 ) ) ).
% swap_simp
thf(fact_56_swap__simp,axiom,
! [X3: rat,Y: rat] :
( ( product_swap_rat_rat @ ( product_Pair_rat_rat @ X3 @ Y ) )
= ( product_Pair_rat_rat @ Y @ X3 ) ) ).
% swap_simp
thf(fact_57_swap__simp,axiom,
! [X3: nat,Y: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y ) )
= ( product_Pair_nat_nat @ Y @ X3 ) ) ).
% swap_simp
thf(fact_58_swap__simp,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( produc1958965069685172853ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
= ( produc6347872285926497493ctxt_a @ Y @ X3 ) ) ).
% swap_simp
thf(fact_59_apsnd__conv,axiom,
! [F2: set_Ground_gmctxt_a > set_Ground_gmctxt_a,X3: set_Ground_gmctxt_a,Y: set_Ground_gmctxt_a] :
( ( produc7699379982755277692ctxt_a @ F2 @ ( produc3704910457525973569ctxt_a @ X3 @ Y ) )
= ( produc3704910457525973569ctxt_a @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_60_apsnd__conv,axiom,
! [F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a,X3: set_Pr8705490787052945405ctxt_a,Y: set_Pr8705490787052945405ctxt_a] :
( ( produc153265931538179903ctxt_a @ F2 @ ( produc795262113445460951ctxt_a @ X3 @ Y ) )
= ( produc795262113445460951ctxt_a @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_61_apsnd__conv,axiom,
! [F2: set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ( produc8013639195810889106et_nat @ F2 @ ( produc4532415448927165861et_nat @ X3 @ Y ) )
= ( produc4532415448927165861et_nat @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_62_apsnd__conv,axiom,
! [F2: rat > rat,X3: rat,Y: rat] :
( ( produc2915153416496185480at_rat @ F2 @ ( product_Pair_rat_rat @ X3 @ Y ) )
= ( product_Pair_rat_rat @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_63_apsnd__conv,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( produc3094765310956027504at_nat @ F2 @ ( product_Pair_nat_nat @ X3 @ Y ) )
= ( product_Pair_nat_nat @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_64_apsnd__conv,axiom,
! [F2: ground_gmctxt_a > ground_gmctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( produc4177324639448251226ctxt_a @ F2 @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
= ( produc6347872285926497493ctxt_a @ X3 @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_65_swap__swap,axiom,
! [P: produc8175205341597284509ctxt_a] :
( ( produc1958965069685172853ctxt_a @ ( produc1958965069685172853ctxt_a @ P ) )
= P ) ).
% swap_swap
thf(fact_66_curryD,axiom,
! [F2: produc301344754507701001ctxt_a > $o,A: set_Ground_gmctxt_a,B: set_Ground_gmctxt_a] :
( ( produc3700491911096464995xt_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc3704910457525973569ctxt_a @ A @ B ) ) ) ).
% curryD
thf(fact_67_curryD,axiom,
! [F2: produc5919946334080622439ctxt_a > $o,A: set_Pr8705490787052945405ctxt_a,B: set_Pr8705490787052945405ctxt_a] :
( ( produc2410358161934032845xt_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc795262113445460951ctxt_a @ A @ B ) ) ) ).
% curryD
thf(fact_68_curryD,axiom,
! [F2: produc7819656566062154093et_nat > $o,A: set_nat,B: set_nat] :
( ( produc8481458855017059839_nat_o @ F2 @ A @ B )
=> ( F2 @ ( produc4532415448927165861et_nat @ A @ B ) ) ) ).
% curryD
thf(fact_69_curryD,axiom,
! [F2: product_prod_rat_rat > $o,A: rat,B: rat] :
( ( produc9054346771070301275_rat_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_rat_rat @ A @ B ) ) ) ).
% curryD
thf(fact_70_curryD,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryD
thf(fact_71_curryD,axiom,
! [F2: produc8175205341597284509ctxt_a > $o,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( produc1032961633260156623xt_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ).
% curryD
thf(fact_72_curryE,axiom,
! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
=> ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryE
thf(fact_73_curryE,axiom,
! [F2: produc8175205341597284509ctxt_a > $o,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( produc1032961633260156623xt_a_o @ F2 @ A @ B )
=> ( F2 @ ( produc6347872285926497493ctxt_a @ A @ B ) ) ) ).
% curryE
thf(fact_74_pair__in__swap__image,axiom,
! [Y: ground_gmctxt_a,X3: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ X3 ) @ ( image_4216501698144419397ctxt_a @ produc1958965069685172853ctxt_a @ A5 ) )
= ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_75_sup__gmctxt_Ocases,axiom,
! [X3: produc8175205341597284509ctxt_a] :
( ! [D2: ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ground_GMHole_a @ D2 ) )
=> ( ! [V: a,Va: list_Ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ( ground_GMFun_a @ V @ Va ) @ ground_GMHole_a ) )
=> ~ ! [F3: a,Cs: list_Ground_gmctxt_a,G: a,Ds: list_Ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ( ground_GMFun_a @ F3 @ Cs ) @ ( ground_GMFun_a @ G @ Ds ) ) ) ) ) ).
% sup_gmctxt.cases
thf(fact_76_sup__gmctxt__args_Ocases,axiom,
! [X3: produc8175205341597284509ctxt_a] :
( ! [D2: ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ground_GMHole_a @ D2 ) )
=> ( ! [V: a,Va: list_Ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ( ground_GMFun_a @ V @ Va ) @ ground_GMHole_a ) )
=> ~ ! [F3: a,Cs: list_Ground_gmctxt_a,G: a,Ds: list_Ground_gmctxt_a] :
( X3
!= ( produc6347872285926497493ctxt_a @ ( ground_GMFun_a @ F3 @ Cs ) @ ( ground_GMFun_a @ G @ Ds ) ) ) ) ) ).
% sup_gmctxt_args.cases
thf(fact_77_mem__Collect__eq,axiom,
! [A: produc8175205341597284509ctxt_a,P2: produc8175205341597284509ctxt_a > $o] :
( ( member461722496487281478ctxt_a @ A @ ( collec5218855173738699400ctxt_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ( collec5218855173738699400ctxt_a
@ ^ [X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A5: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_81_mem__case__prodE,axiom,
! [Z: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a > ground_gmctxt_a > set_Pr8705490787052945405ctxt_a,P: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Z @ ( produc4706414670047607955ctxt_a @ C2 @ P ) )
=> ~ ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( P
= ( produc6347872285926497493ctxt_a @ X @ Y3 ) )
=> ~ ( member461722496487281478ctxt_a @ Z @ ( C2 @ X @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_82_mem__case__prodE,axiom,
! [Z: nat,C2: ground_gmctxt_a > ground_gmctxt_a > set_nat,P: produc8175205341597284509ctxt_a] :
( ( member_nat @ Z @ ( produc2115147248861261424et_nat @ C2 @ P ) )
=> ~ ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( P
= ( produc6347872285926497493ctxt_a @ X @ Y3 ) )
=> ~ ( member_nat @ Z @ ( C2 @ X @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_83_image__eqI,axiom,
! [B: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ B @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_84_image__eqI,axiom,
! [B: nat,F2: produc8175205341597284509ctxt_a > nat,X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member_nat @ B @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_85_image__eqI,axiom,
! [B: rat,F2: nat > rat,X3: nat,A5: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_86_image__eqI,axiom,
! [B: produc8175205341597284509ctxt_a,F2: nat > produc8175205341597284509ctxt_a,X3: nat,A5: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ B @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_87_image__eqI,axiom,
! [B: nat,F2: nat > nat,X3: nat,A5: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_88_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_rat @ C @ A5 ) )
= ( Inf @ ( image_nat_rat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_89_Inf_OINF__cong,axiom,
! [A5: set_nat,B5: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A5 ) )
= ( Inf @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_90_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_rat @ C @ A5 ) )
= ( Sup @ ( image_nat_rat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_91_Sup_OSUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A5 ) )
= ( Sup @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_92_imageI,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ X3 ) @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_93_imageI,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > nat] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_94_imageI,axiom,
! [X3: nat,A5: set_nat,F2: nat > rat] :
( ( member_nat @ X3 @ A5 )
=> ( member_rat @ ( F2 @ X3 ) @ ( image_nat_rat @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_95_imageI,axiom,
! [X3: nat,A5: set_nat,F2: nat > produc8175205341597284509ctxt_a] :
( ( member_nat @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ X3 ) @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_96_imageI,axiom,
! [X3: nat,A5: set_nat,F2: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A5 ) ) ) ).
% imageI
thf(fact_97_image__iff,axiom,
! [Z: rat,F2: nat > rat,A5: set_nat] :
( ( member_rat @ Z @ ( image_nat_rat @ F2 @ A5 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A5 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_98_image__iff,axiom,
! [Z: nat,F2: nat > nat,A5: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F2 @ A5 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A5 )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_99_bex__imageD,axiom,
! [F2: nat > rat,A5: set_nat,P2: rat > $o] :
( ? [X5: rat] :
( ( member_rat @ X5 @ ( image_nat_rat @ F2 @ A5 ) )
& ( P2 @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P2 @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_100_bex__imageD,axiom,
! [F2: nat > nat,A5: set_nat,P2: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F2 @ A5 ) )
& ( P2 @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A5 )
& ( P2 @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_101_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > rat,G2: nat > rat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_rat @ F2 @ M )
= ( image_nat_rat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_102_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > nat,G2: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_nat @ F2 @ M )
= ( image_nat_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_103_ball__imageD,axiom,
! [F2: nat > rat,A5: set_nat,P2: rat > $o] :
( ! [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F2 @ A5 ) )
=> ( P2 @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( P2 @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_104_ball__imageD,axiom,
! [F2: nat > nat,A5: set_nat,P2: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F2 @ A5 ) )
=> ( P2 @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( P2 @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_105_rev__image__eqI,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member461722496487281478ctxt_a @ B @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B: nat,F2: produc8175205341597284509ctxt_a > nat] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_107_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: rat,F2: nat > rat] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_108_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: produc8175205341597284509ctxt_a,F2: nat > produc8175205341597284509ctxt_a] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member461722496487281478ctxt_a @ B @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_109_rev__image__eqI,axiom,
! [X3: nat,A5: set_nat,B: nat,F2: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_110_UNIV__I,axiom,
! [X3: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X3 @ top_to2898181658028262733ctxt_a ) ).
% UNIV_I
thf(fact_111_UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_112_UNIV__I,axiom,
! [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ).
% UNIV_I
thf(fact_113_UNIV__witness,axiom,
? [X: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X @ top_to2898181658028262733ctxt_a ) ).
% UNIV_witness
thf(fact_114_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_115_UNIV__witness,axiom,
? [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_116_UNIV__eq__I,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X @ A5 )
=> ( top_to2898181658028262733ctxt_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_117_UNIV__eq__I,axiom,
! [A5: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A5 )
=> ( top_top_set_nat = A5 ) ) ).
% UNIV_eq_I
thf(fact_118_UNIV__eq__I,axiom,
! [A5: set_rat] :
( ! [X: rat] : ( member_rat @ X @ A5 )
=> ( top_top_set_rat = A5 ) ) ).
% UNIV_eq_I
thf(fact_119_range__eqI,axiom,
! [B: rat,F2: nat > rat,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_120_range__eqI,axiom,
! [B: produc8175205341597284509ctxt_a,F2: nat > produc8175205341597284509ctxt_a,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member461722496487281478ctxt_a @ B @ ( image_43530617296285190ctxt_a @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_121_range__eqI,axiom,
! [B: nat,F2: nat > nat,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_122_range__eqI,axiom,
! [B: produc8175205341597284509ctxt_a,F2: rat > produc8175205341597284509ctxt_a,X3: rat] :
( ( B
= ( F2 @ X3 ) )
=> ( member461722496487281478ctxt_a @ B @ ( image_9112142474756165118ctxt_a @ F2 @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_123_range__eqI,axiom,
! [B: nat,F2: rat > nat,X3: rat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_rat_nat @ F2 @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_124_rangeI,axiom,
! [F2: nat > rat,X3: nat] : ( member_rat @ ( F2 @ X3 ) @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_125_rangeI,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,X3: nat] : ( member461722496487281478ctxt_a @ ( F2 @ X3 ) @ ( image_43530617296285190ctxt_a @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_126_rangeI,axiom,
! [F2: nat > nat,X3: nat] : ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_127_rangeI,axiom,
! [F2: rat > produc8175205341597284509ctxt_a,X3: rat] : ( member461722496487281478ctxt_a @ ( F2 @ X3 ) @ ( image_9112142474756165118ctxt_a @ F2 @ top_top_set_rat ) ) ).
% rangeI
thf(fact_128_rangeI,axiom,
! [F2: rat > nat,X3: rat] : ( member_nat @ ( F2 @ X3 ) @ ( image_rat_nat @ F2 @ top_top_set_rat ) ) ).
% rangeI
thf(fact_129_iso__tuple__UNIV__I,axiom,
! [X3: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X3 @ top_to2898181658028262733ctxt_a ) ).
% iso_tuple_UNIV_I
thf(fact_130_iso__tuple__UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_131_iso__tuple__UNIV__I,axiom,
! [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_132_surj__def,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X4: nat] :
( Y4
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_133_surj__def,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y4: rat] :
? [X4: nat] :
( Y4
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_134_surj__def,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X4: rat] :
( Y4
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_135_surj__def,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y4: rat] :
? [X4: rat] :
( Y4
= ( F2 @ X4 ) ) ) ) ).
% surj_def
thf(fact_136_surjI,axiom,
! [G2: nat > nat,F2: nat > nat] :
( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_137_surjI,axiom,
! [G2: nat > rat,F2: rat > nat] :
( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( image_nat_rat @ G2 @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_138_surjI,axiom,
! [G2: rat > nat,F2: nat > rat] :
( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( image_rat_nat @ G2 @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_139_surjI,axiom,
! [G2: rat > rat,F2: rat > rat] :
( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( image_rat_rat @ G2 @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_140_surjE,axiom,
! [F2: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y
!= ( F2 @ X ) ) ) ).
% surjE
thf(fact_141_surjE,axiom,
! [F2: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X: nat] :
( Y
!= ( F2 @ X ) ) ) ).
% surjE
thf(fact_142_surjE,axiom,
! [F2: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X: rat] :
( Y
!= ( F2 @ X ) ) ) ).
% surjE
thf(fact_143_surjE,axiom,
! [F2: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X: rat] :
( Y
!= ( F2 @ X ) ) ) ).
% surjE
thf(fact_144_surjD,axiom,
! [F2: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y
= ( F2 @ X ) ) ) ).
% surjD
thf(fact_145_surjD,axiom,
! [F2: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ? [X: nat] :
( Y
= ( F2 @ X ) ) ) ).
% surjD
thf(fact_146_surjD,axiom,
! [F2: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ? [X: rat] :
( Y
= ( F2 @ X ) ) ) ).
% surjD
thf(fact_147_surjD,axiom,
! [F2: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ? [X: rat] :
( Y
= ( F2 @ X ) ) ) ).
% surjD
thf(fact_148_map__prod__surj,axiom,
! [F2: nat > nat,G2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F2 @ G2 ) @ top_to4669805908274784177at_nat )
= top_to4669805908274784177at_nat ) ) ) ).
% map_prod_surj
thf(fact_149_map__prod__surj,axiom,
! [F2: nat > nat,G2: nat > rat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_rat @ G2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_1660685944454060877at_rat @ ( produc6342756635244135234at_rat @ F2 @ G2 ) @ top_to4669805908274784177at_nat )
= top_to7513191607651882425at_rat ) ) ) ).
% map_prod_surj
thf(fact_150_map__prod__surj,axiom,
! [F2: nat > nat,G2: rat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_rat_nat @ G2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_7659725744217795901at_nat @ ( produc6661332051395765058at_nat @ F2 @ G2 ) @ top_to7513191607651882425at_rat )
= top_to4669805908274784177at_nat ) ) ) ).
% map_prod_surj
thf(fact_151_map__prod__surj,axiom,
! [F2: nat > nat,G2: rat > rat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_rat_rat @ G2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_4151497185824399173at_rat @ ( produc6026201991309269322at_rat @ F2 @ G2 ) @ top_to7513191607651882425at_rat )
= top_to7513191607651882425at_rat ) ) ) ).
% map_prod_surj
thf(fact_152_map__prod__surj,axiom,
! [F2: nat > rat,G2: nat > nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_4852359858912591693at_nat @ ( produc7749959504892150594at_nat @ F2 @ G2 ) @ top_to4669805908274784177at_nat )
= top_to269121717765781945at_nat ) ) ) ).
% map_prod_surj
thf(fact_153_map__prod__surj,axiom,
! [F2: nat > rat,G2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ( image_nat_rat @ G2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_1344131300519194965at_rat @ ( produc7114829444805654858at_rat @ F2 @ G2 ) @ top_to4669805908274784177at_nat )
= top_to3112507417142880193at_rat ) ) ) ).
% map_prod_surj
thf(fact_154_map__prod__surj,axiom,
! [F2: nat > rat,G2: rat > nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ( image_rat_nat @ G2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_7343171100282929989at_nat @ ( produc7433404860957284682at_nat @ F2 @ G2 ) @ top_to7513191607651882425at_rat )
= top_to269121717765781945at_nat ) ) ) ).
% map_prod_surj
thf(fact_155_map__prod__surj,axiom,
! [F2: nat > rat,G2: rat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ( image_rat_rat @ G2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_3834942541889533261at_rat @ ( produc6798274800870788946at_rat @ F2 @ G2 ) @ top_to7513191607651882425at_rat )
= top_to3112507417142880193at_rat ) ) ) ).
% map_prod_surj
thf(fact_156_map__prod__surj,axiom,
! [F2: rat > nat,G2: nat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_7033956296883226941at_nat @ ( produc7109947960609612610at_nat @ F2 @ G2 ) @ top_to269121717765781945at_nat )
= top_to4669805908274784177at_nat ) ) ) ).
% map_prod_surj
thf(fact_157_map__prod__surj,axiom,
! [F2: rat > nat,G2: nat > rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( ( image_nat_rat @ G2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_3525727738489830213at_rat @ ( produc6474817900523116874at_rat @ F2 @ G2 ) @ top_to269121717765781945at_nat )
= top_to7513191607651882425at_rat ) ) ) ).
% map_prod_surj
thf(fact_158_map__prod__imageI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a,G2: ground_gmctxt_a > ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ A ) @ ( G2 @ B ) ) @ ( image_4216501698144419397ctxt_a @ ( produc2051271937606247026ctxt_a @ F2 @ G2 ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_159_map__prod__simp,axiom,
! [F2: ground_gmctxt_a > ground_gmctxt_a,G2: ground_gmctxt_a > ground_gmctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( produc2051271937606247026ctxt_a @ F2 @ G2 @ ( produc6347872285926497493ctxt_a @ A @ B ) )
= ( produc6347872285926497493ctxt_a @ ( F2 @ A ) @ ( G2 @ B ) ) ) ).
% map_prod_simp
thf(fact_160_prod_Ocollapse,axiom,
! [Prod: produc8175205341597284509ctxt_a] :
( ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ Prod ) @ ( produc3308216634374912259ctxt_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_161_Product__Type_OCollect__case__prodD,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ X3 @ ( collec5218855173738699400ctxt_a @ ( produc4116768209347204654xt_a_o @ A5 ) ) )
=> ( A5 @ ( produc844137399493384385ctxt_a @ X3 ) @ ( produc3308216634374912259ctxt_a @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_162_surjective__pairing,axiom,
! [T: produc8175205341597284509ctxt_a] :
( T
= ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ T ) @ ( produc3308216634374912259ctxt_a @ T ) ) ) ).
% surjective_pairing
thf(fact_163_prod_Oexhaust__sel,axiom,
! [Prod: produc8175205341597284509ctxt_a] :
( Prod
= ( produc6347872285926497493ctxt_a @ ( produc844137399493384385ctxt_a @ Prod ) @ ( produc3308216634374912259ctxt_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_164_top__empty__eq,axiom,
( top_to5133955659620539344xt_a_o
= ( ^ [X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ top_to2898181658028262733ctxt_a ) ) ) ).
% top_empty_eq
thf(fact_165_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_166_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X4: rat] : ( member_rat @ X4 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_167_prod_Oswap__def,axiom,
( produc1958965069685172853ctxt_a
= ( ^ [P3: produc8175205341597284509ctxt_a] : ( produc6347872285926497493ctxt_a @ ( produc3308216634374912259ctxt_a @ P3 ) @ ( produc844137399493384385ctxt_a @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_168_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_169_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_170_snd__conv,axiom,
! [X1: ground_gmctxt_a,X2: ground_gmctxt_a] :
( ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_171_snd__eqD,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_172_fst__conv,axiom,
! [X1: ground_gmctxt_a,X2: ground_gmctxt_a] :
( ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_173_fst__eqD,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,A: ground_gmctxt_a] :
( ( ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_174_prod__fun__imageE,axiom,
! [C2: produc8175205341597284509ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a,G2: ground_gmctxt_a > ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( image_4216501698144419397ctxt_a @ ( produc2051271937606247026ctxt_a @ F2 @ G2 ) @ R2 ) )
=> ~ ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( C2
= ( produc6347872285926497493ctxt_a @ ( F2 @ X ) @ ( G2 @ Y3 ) ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_175_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: ground_gmctxt_a > ground_gmctxt_a > $o,X3: ground_gmctxt_a,Y: ground_gmctxt_a,A: produc8175205341597284509ctxt_a] :
( ( P2 @ X3 @ Y )
=> ( ( A
= ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
=> ( P2 @ ( produc844137399493384385ctxt_a @ A ) @ ( produc3308216634374912259ctxt_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_176_conjI__realizer,axiom,
! [P2: ground_gmctxt_a > $o,P: ground_gmctxt_a,Q: ground_gmctxt_a > $o,Q2: ground_gmctxt_a] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ P @ Q2 ) ) )
& ( Q @ ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_177_exI__realizer,axiom,
! [P2: ground_gmctxt_a > ground_gmctxt_a > $o,Y: ground_gmctxt_a,X3: ground_gmctxt_a] :
( ( P2 @ Y @ X3 )
=> ( P2 @ ( produc3308216634374912259ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) ) @ ( produc844137399493384385ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_178_eq__snd__iff,axiom,
! [B: ground_gmctxt_a,P: produc8175205341597284509ctxt_a] :
( ( B
= ( produc3308216634374912259ctxt_a @ P ) )
= ( ? [A4: ground_gmctxt_a] :
( P
= ( produc6347872285926497493ctxt_a @ A4 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_179_sndI,axiom,
! [X3: produc8175205341597284509ctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( X3
= ( produc6347872285926497493ctxt_a @ Y @ Z ) )
=> ( ( produc3308216634374912259ctxt_a @ X3 )
= Z ) ) ).
% sndI
thf(fact_180_eq__fst__iff,axiom,
! [A: ground_gmctxt_a,P: produc8175205341597284509ctxt_a] :
( ( A
= ( produc844137399493384385ctxt_a @ P ) )
= ( ? [B4: ground_gmctxt_a] :
( P
= ( produc6347872285926497493ctxt_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_181_apfst__conv,axiom,
! [F2: ground_gmctxt_a > ground_gmctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( produc200381826416914200ctxt_a @ F2 @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) )
= ( produc6347872285926497493ctxt_a @ ( F2 @ X3 ) @ Y ) ) ).
% apfst_conv
thf(fact_182_fstI,axiom,
! [X3: produc8175205341597284509ctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( X3
= ( produc6347872285926497493ctxt_a @ Y @ Z ) )
=> ( ( produc844137399493384385ctxt_a @ X3 )
= Y ) ) ).
% fstI
thf(fact_183_apfst__convE,axiom,
! [Q2: produc8175205341597284509ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a,P: produc8175205341597284509ctxt_a] :
( ( Q2
= ( produc200381826416914200ctxt_a @ F2 @ P ) )
=> ~ ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( P
= ( produc6347872285926497493ctxt_a @ X @ Y3 ) )
=> ( Q2
!= ( produc6347872285926497493ctxt_a @ ( F2 @ X ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_184_image2__eqI,axiom,
! [B: ground_gmctxt_a,F2: produc8175205341597284509ctxt_a > ground_gmctxt_a,X3: produc8175205341597284509ctxt_a,C2: ground_gmctxt_a,G2: produc8175205341597284509ctxt_a > ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( B
= ( F2 @ X3 ) )
=> ( ( C2
= ( G2 @ X3 ) )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( bNF_Gr5342263052909934465ctxt_a @ A5 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_185_image2__eqI,axiom,
! [B: ground_gmctxt_a,F2: nat > ground_gmctxt_a,X3: nat,C2: ground_gmctxt_a,G2: nat > ground_gmctxt_a,A5: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( C2
= ( G2 @ X3 ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( bNF_Gr3868524873790170984ctxt_a @ A5 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_186_asymI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ X ) @ R ) )
=> ( asym_o1385093689995826796ctxt_a @ top_to3933008015346724740ctxt_a @ R ) ) ).
% asymI
thf(fact_187_asymI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) )
=> ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% asymI
thf(fact_188_asymI,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ! [X: rat,Y3: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X @ Y3 ) @ R )
=> ~ ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Y3 @ X ) @ R ) )
=> ( asym_on_rat @ top_top_set_rat @ R ) ) ).
% asymI
thf(fact_189_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_190_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_191_subsetI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( member461722496487281478ctxt_a @ X @ B5 ) )
=> ( ord_le2690539916662146973ctxt_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_192_subsetI,axiom,
! [A5: set_nat,B5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B5 ) )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).
% subsetI
thf(fact_193_asym__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a,Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y3 @ A5 )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Y3 ) @ R )
=> ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y3 @ X ) @ R ) ) ) )
=> ( asym_o3378537592941112015ctxt_a @ A5 @ R ) ) ).
% asym_onI
thf(fact_194_asym__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) ) ) )
=> ( asym_on_nat @ A5 @ R ) ) ).
% asym_onI
thf(fact_195_asym__onI,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ A5 )
=> ( ( member3306386364193653941ctxt_a @ Y3 @ A5 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ X ) @ R ) ) ) )
=> ( asym_o1385093689995826796ctxt_a @ A5 @ R ) ) ).
% asym_onI
thf(fact_196_subset__iff,axiom,
( ord_le2690539916662146973ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
! [T2: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ T2 @ A6 )
=> ( member461722496487281478ctxt_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_197_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_198_subset__eq,axiom,
( ord_le2690539916662146973ctxt_a
= ( ^ [A6: set_Pr8705490787052945405ctxt_a,B6: set_Pr8705490787052945405ctxt_a] :
! [X4: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X4 @ A6 )
=> ( member461722496487281478ctxt_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_199_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_200_subsetD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,C2: produc8175205341597284509ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_201_subsetD,axiom,
! [A5: set_nat,B5: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_202_in__mono,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( member461722496487281478ctxt_a @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_203_in__mono,axiom,
! [A5: set_nat,B5: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_204_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_205_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_206_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_207_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_209_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_210_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_211_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_212_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_213_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_214_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_215_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_216_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_217_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_218_order__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_219_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_220_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_221_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_222_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_223_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_224_le__cases3,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_225_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_226_subrelI,axiom,
! [R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ S ) )
=> ( ord_le2690539916662146973ctxt_a @ R @ S ) ) ).
% subrelI
thf(fact_227_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_228_top__greatest,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% top_greatest
thf(fact_229_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_230_top_Oextremum__unique,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
= ( A = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_231_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_232_top_Oextremum__uniqueI,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A )
=> ( A = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_233_image__mono,axiom,
! [A5: set_nat,B5: set_nat,F2: nat > rat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_234_image__mono,axiom,
! [A5: set_nat,B5: set_nat,F2: nat > nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_235_image__subsetI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ X ) @ B5 ) )
=> ( ord_le2690539916662146973ctxt_a @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_236_image__subsetI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > nat,B5: set_nat] :
( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( member_nat @ ( F2 @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_237_image__subsetI,axiom,
! [A5: set_nat,F2: nat > rat,B5: set_rat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_rat @ ( F2 @ X ) @ B5 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_238_image__subsetI,axiom,
! [A5: set_nat,F2: nat > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ X ) @ B5 ) )
=> ( ord_le2690539916662146973ctxt_a @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_239_image__subsetI,axiom,
! [A5: set_nat,F2: nat > nat,B5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ ( F2 @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_240_subset__imageE,axiom,
! [B5: set_rat,F2: nat > rat,A5: set_nat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A5 )
=> ( B5
!= ( image_nat_rat @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_241_subset__imageE,axiom,
! [B5: set_nat,F2: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A5 )
=> ( B5
!= ( image_nat_nat @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_242_image__subset__iff,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_rat @ ( F2 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_243_image__subset__iff,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ ( F2 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_244_subset__image__iff,axiom,
! [B5: set_rat,F2: nat > rat,A5: set_nat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A5 )
& ( B5
= ( image_nat_rat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_245_subset__image__iff,axiom,
! [B5: set_nat,F2: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A5 )
& ( B5
= ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_246_subset__UNIV,axiom,
! [A5: set_nat] : ( ord_less_eq_set_nat @ A5 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_247_subset__UNIV,axiom,
! [A5: set_rat] : ( ord_less_eq_set_rat @ A5 @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_248_asym__on__def,axiom,
( asym_o1385093689995826796ctxt_a
= ( ^ [A6: set_Ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
! [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A6 )
=> ! [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A6 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ).
% asym_on_def
thf(fact_249_asym__onD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a,X3: produc8175205341597284509ctxt_a,Y: produc8175205341597284509ctxt_a] :
( ( asym_o3378537592941112015ctxt_a @ A5 @ R )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y @ A5 )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X3 @ Y ) @ R )
=> ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_250_asym__onD,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ( asym_on_nat @ A5 @ R )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_251_asym__onD,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( asym_o1385093689995826796ctxt_a @ A5 @ R )
=> ( ( member3306386364193653941ctxt_a @ X3 @ A5 )
=> ( ( member3306386364193653941ctxt_a @ Y @ A5 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ X3 ) @ R ) ) ) ) ) ).
% asym_onD
thf(fact_252_range__subsetD,axiom,
! [F2: nat > rat,B5: set_rat,I: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) @ B5 )
=> ( member_rat @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_253_range__subsetD,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,I: nat] :
( ( ord_le2690539916662146973ctxt_a @ ( image_43530617296285190ctxt_a @ F2 @ top_top_set_nat ) @ B5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_254_range__subsetD,axiom,
! [F2: nat > nat,B5: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ B5 )
=> ( member_nat @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_255_range__subsetD,axiom,
! [F2: rat > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,I: rat] :
( ( ord_le2690539916662146973ctxt_a @ ( image_9112142474756165118ctxt_a @ F2 @ top_top_set_rat ) @ B5 )
=> ( member461722496487281478ctxt_a @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_256_range__subsetD,axiom,
! [F2: rat > nat,B5: set_nat,I: rat] :
( ( ord_less_eq_set_nat @ ( image_rat_nat @ F2 @ top_top_set_rat ) @ B5 )
=> ( member_nat @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_257_asym__iff,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ( asym_o1385093689995826796ctxt_a @ top_to3933008015346724740ctxt_a @ R )
= ( ! [X4: ground_gmctxt_a,Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_258_asym__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
= ( ! [X4: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_259_asym__iff,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ( asym_on_rat @ top_top_set_rat @ R )
= ( ! [X4: rat,Y4: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X4 @ Y4 ) @ R )
=> ~ ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Y4 @ X4 ) @ R ) ) ) ) ).
% asym_iff
thf(fact_260_asymD,axiom,
! [R: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( asym_o1385093689995826796ctxt_a @ top_to3933008015346724740ctxt_a @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_261_asymD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ( asym_on_nat @ top_top_set_nat @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_262_asymD,axiom,
! [R: set_Pr8928021450653196913at_rat,X3: rat,Y: rat] :
( ( asym_on_rat @ top_top_set_rat @ R )
=> ( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X3 @ Y ) @ R )
=> ~ ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Y @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_263_all__subset__image,axiom,
! [F2: nat > rat,A5: set_nat,P2: set_rat > $o] :
( ( ! [B6: set_rat] :
( ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
=> ( P2 @ ( image_nat_rat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_264_all__subset__image,axiom,
! [F2: nat > nat,A5: set_nat,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
=> ( P2 @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_265_relChain__def,axiom,
( bNF_Ca1214501019221039892_a_nat
= ( ^ [R3: set_Pr8705490787052945405ctxt_a,As: ground_gmctxt_a > nat] :
! [I2: ground_gmctxt_a,J: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ I2 @ J ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_266_asym__inv__image,axiom,
! [R2: set_Pr1261947904930325089at_nat,F2: nat > nat] :
( ( asym_on_nat @ top_top_set_nat @ R2 )
=> ( asym_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R2 @ F2 ) ) ) ).
% asym_inv_image
thf(fact_267_asym__inv__image,axiom,
! [R2: set_Pr1261947904930325089at_nat,F2: rat > nat] :
( ( asym_on_nat @ top_top_set_nat @ R2 )
=> ( asym_on_rat @ top_top_set_rat @ ( inv_image_nat_rat @ R2 @ F2 ) ) ) ).
% asym_inv_image
thf(fact_268_asym__inv__image,axiom,
! [R2: set_Pr8928021450653196913at_rat,F2: nat > rat] :
( ( asym_on_rat @ top_top_set_rat @ R2 )
=> ( asym_on_nat @ top_top_set_nat @ ( inv_image_rat_nat @ R2 @ F2 ) ) ) ).
% asym_inv_image
thf(fact_269_asym__inv__image,axiom,
! [R2: set_Pr8928021450653196913at_rat,F2: rat > rat] :
( ( asym_on_rat @ top_top_set_rat @ R2 )
=> ( asym_on_rat @ top_top_set_rat @ ( inv_image_rat_rat @ R2 @ F2 ) ) ) ).
% asym_inv_image
thf(fact_270_Greatest__equality,axiom,
! [P2: nat > $o,X3: nat] :
( ( P2 @ X3 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( order_Greatest_nat @ P2 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_271_GreatestI2__order,axiom,
! [P2: nat > $o,X3: nat,Q: nat > $o] :
( ( P2 @ X3 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ! [X: nat] :
( ( P2 @ X )
=> ( ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_272_asym__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( asym_on_nat @ top_top_set_nat @ R_A )
=> ( ( asym_on_nat @ top_top_set_nat @ R_B )
=> ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_273_asym__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8928021450653196913at_rat] :
( ( asym_on_nat @ top_top_set_nat @ R_A )
=> ( ( asym_on_rat @ top_top_set_rat @ R_B )
=> ( asym_o5038356120367172923at_rat @ top_to7513191607651882425at_rat @ ( lex_prod_nat_rat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_274_asym__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr1261947904930325089at_nat] :
( ( asym_on_rat @ top_top_set_rat @ R_A )
=> ( ( asym_on_nat @ top_top_set_nat @ R_B )
=> ( asym_o8230030034825703739at_nat @ top_to269121717765781945at_nat @ ( lex_prod_rat_nat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_275_asym__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr8928021450653196913at_rat] :
( ( asym_on_rat @ top_top_set_rat @ R_A )
=> ( ( asym_on_rat @ top_top_set_rat @ R_B )
=> ( asym_o4721801476432307011at_rat @ top_to3112507417142880193at_rat @ ( lex_prod_rat_rat @ R_A @ R_B ) ) ) ) ).
% asym_lex_prod
thf(fact_276_surj__Compl__image__subset,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F2 @ A5 ) ) @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_277_surj__Compl__image__subset,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A5 ) ) @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_278_surj__Compl__image__subset,axiom,
! [F2: rat > nat,A5: set_rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F2 @ A5 ) ) @ ( image_rat_nat @ F2 @ ( uminus2201863774496077783et_rat @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_279_surj__Compl__image__subset,axiom,
! [F2: rat > rat,A5: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F2 @ A5 ) ) @ ( image_rat_rat @ F2 @ ( uminus2201863774496077783et_rat @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_280_ComplI,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ ( uminus5163801920556549748ctxt_a @ A5 ) ) ) ).
% ComplI
thf(fact_281_ComplI,axiom,
! [C2: nat,A5: set_nat] :
( ~ ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A5 ) ) ) ).
% ComplI
thf(fact_282_Compl__iff,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( uminus5163801920556549748ctxt_a @ A5 ) )
= ( ~ ( member461722496487281478ctxt_a @ C2 @ A5 ) ) ) ).
% Compl_iff
thf(fact_283_Compl__iff,axiom,
! [C2: nat,A5: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A5 ) )
= ( ~ ( member_nat @ C2 @ A5 ) ) ) ).
% Compl_iff
thf(fact_284_in__lex__prod,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A2: ground_gmctxt_a,B2: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( produc6347872285926497493ctxt_a @ A2 @ B2 ) ) @ ( lex_pr7880170277141576941ctxt_a @ R @ S ) )
= ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_285_in__inv__image,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( inv_im1971831381409032464ctxt_a @ R @ F2 ) )
= ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_286_surj__uminus,axiom,
( ( image_rat_rat @ uminus_uminus_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_uminus
thf(fact_287_ComplD,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( uminus5163801920556549748ctxt_a @ A5 ) )
=> ~ ( member461722496487281478ctxt_a @ C2 @ A5 ) ) ).
% ComplD
thf(fact_288_ComplD,axiom,
! [C2: nat,A5: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A5 ) )
=> ~ ( member_nat @ C2 @ A5 ) ) ).
% ComplD
thf(fact_289_trancl__map,axiom,
! [R: set_Pr8705490787052945405ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ ( transi3225167780237802813ctxt_a @ S ) ) ) ) ).
% trancl_map
thf(fact_290_converse__trancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ B ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_291_trancl__trans__induct,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ( P2 @ X @ Y3 ) )
=> ( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( P2 @ X @ Y3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( P2 @ Y3 @ Z3 )
=> ( P2 @ X @ Z3 ) ) ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_292_trancl__into__trancl2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_293_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_294_irrefl__trancl__rD,axiom,
! [R: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ! [X: ground_gmctxt_a] :
~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ X ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ R )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_295_converse__tranclE,axiom,
! [X3: ground_gmctxt_a,Z: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ R )
=> ~ ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_296_r__r__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R2 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_297_trancl__induct2,axiom,
! [Ax: ground_gmctxt_a,Ay: ground_gmctxt_a,Bx: ground_gmctxt_a,By: ground_gmctxt_a,R: set_Pr490164551564945991ctxt_a,P2: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ Bx @ By ) ) @ ( transi6952278968286904510ctxt_a @ R ) )
=> ( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) ) @ R )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a,Aa: ground_gmctxt_a,Ba: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) ) @ ( transi6952278968286904510ctxt_a @ R ) )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) @ ( produc6347872285926497493ctxt_a @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_298_trancl_Ocases,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ R )
=> ~ ! [B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ B3 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_299_trancl_Osimps,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( transi3225167780237802813ctxt_a @ R ) )
= ( ? [A4: ground_gmctxt_a,B4: ground_gmctxt_a] :
( ( A1 = A4 )
& ( A22 = B4 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A4 @ B4 ) @ R ) )
| ? [A4: ground_gmctxt_a,B4: ground_gmctxt_a,C4: ground_gmctxt_a] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A4 @ B4 ) @ ( transi3225167780237802813ctxt_a @ R ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_300_trancl_Or__into__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_301_tranclE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ~ ! [C5: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C5 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ C5 @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_302_trancl__trans,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,Z: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% trancl_trans
thf(fact_303_trancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y3 ) @ R )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y3 ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% trancl_induct
thf(fact_304_same__fstI,axiom,
! [P2: ground_gmctxt_a > $o,X3: ground_gmctxt_a,Y7: ground_gmctxt_a,Y: ground_gmctxt_a,R2: ground_gmctxt_a > set_Pr8705490787052945405ctxt_a] :
( ( P2 @ X3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y7 @ Y ) @ ( R2 @ X3 ) )
=> ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y7 ) @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) ) @ ( same_f3606491088509793834ctxt_a @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_305_image__Fpow__mono,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A5 ) ) @ ( finite_Fpow_rat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_306_image__Fpow__mono,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A5 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_307_inj__image__Compl__subset,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A5 ) ) @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_308_inj__image__Compl__subset,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A5 ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_309_transp__relcompp,axiom,
! [R: nat > nat > $o] :
( ( transp_on_nat @ top_top_set_nat @ R )
= ( ord_le2646555220125990790_nat_o @ ( relcompp_nat_nat_nat @ R @ R ) @ R ) ) ).
% transp_relcompp
thf(fact_310_transp__relcompp,axiom,
! [R: rat > rat > $o] :
( ( transp_on_rat @ top_top_set_rat @ R )
= ( ord_le4717968354871517046_rat_o @ ( relcompp_rat_rat_rat @ R @ R ) @ R ) ) ).
% transp_relcompp
thf(fact_311_transp__relcompp__less__eq,axiom,
! [R: nat > nat > $o] :
( ( transp_on_nat @ top_top_set_nat @ R )
=> ( ord_le2646555220125990790_nat_o @ ( relcompp_nat_nat_nat @ R @ R ) @ R ) ) ).
% transp_relcompp_less_eq
thf(fact_312_transp__relcompp__less__eq,axiom,
! [R: rat > rat > $o] :
( ( transp_on_rat @ top_top_set_rat @ R )
=> ( ord_le4717968354871517046_rat_o @ ( relcompp_rat_rat_rat @ R @ R ) @ R ) ) ).
% transp_relcompp_less_eq
thf(fact_313_acyclic__def,axiom,
( transi8399219972279719055ctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a] :
! [X4: ground_gmctxt_a] :
~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ X4 ) @ ( transi3225167780237802813ctxt_a @ R3 ) ) ) ) ).
% acyclic_def
thf(fact_314_UnCI,axiom,
! [C2: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( ~ ( member461722496487281478ctxt_a @ C2 @ B5 )
=> ( member461722496487281478ctxt_a @ C2 @ A5 ) )
=> ( member461722496487281478ctxt_a @ C2 @ ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_315_UnCI,axiom,
! [C2: nat,B5: set_nat,A5: set_nat] :
( ( ~ ( member_nat @ C2 @ B5 )
=> ( member_nat @ C2 @ A5 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_316_Un__iff,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) )
= ( ( member461722496487281478ctxt_a @ C2 @ A5 )
| ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_317_Un__iff,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
= ( ( member_nat @ C2 @ A5 )
| ( member_nat @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_318_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
=> ( ( ( image_nat_rat @ F2 @ A5 )
= ( image_nat_rat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_319_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
=> ( ( ( image_nat_nat @ F2 @ A5 )
= ( image_nat_nat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_320_transp__onD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,X3: produc8175205341597284509ctxt_a,Y: produc8175205341597284509ctxt_a,Z: produc8175205341597284509ctxt_a] :
( ( transp5438720461609813581ctxt_a @ A5 @ R2 )
=> ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y @ A5 )
=> ( ( member461722496487281478ctxt_a @ Z @ A5 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) ) ) ) ) ) ).
% transp_onD
thf(fact_321_transp__onD,axiom,
! [A5: set_nat,R2: nat > nat > $o,X3: nat,Y: nat,Z: nat] :
( ( transp_on_nat @ A5 @ R2 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( ( member_nat @ Z @ A5 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) ) ) ) ) ) ).
% transp_onD
thf(fact_322_transp__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o] :
( ! [X: produc8175205341597284509ctxt_a,Y3: produc8175205341597284509ctxt_a,Z3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y3 @ A5 )
=> ( ( member461722496487281478ctxt_a @ Z3 @ A5 )
=> ( ( R2 @ X @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X @ Z3 ) ) ) ) ) )
=> ( transp5438720461609813581ctxt_a @ A5 @ R2 ) ) ).
% transp_onI
thf(fact_323_transp__onI,axiom,
! [A5: set_nat,R2: nat > nat > $o] :
( ! [X: nat,Y3: nat,Z3: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( member_nat @ Z3 @ A5 )
=> ( ( R2 @ X @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X @ Z3 ) ) ) ) ) )
=> ( transp_on_nat @ A5 @ R2 ) ) ).
% transp_onI
thf(fact_324_UnE,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) )
=> ( ~ ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_325_UnE,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
=> ( ~ ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_326_UnI1,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_327_UnI1,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_328_UnI2,axiom,
! [C2: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ B5 )
=> ( member461722496487281478ctxt_a @ C2 @ ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_329_UnI2,axiom,
! [C2: nat,B5: set_nat,A5: set_nat] :
( ( member_nat @ C2 @ B5 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_330_inj__on__image__Fpow,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_331_inj__on__image__Fpow,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_332_transp__on__le,axiom,
! [A5: set_nat] : ( transp_on_nat @ A5 @ ord_less_eq_nat ) ).
% transp_on_le
thf(fact_333_transp__equality,axiom,
( transp_on_nat @ top_top_set_nat
@ ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) ) ).
% transp_equality
thf(fact_334_transp__equality,axiom,
( transp_on_rat @ top_top_set_rat
@ ^ [Y5: rat,Z2: rat] : ( Y5 = Z2 ) ) ).
% transp_equality
thf(fact_335_transp__def,axiom,
! [R2: nat > nat > $o] :
( ( transp_on_nat @ top_top_set_nat @ R2 )
= ( ! [X4: nat,Y4: nat,Z4: nat] :
( ( R2 @ X4 @ Y4 )
=> ( ( R2 @ Y4 @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) ) ) ) ).
% transp_def
thf(fact_336_transp__def,axiom,
! [R2: rat > rat > $o] :
( ( transp_on_rat @ top_top_set_rat @ R2 )
= ( ! [X4: rat,Y4: rat,Z4: rat] :
( ( R2 @ X4 @ Y4 )
=> ( ( R2 @ Y4 @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) ) ) ) ).
% transp_def
thf(fact_337_transpI,axiom,
! [R2: nat > nat > $o] :
( ! [X: nat,Y3: nat,Z3: nat] :
( ( R2 @ X @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( transp_on_nat @ top_top_set_nat @ R2 ) ) ).
% transpI
thf(fact_338_transpI,axiom,
! [R2: rat > rat > $o] :
( ! [X: rat,Y3: rat,Z3: rat] :
( ( R2 @ X @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( transp_on_rat @ top_top_set_rat @ R2 ) ) ).
% transpI
thf(fact_339_transpE,axiom,
! [R: nat > nat > $o,X3: nat,Y: nat,Z: nat] :
( ( transp_on_nat @ top_top_set_nat @ R )
=> ( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X3 @ Z ) ) ) ) ).
% transpE
thf(fact_340_transpE,axiom,
! [R: rat > rat > $o,X3: rat,Y: rat,Z: rat] :
( ( transp_on_rat @ top_top_set_rat @ R )
=> ( ( R @ X3 @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X3 @ Z ) ) ) ) ).
% transpE
thf(fact_341_transpD,axiom,
! [R2: nat > nat > $o,X3: nat,Y: nat,Z: nat] :
( ( transp_on_nat @ top_top_set_nat @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) ) ) ).
% transpD
thf(fact_342_transpD,axiom,
! [R2: rat > rat > $o,X3: rat,Y: rat,Z: rat] :
( ( transp_on_rat @ top_top_set_rat @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) ) ) ).
% transpD
thf(fact_343_image__Un,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] :
( ( image_nat_rat @ F2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
= ( sup_sup_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_344_image__Un,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( image_nat_nat @ F2 @ ( sup_sup_set_nat @ A5 @ B5 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_345_Un__UNIV__right,axiom,
! [A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_346_Un__UNIV__right,axiom,
! [A5: set_rat] :
( ( sup_sup_set_rat @ A5 @ top_top_set_rat )
= top_top_set_rat ) ).
% Un_UNIV_right
thf(fact_347_Un__UNIV__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B5 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_348_Un__UNIV__left,axiom,
! [B5: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ B5 )
= top_top_set_rat ) ).
% Un_UNIV_left
thf(fact_349_range__ex1__eq,axiom,
! [F2: nat > rat,B: rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y4: nat] :
( ( B
= ( F2 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_350_range__ex1__eq,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ top_top_set_nat )
=> ( ( member461722496487281478ctxt_a @ B @ ( image_43530617296285190ctxt_a @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y4: nat] :
( ( B
= ( F2 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_351_range__ex1__eq,axiom,
! [F2: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y4: nat] :
( ( B
= ( F2 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_352_range__ex1__eq,axiom,
! [F2: rat > produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( inj_on8318268736959861138ctxt_a @ F2 @ top_top_set_rat )
=> ( ( member461722496487281478ctxt_a @ B @ ( image_9112142474756165118ctxt_a @ F2 @ top_top_set_rat ) )
= ( ? [X4: rat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y4: rat] :
( ( B
= ( F2 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_353_range__ex1__eq,axiom,
! [F2: rat > nat,B: nat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( member_nat @ B @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
= ( ? [X4: rat] :
( ( B
= ( F2 @ X4 ) )
& ! [Y4: rat] :
( ( B
= ( F2 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_354_inj__image__eq__iff,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F2 @ A5 )
= ( image_nat_rat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_355_inj__image__eq__iff,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F2 @ A5 )
= ( image_nat_nat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_356_inj__image__mem__iff,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on4839580843290501553ctxt_a @ F2 @ top_to2898181658028262733ctxt_a )
=> ( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) )
= ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_357_inj__image__mem__iff,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on5187228193288102844_a_nat @ F2 @ top_to2898181658028262733ctxt_a )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
= ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_358_inj__image__mem__iff,axiom,
! [F2: nat > rat,A: nat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_359_inj__image__mem__iff,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A: nat,A5: set_nat] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ top_top_set_nat )
=> ( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_360_inj__image__mem__iff,axiom,
! [F2: nat > nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_361_inj__image__mem__iff,axiom,
! [F2: rat > produc8175205341597284509ctxt_a,A: rat,A5: set_rat] :
( ( inj_on8318268736959861138ctxt_a @ F2 @ top_top_set_rat )
=> ( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_9112142474756165118ctxt_a @ F2 @ A5 ) )
= ( member_rat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_362_inj__image__mem__iff,axiom,
! [F2: rat > nat,A: rat,A5: set_rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_rat_nat @ F2 @ A5 ) )
= ( member_rat @ A @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_363_inj__on__image__eq__iff,axiom,
! [F2: nat > rat,C: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ C )
=> ( ( ord_less_eq_set_nat @ A5 @ C )
=> ( ( ord_less_eq_set_nat @ B5 @ C )
=> ( ( ( image_nat_rat @ F2 @ A5 )
= ( image_nat_rat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_364_inj__on__image__eq__iff,axiom,
! [F2: nat > nat,C: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ C )
=> ( ( ord_less_eq_set_nat @ A5 @ C )
=> ( ( ord_less_eq_set_nat @ B5 @ C )
=> ( ( ( image_nat_nat @ F2 @ A5 )
= ( image_nat_nat @ F2 @ B5 ) )
= ( A5 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_365_inj__on__image__mem__iff,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on4839580843290501553ctxt_a @ F2 @ B5 )
=> ( ( member461722496487281478ctxt_a @ A @ B5 )
=> ( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) )
= ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_366_inj__on__image__mem__iff,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,B5: set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on5187228193288102844_a_nat @ F2 @ B5 )
=> ( ( member461722496487281478ctxt_a @ A @ B5 )
=> ( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
= ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_367_inj__on__image__mem__iff,axiom,
! [F2: nat > rat,B5: set_nat,A: nat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_368_inj__on__image__mem__iff,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,B5: set_nat,A: nat,A5: set_nat] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_369_inj__on__image__mem__iff,axiom,
! [F2: nat > nat,B5: set_nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A5 ) )
= ( member_nat @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_370_Compl__partition,axiom,
! [A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ ( uminus5710092332889474511et_nat @ A5 ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_371_Compl__partition,axiom,
! [A5: set_rat] :
( ( sup_sup_set_rat @ A5 @ ( uminus2201863774496077783et_rat @ A5 ) )
= top_top_set_rat ) ).
% Compl_partition
thf(fact_372_Compl__partition2,axiom,
! [A5: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A5 ) @ A5 )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_373_Compl__partition2,axiom,
! [A5: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ A5 ) @ A5 )
= top_top_set_rat ) ).
% Compl_partition2
thf(fact_374_inj__image__subset__iff,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) )
= ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_375_inj__image__subset__iff,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) )
= ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_376_acyclicI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a] :
~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ X ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( transi8399219972279719055ctxt_a @ R ) ) ).
% acyclicI
thf(fact_377_sup__compl__top__left1,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ ( sup_sup_set_nat @ X3 @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_378_sup__compl__top__left1,axiom,
! [X3: set_rat,Y: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ ( sup_sup_set_rat @ X3 @ Y ) )
= top_top_set_rat ) ).
% sup_compl_top_left1
thf(fact_379_sup__compl__top__left2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_380_sup__compl__top__left2,axiom,
! [X3: set_rat,Y: set_rat] :
( ( sup_sup_set_rat @ X3 @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ Y ) )
= top_top_set_rat ) ).
% sup_compl_top_left2
thf(fact_381_boolean__algebra_Odisj__cancel__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ X3 )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_382_boolean__algebra_Odisj__cancel__left,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ X3 )
= top_top_set_rat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_383_boolean__algebra_Odisj__cancel__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ X3 ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_384_boolean__algebra_Odisj__cancel__right,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ X3 @ ( uminus2201863774496077783et_rat @ X3 ) )
= top_top_set_rat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_385_sup__top__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X3 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_386_sup__top__left,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ X3 )
= top_top_set_rat ) ).
% sup_top_left
thf(fact_387_sup__top__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_388_sup__top__right,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ X3 @ top_top_set_rat )
= top_top_set_rat ) ).
% sup_top_right
thf(fact_389_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X3 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_390_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ top_top_set_rat @ X3 )
= top_top_set_rat ) ).
% boolean_algebra.disj_one_left
thf(fact_391_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_392_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_rat] :
( ( sup_sup_set_rat @ X3 @ top_top_set_rat )
= top_top_set_rat ) ).
% boolean_algebra.disj_one_right
thf(fact_393_sup__cancel__left2,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ A ) @ ( sup_sup_set_nat @ X3 @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_394_sup__cancel__left2,axiom,
! [X3: set_rat,A: set_rat,B: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ A ) @ ( sup_sup_set_rat @ X3 @ B ) )
= top_top_set_rat ) ).
% sup_cancel_left2
thf(fact_395_sup__cancel__left1,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ A ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_396_sup__cancel__left1,axiom,
! [X3: set_rat,A: set_rat,B: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ X3 @ A ) @ ( sup_sup_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ B ) )
= top_top_set_rat ) ).
% sup_cancel_left1
thf(fact_397_sup__shunt,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y ) ) ).
% sup_shunt
thf(fact_398_sup__shunt,axiom,
! [X3: set_rat,Y: set_rat] :
( ( ( sup_sup_set_rat @ X3 @ Y )
= top_top_set_rat )
= ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ X3 ) @ Y ) ) ).
% sup_shunt
thf(fact_399_subset__image__inj,axiom,
! [S3: set_rat,F2: nat > rat,T3: set_nat] :
( ( ord_less_eq_set_rat @ S3 @ ( image_nat_rat @ F2 @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_rat @ F2 @ U )
& ( S3
= ( image_nat_rat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_400_subset__image__inj,axiom,
! [S3: set_nat,F2: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ ( image_nat_nat @ F2 @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_nat @ F2 @ U )
& ( S3
= ( image_nat_nat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_401_image__Pow__mono,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A5 ) ) @ ( pow_rat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_402_image__Pow__mono,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A5 ) ) @ ( pow_nat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_403_acyclic__irrefl,axiom,
( transi3139058337892597825ic_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat] : ( irrefl_on_nat @ top_top_set_nat @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% acyclic_irrefl
thf(fact_404_acyclic__irrefl,axiom,
( transi2503928277806102089ic_rat
= ( ^ [R3: set_Pr8928021450653196913at_rat] : ( irrefl_on_rat @ top_top_set_rat @ ( transi5628869978870870775cl_rat @ R3 ) ) ) ) ).
% acyclic_irrefl
thf(fact_405_the__inv__into__into,axiom,
! [F2: nat > rat,A5: set_nat,X3: rat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_rat @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_406_the__inv__into__into,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on4839580843290501553ctxt_a @ F2 @ A5 )
=> ( ( member461722496487281478ctxt_a @ X3 @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) )
=> ( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( member461722496487281478ctxt_a @ ( the_in7972105333848070131ctxt_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_407_the__inv__into__into,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A5: set_nat,X3: produc8175205341597284509ctxt_a,B5: set_nat] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ A5 )
=> ( ( member461722496487281478ctxt_a @ X3 @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( member_nat @ ( the_in1598212552454528472ctxt_a @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_408_the__inv__into__into,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,A5: set_Pr8705490787052945405ctxt_a,X3: nat,B5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on5187228193288102844_a_nat @ F2 @ A5 )
=> ( ( member_nat @ X3 @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
=> ( ( ord_le2690539916662146973ctxt_a @ A5 @ B5 )
=> ( member461722496487281478ctxt_a @ ( the_in7535783866242650106_a_nat @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_409_the__inv__into__into,axiom,
! [F2: nat > nat,A5: set_nat,X3: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A5 @ F2 @ X3 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_410_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_411_Pow__UNIV,axiom,
( ( pow_rat @ top_top_set_rat )
= top_top_set_set_rat ) ).
% Pow_UNIV
thf(fact_412_the__inv__into__onto,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A5 @ F2 ) @ ( image_nat_rat @ F2 @ A5 ) )
= A5 ) ) ).
% the_inv_into_onto
thf(fact_413_the__inv__into__onto,axiom,
! [F2: rat > nat,A5: set_rat] :
( ( inj_on_rat_nat @ F2 @ A5 )
=> ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A5 @ F2 ) @ ( image_rat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% the_inv_into_onto
thf(fact_414_the__inv__into__onto,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A5 @ F2 ) @ ( image_nat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% the_inv_into_onto
thf(fact_415_wf__imp__irrefl,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R )
=> ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% wf_imp_irrefl
thf(fact_416_wf__imp__irrefl,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ( wf_rat @ R )
=> ( irrefl_on_rat @ top_top_set_rat @ R ) ) ).
% wf_imp_irrefl
thf(fact_417_wf__if__convertible__to__wf,axiom,
! [S: set_Pr8705490787052945405ctxt_a,R: set_Pr8705490787052945405ctxt_a,F2: ground_gmctxt_a > ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ S )
=> ( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ S ) )
=> ( wf_Ground_gmctxt_a @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_418_wf__induct__rule,axiom,
! [R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o,A: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ( ! [X: ground_gmctxt_a] :
( ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y6 @ X ) @ R )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ) ).
% wf_induct_rule
thf(fact_419_wf__eq__minimal,axiom,
( wf_Pro4728511896666056454ctxt_a
= ( ^ [R3: set_Pr490164551564945991ctxt_a] :
! [Q3: set_Pr8705490787052945405ctxt_a] :
( ? [X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ Q3 )
=> ? [X4: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X4 @ Q3 )
& ! [Y4: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y4 @ X4 ) @ R3 )
=> ~ ( member461722496487281478ctxt_a @ Y4 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_420_wf__eq__minimal,axiom,
( wf_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat] :
! [Q3: set_nat] :
( ? [X4: nat] : ( member_nat @ X4 @ Q3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ Q3 )
& ! [Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X4 ) @ R3 )
=> ~ ( member_nat @ Y4 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_421_wf__eq__minimal,axiom,
( wf_Ground_gmctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a] :
! [Q3: set_Ground_gmctxt_a] :
( ? [X4: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X4 @ Q3 )
=> ? [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ Q3 )
& ! [Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R3 )
=> ~ ( member3306386364193653941ctxt_a @ Y4 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_422_wf__not__refl,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A ) @ R ) ) ).
% wf_not_refl
thf(fact_423_wf__not__sym,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a,X3: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ X3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ A ) @ R ) ) ) ).
% wf_not_sym
thf(fact_424_wf__irrefl,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A ) @ R ) ) ).
% wf_irrefl
thf(fact_425_wf__induct,axiom,
! [R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o,A: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ( ! [X: ground_gmctxt_a] :
( ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y6 @ X ) @ R )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ) ).
% wf_induct
thf(fact_426_wf__asym,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a,X3: ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ X3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ A ) @ R ) ) ) ).
% wf_asym
thf(fact_427_wfUNIVI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [P4: ground_gmctxt_a > $o,X: ground_gmctxt_a] :
( ! [Xa: ground_gmctxt_a] :
( ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Xa ) @ R )
=> ( P4 @ Y3 ) )
=> ( P4 @ Xa ) )
=> ( P4 @ X ) )
=> ( wf_Ground_gmctxt_a @ R ) ) ).
% wfUNIVI
thf(fact_428_wfI__min,axiom,
! [R2: set_Pr490164551564945991ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a,Q4: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ Q4 )
=> ? [Xa: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Xa @ Q4 )
& ! [Y3: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y3 @ Xa ) @ R2 )
=> ~ ( member461722496487281478ctxt_a @ Y3 @ Q4 ) ) ) )
=> ( wf_Pro4728511896666056454ctxt_a @ R2 ) ) ).
% wfI_min
thf(fact_429_wfI__min,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Q4: set_nat] :
( ( member_nat @ X @ Q4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ Q4 )
& ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa ) @ R2 )
=> ~ ( member_nat @ Y3 @ Q4 ) ) ) )
=> ( wf_nat @ R2 ) ) ).
% wfI_min
thf(fact_430_wfI__min,axiom,
! [R2: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Q4: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ Q4 )
=> ? [Xa: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa @ Q4 )
& ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Xa ) @ R2 )
=> ~ ( member3306386364193653941ctxt_a @ Y3 @ Q4 ) ) ) )
=> ( wf_Ground_gmctxt_a @ R2 ) ) ).
% wfI_min
thf(fact_431_wfE__min,axiom,
! [R2: set_Pr490164551564945991ctxt_a,X3: produc8175205341597284509ctxt_a,Q: set_Pr8705490787052945405ctxt_a] :
( ( wf_Pro4728511896666056454ctxt_a @ R2 )
=> ( ( member461722496487281478ctxt_a @ X3 @ Q )
=> ~ ! [Z3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Z3 @ Q )
=> ~ ! [Y6: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y6 @ Z3 ) @ R2 )
=> ~ ( member461722496487281478ctxt_a @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_432_wfE__min,axiom,
! [R2: set_Pr1261947904930325089at_nat,X3: nat,Q: set_nat] :
( ( wf_nat @ R2 )
=> ( ( member_nat @ X3 @ Q )
=> ~ ! [Z3: nat] :
( ( member_nat @ Z3 @ Q )
=> ~ ! [Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Z3 ) @ R2 )
=> ~ ( member_nat @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_433_wfE__min,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Q: set_Ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R2 )
=> ( ( member3306386364193653941ctxt_a @ X3 @ Q )
=> ~ ! [Z3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z3 @ Q )
=> ~ ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y6 @ Z3 ) @ R2 )
=> ~ ( member3306386364193653941ctxt_a @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_434_wf__def,axiom,
( wf_Ground_gmctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a] :
! [P5: ground_gmctxt_a > $o] :
( ! [X4: ground_gmctxt_a] :
( ! [Y4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R3 )
=> ( P5 @ Y4 ) )
=> ( P5 @ X4 ) )
=> ! [X6: ground_gmctxt_a] : ( P5 @ X6 ) ) ) ) ).
% wf_def
thf(fact_435_irrefl__onD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a,A: produc8175205341597284509ctxt_a] :
( ( irrefl7400473109832159123ctxt_a @ A5 @ R )
=> ( ( member461722496487281478ctxt_a @ A @ A5 )
=> ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_436_irrefl__onD,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( irrefl_on_nat @ A5 @ R )
=> ( ( member_nat @ A @ A5 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_437_irrefl__onD,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a] :
( ( irrefl2796304932565638824ctxt_a @ A5 @ R )
=> ( ( member3306386364193653941ctxt_a @ A @ A5 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A ) @ R ) ) ) ).
% irrefl_onD
thf(fact_438_irrefl__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a] :
( ! [A3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ A3 @ A5 )
=> ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ A3 @ A3 ) @ R ) )
=> ( irrefl7400473109832159123ctxt_a @ A5 @ R ) ) ).
% irrefl_onI
thf(fact_439_irrefl__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A5 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R ) )
=> ( irrefl_on_nat @ A5 @ R ) ) ).
% irrefl_onI
thf(fact_440_irrefl__onI,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ! [A3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ A3 @ A5 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ A3 ) @ R ) )
=> ( irrefl2796304932565638824ctxt_a @ A5 @ R ) ) ).
% irrefl_onI
thf(fact_441_irrefl__on__def,axiom,
( irrefl2796304932565638824ctxt_a
= ( ^ [A6: set_Ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
! [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A6 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ X4 ) @ R3 ) ) ) ) ).
% irrefl_on_def
thf(fact_442_irrefl__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( irrefl_on_nat @ top_top_set_nat @ R_A )
=> ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
=> ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_443_irrefl__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8928021450653196913at_rat] :
( ( irrefl_on_nat @ top_top_set_nat @ R_A )
=> ( ( irrefl_on_rat @ top_top_set_rat @ R_B )
=> ( irrefl3900030141358088703at_rat @ top_to7513191607651882425at_rat @ ( lex_prod_nat_rat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_444_irrefl__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr1261947904930325089at_nat] :
( ( irrefl_on_rat @ top_top_set_rat @ R_A )
=> ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
=> ( irrefl7091704055816619519at_nat @ top_to269121717765781945at_nat @ ( lex_prod_rat_nat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_445_irrefl__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr8928021450653196913at_rat] :
( ( irrefl_on_rat @ top_top_set_rat @ R_A )
=> ( ( irrefl_on_rat @ top_top_set_rat @ R_B )
=> ( irrefl3583475497423222791at_rat @ top_to3112507417142880193at_rat @ ( lex_prod_rat_rat @ R_A @ R_B ) ) ) ) ).
% irrefl_lex_prod
thf(fact_446_wf__imp__asym,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R )
=> ( asym_on_nat @ top_top_set_nat @ R ) ) ).
% wf_imp_asym
thf(fact_447_wf__imp__asym,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ( wf_rat @ R )
=> ( asym_on_rat @ top_top_set_rat @ R ) ) ).
% wf_imp_asym
thf(fact_448_image__Pow__surj,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( ( image_nat_rat @ F2 @ A5 )
= B5 )
=> ( ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A5 ) )
= ( pow_rat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_449_image__Pow__surj,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( ( image_nat_nat @ F2 @ A5 )
= B5 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A5 ) )
= ( pow_nat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_450_irreflD,axiom,
! [R: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a] :
( ( irrefl2796304932565638824ctxt_a @ top_to3933008015346724740ctxt_a @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_451_irreflD,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat] :
( ( irrefl_on_nat @ top_top_set_nat @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_452_irreflD,axiom,
! [R: set_Pr8928021450653196913at_rat,X3: rat] :
( ( irrefl_on_rat @ top_top_set_rat @ R )
=> ~ ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X3 @ X3 ) @ R ) ) ).
% irreflD
thf(fact_453_irreflI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [A3: ground_gmctxt_a] :
~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ A3 ) @ R )
=> ( irrefl2796304932565638824ctxt_a @ top_to3933008015346724740ctxt_a @ R ) ) ).
% irreflI
thf(fact_454_irreflI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [A3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
=> ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).
% irreflI
thf(fact_455_irreflI,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ! [A3: rat] :
~ ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ A3 @ A3 ) @ R )
=> ( irrefl_on_rat @ top_top_set_rat @ R ) ) ).
% irreflI
thf(fact_456_f__the__inv__into__f,axiom,
! [F2: nat > rat,A5: set_nat,Y: rat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( member_rat @ Y @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ( F2 @ ( the_inv_into_nat_rat @ A5 @ F2 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_457_f__the__inv__into__f,axiom,
! [F2: nat > nat,A5: set_nat,Y: nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ( F2 @ ( the_inv_into_nat_nat @ A5 @ F2 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_458_inj__on__the__inv__into,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A5 @ F2 ) @ ( image_nat_rat @ F2 @ A5 ) ) ) ).
% inj_on_the_inv_into
thf(fact_459_inj__on__the__inv__into,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A5 @ F2 ) @ ( image_nat_nat @ F2 @ A5 ) ) ) ).
% inj_on_the_inv_into
thf(fact_460_inj__on__image__Pow,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ ( pow_nat @ A5 ) ) ) ).
% inj_on_image_Pow
thf(fact_461_inj__on__image__Pow,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( pow_nat @ A5 ) ) ) ).
% inj_on_image_Pow
thf(fact_462_total__inv__image,axiom,
! [F2: nat > nat,R: set_Pr1261947904930325089at_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( total_on_nat @ top_top_set_nat @ R )
=> ( total_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_463_total__inv__image,axiom,
! [F2: nat > rat,R: set_Pr8928021450653196913at_rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( total_on_rat @ top_top_set_rat @ R )
=> ( total_on_nat @ top_top_set_nat @ ( inv_image_rat_nat @ R @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_464_total__inv__image,axiom,
! [F2: rat > nat,R: set_Pr1261947904930325089at_nat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( total_on_nat @ top_top_set_nat @ R )
=> ( total_on_rat @ top_top_set_rat @ ( inv_image_nat_rat @ R @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_465_total__inv__image,axiom,
! [F2: rat > rat,R: set_Pr8928021450653196913at_rat] :
( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
=> ( ( total_on_rat @ top_top_set_rat @ R )
=> ( total_on_rat @ top_top_set_rat @ ( inv_image_rat_rat @ R @ F2 ) ) ) ) ).
% total_inv_image
thf(fact_466_inj__on__image,axiom,
! [F2: nat > rat,A5: set_set_nat] :
( ( inj_on_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ A5 ) )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ A5 ) ) ).
% inj_on_image
thf(fact_467_inj__on__image,axiom,
! [F2: nat > nat,A5: set_set_nat] :
( ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ A5 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ A5 ) ) ).
% inj_on_image
thf(fact_468_vimage__subsetI,axiom,
! [F2: nat > rat,B5: set_rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F2 @ B5 ) @ A5 ) ) ) ).
% vimage_subsetI
thf(fact_469_vimage__subsetI,axiom,
! [F2: nat > nat,B5: set_nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ B5 ) @ A5 ) ) ) ).
% vimage_subsetI
thf(fact_470_inv__into__image__cancel,axiom,
! [F2: nat > rat,A5: set_nat,S3: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ S3 @ A5 )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A5 @ F2 ) @ ( image_nat_rat @ F2 @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_471_inv__into__image__cancel,axiom,
! [F2: rat > nat,A5: set_rat,S3: set_rat] :
( ( inj_on_rat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_rat @ S3 @ A5 )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A5 @ F2 ) @ ( image_rat_nat @ F2 @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_472_inv__into__image__cancel,axiom,
! [F2: nat > nat,A5: set_nat,S3: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ S3 @ A5 )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ F2 ) @ ( image_nat_nat @ F2 @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_473_Union__iff,axiom,
! [A5: produc8175205341597284509ctxt_a,C: set_se6366015629438403421ctxt_a] :
( ( member461722496487281478ctxt_a @ A5 @ ( comple1483637506433994474ctxt_a @ C ) )
= ( ? [X4: set_Pr8705490787052945405ctxt_a] :
( ( member6521329796374918822ctxt_a @ X4 @ C )
& ( member461722496487281478ctxt_a @ A5 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_474_Union__iff,axiom,
! [A5: nat,C: set_set_nat] :
( ( member_nat @ A5 @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ C )
& ( member_nat @ A5 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_475_UnionI,axiom,
! [X7: set_Pr8705490787052945405ctxt_a,C: set_se6366015629438403421ctxt_a,A5: produc8175205341597284509ctxt_a] :
( ( member6521329796374918822ctxt_a @ X7 @ C )
=> ( ( member461722496487281478ctxt_a @ A5 @ X7 )
=> ( member461722496487281478ctxt_a @ A5 @ ( comple1483637506433994474ctxt_a @ C ) ) ) ) ).
% UnionI
thf(fact_476_UnionI,axiom,
! [X7: set_nat,C: set_set_nat,A5: nat] :
( ( member_set_nat @ X7 @ C )
=> ( ( member_nat @ A5 @ X7 )
=> ( member_nat @ A5 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_477_vimage__eq,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ ( vimage1911517659246870011ctxt_a @ F2 @ B5 ) )
= ( member461722496487281478ctxt_a @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_478_vimage__eq,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > nat,B5: set_nat] :
( ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ B5 ) )
= ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_479_vimage__eq,axiom,
! [A: nat,F2: nat > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member_nat @ A @ ( vimage172744354138440144ctxt_a @ F2 @ B5 ) )
= ( member461722496487281478ctxt_a @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_480_vimage__eq,axiom,
! [A: nat,F2: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) )
= ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimage_eq
thf(fact_481_vimageI,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A: produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ( F2 @ A )
= B )
=> ( ( member461722496487281478ctxt_a @ B @ B5 )
=> ( member461722496487281478ctxt_a @ A @ ( vimage1911517659246870011ctxt_a @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_482_vimageI,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A: nat,B: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ( F2 @ A )
= B )
=> ( ( member461722496487281478ctxt_a @ B @ B5 )
=> ( member_nat @ A @ ( vimage172744354138440144ctxt_a @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_483_vimageI,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,A: produc8175205341597284509ctxt_a,B: nat,B5: set_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_484_vimageI,axiom,
! [F2: nat > nat,A: nat,B: nat,B5: set_nat] :
( ( ( F2 @ A )
= B )
=> ( ( member_nat @ B @ B5 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) ) ) ) ).
% vimageI
thf(fact_485_vimage__UNIV,axiom,
! [F2: nat > nat] :
( ( vimage_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_486_vimage__UNIV,axiom,
! [F2: rat > nat] :
( ( vimage_rat_nat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% vimage_UNIV
thf(fact_487_vimage__UNIV,axiom,
! [F2: nat > rat] :
( ( vimage_nat_rat @ F2 @ top_top_set_rat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_488_vimage__UNIV,axiom,
! [F2: rat > rat] :
( ( vimage_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat ) ).
% vimage_UNIV
thf(fact_489_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_490_Sup__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Sup_UNIV
thf(fact_491_UnionE,axiom,
! [A5: produc8175205341597284509ctxt_a,C: set_se6366015629438403421ctxt_a] :
( ( member461722496487281478ctxt_a @ A5 @ ( comple1483637506433994474ctxt_a @ C ) )
=> ~ ! [X8: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A5 @ X8 )
=> ~ ( member6521329796374918822ctxt_a @ X8 @ C ) ) ) ).
% UnionE
thf(fact_492_UnionE,axiom,
! [A5: nat,C: set_set_nat] :
( ( member_nat @ A5 @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X8: set_nat] :
( ( member_nat @ A5 @ X8 )
=> ~ ( member_set_nat @ X8 @ C ) ) ) ).
% UnionE
thf(fact_493_vimageI2,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ A5 )
=> ( member461722496487281478ctxt_a @ A @ ( vimage1911517659246870011ctxt_a @ F2 @ A5 ) ) ) ).
% vimageI2
thf(fact_494_vimageI2,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A: nat,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F2 @ A ) @ A5 )
=> ( member_nat @ A @ ( vimage172744354138440144ctxt_a @ F2 @ A5 ) ) ) ).
% vimageI2
thf(fact_495_vimageI2,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,A: produc8175205341597284509ctxt_a,A5: set_nat] :
( ( member_nat @ ( F2 @ A ) @ A5 )
=> ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ A5 ) ) ) ).
% vimageI2
thf(fact_496_vimageI2,axiom,
! [F2: nat > nat,A: nat,A5: set_nat] :
( ( member_nat @ ( F2 @ A ) @ A5 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A5 ) ) ) ).
% vimageI2
thf(fact_497_vimageE,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ ( vimage1911517659246870011ctxt_a @ F2 @ B5 ) )
=> ( member461722496487281478ctxt_a @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_498_vimageE,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > nat,B5: set_nat] :
( ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ B5 ) )
=> ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_499_vimageE,axiom,
! [A: nat,F2: nat > produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member_nat @ A @ ( vimage172744354138440144ctxt_a @ F2 @ B5 ) )
=> ( member461722496487281478ctxt_a @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_500_vimageE,axiom,
! [A: nat,F2: nat > nat,B5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ B5 ) )
=> ( member_nat @ ( F2 @ A ) @ B5 ) ) ).
% vimageE
thf(fact_501_vimageD,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ ( vimage1911517659246870011ctxt_a @ F2 @ A5 ) )
=> ( member461722496487281478ctxt_a @ ( F2 @ A ) @ A5 ) ) ).
% vimageD
thf(fact_502_vimageD,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > nat,A5: set_nat] :
( ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ A5 ) )
=> ( member_nat @ ( F2 @ A ) @ A5 ) ) ).
% vimageD
thf(fact_503_vimageD,axiom,
! [A: nat,F2: nat > produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member_nat @ A @ ( vimage172744354138440144ctxt_a @ F2 @ A5 ) )
=> ( member461722496487281478ctxt_a @ ( F2 @ A ) @ A5 ) ) ).
% vimageD
thf(fact_504_vimageD,axiom,
! [A: nat,F2: nat > nat,A5: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ A5 ) )
=> ( member_nat @ ( F2 @ A ) @ A5 ) ) ).
% vimageD
thf(fact_505_f__inv__into__f,axiom,
! [Y: rat,F2: nat > rat,A5: set_nat] :
( ( member_rat @ Y @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ( F2 @ ( hilber2998747136712319222at_rat @ A5 @ F2 @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_506_f__inv__into__f,axiom,
! [Y: nat,F2: nat > nat,A5: set_nat] :
( ( member_nat @ Y @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ A5 @ F2 @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_507_inv__into__into,axiom,
! [X3: rat,F2: nat > rat,A5: set_nat] :
( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( member_nat @ ( hilber2998747136712319222at_rat @ A5 @ F2 @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_508_inv__into__into,axiom,
! [X3: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ ( image_4216501698144419397ctxt_a @ F2 @ A5 ) )
=> ( member461722496487281478ctxt_a @ ( hilber1117725407648346380ctxt_a @ A5 @ F2 @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_509_inv__into__into,axiom,
! [X3: produc8175205341597284509ctxt_a,F2: nat > produc8175205341597284509ctxt_a,A5: set_nat] :
( ( member461722496487281478ctxt_a @ X3 @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
=> ( member_nat @ ( hilber1942506781820365823ctxt_a @ A5 @ F2 @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_510_inv__into__into,axiom,
! [X3: nat,F2: produc8175205341597284509ctxt_a > nat,A5: set_Pr8705490787052945405ctxt_a] :
( ( member_nat @ X3 @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
=> ( member461722496487281478ctxt_a @ ( hilber7880078095608487457_a_nat @ A5 @ F2 @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_511_inv__into__into,axiom,
! [X3: nat,F2: nat > nat,A5: set_nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A5 @ F2 @ X3 ) @ A5 ) ) ).
% inv_into_into
thf(fact_512_inv__into__injective,axiom,
! [A5: set_nat,F2: nat > rat,X3: rat,Y: rat] :
( ( ( hilber2998747136712319222at_rat @ A5 @ F2 @ X3 )
= ( hilber2998747136712319222at_rat @ A5 @ F2 @ Y ) )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ( member_rat @ Y @ ( image_nat_rat @ F2 @ A5 ) )
=> ( X3 = Y ) ) ) ) ).
% inv_into_injective
thf(fact_513_inv__into__injective,axiom,
! [A5: set_nat,F2: nat > nat,X3: nat,Y: nat] :
( ( ( hilber3633877196798814958at_nat @ A5 @ F2 @ X3 )
= ( hilber3633877196798814958at_nat @ A5 @ F2 @ Y ) )
=> ( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F2 @ A5 ) )
=> ( X3 = Y ) ) ) ) ).
% inv_into_injective
thf(fact_514_SUP__cong,axiom,
! [A5: set_nat,B5: set_nat,C: nat > nat,D: nat > nat] :
( ( A5 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A5 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_515_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_516_Union__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Union_UNIV
thf(fact_517_total__on__def,axiom,
( total_2093374903485932954ctxt_a
= ( ^ [A6: set_Ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
! [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A6 )
=> ! [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A6 )
=> ( ( X4 != Y4 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R3 )
| ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_518_total__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a,Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y3 @ A5 )
=> ( ( X != Y3 )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Y3 ) @ R )
| ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y3 @ X ) @ R ) ) ) ) )
=> ( total_4322320475612085793ctxt_a @ A5 @ R ) ) ).
% total_onI
thf(fact_519_total__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( X != Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) ) ) ) )
=> ( total_on_nat @ A5 @ R ) ) ).
% total_onI
thf(fact_520_total__onI,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ A5 )
=> ( ( member3306386364193653941ctxt_a @ Y3 @ A5 )
=> ( ( X != Y3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
| ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ X ) @ R ) ) ) ) )
=> ( total_2093374903485932954ctxt_a @ A5 @ R ) ) ).
% total_onI
thf(fact_521_surj__image__vimage__eq,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F2 @ ( vimage_nat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_522_surj__image__vimage__eq,axiom,
! [F2: nat > rat,A5: set_rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F2 @ ( vimage_nat_rat @ F2 @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_523_surj__image__vimage__eq,axiom,
! [F2: rat > nat,A5: set_nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F2 @ ( vimage_rat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_524_surj__image__vimage__eq,axiom,
! [F2: rat > rat,A5: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F2 @ ( vimage_rat_rat @ F2 @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_525_image__subset__iff__subset__vimage,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
= ( ord_less_eq_set_nat @ A5 @ ( vimage_nat_rat @ F2 @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_526_image__subset__iff__subset__vimage,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
= ( ord_less_eq_set_nat @ A5 @ ( vimage_nat_nat @ F2 @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_527_image__vimage__subset,axiom,
! [F2: nat > rat,A5: set_rat] : ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( vimage_nat_rat @ F2 @ A5 ) ) @ A5 ) ).
% image_vimage_subset
thf(fact_528_image__vimage__subset,axiom,
! [F2: nat > nat,A5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( vimage_nat_nat @ F2 @ A5 ) ) @ A5 ) ).
% image_vimage_subset
thf(fact_529_surj__imp__inv__eq,axiom,
! [F2: nat > nat,G2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_530_surj__imp__inv__eq,axiom,
! [F2: nat > rat,G2: rat > nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_531_surj__imp__inv__eq,axiom,
! [F2: rat > nat,G2: nat > rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_532_surj__imp__inv__eq,axiom,
! [F2: rat > rat,G2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_533_image__f__inv__f,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F2 @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ A5 ) )
= A5 ) ) ).
% image_f_inv_f
thf(fact_534_image__f__inv__f,axiom,
! [F2: nat > rat,A5: set_rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F2 @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ A5 ) )
= A5 ) ) ).
% image_f_inv_f
thf(fact_535_image__f__inv__f,axiom,
! [F2: rat > nat,A5: set_nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F2 @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ A5 ) )
= A5 ) ) ).
% image_f_inv_f
thf(fact_536_image__f__inv__f,axiom,
! [F2: rat > rat,A5: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F2 @ ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ A5 ) )
= A5 ) ) ).
% image_f_inv_f
thf(fact_537_surj__iff__all,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X4: nat] :
( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_538_surj__iff__all,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
= ( ! [X4: rat] :
( ( F2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_539_surj__iff__all,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
= ( ! [X4: nat] :
( ( F2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_540_surj__iff__all,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
= ( ! [X4: rat] :
( ( F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_541_surj__f__inv__f,axiom,
! [F2: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_542_surj__f__inv__f,axiom,
! [F2: nat > rat,Y: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( F2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_543_surj__f__inv__f,axiom,
! [F2: rat > nat,Y: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( F2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_544_surj__f__inv__f,axiom,
! [F2: rat > rat,Y: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_545_image__inv__into__cancel,axiom,
! [F2: rat > nat,A5: set_rat,A7: set_nat,B7: set_nat] :
( ( ( image_rat_nat @ F2 @ A5 )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_rat_nat @ F2 @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A5 @ F2 ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_546_image__inv__into__cancel,axiom,
! [F2: nat > rat,A5: set_nat,A7: set_rat,B7: set_rat] :
( ( ( image_nat_rat @ F2 @ A5 )
= A7 )
=> ( ( ord_less_eq_set_rat @ B7 @ A7 )
=> ( ( image_nat_rat @ F2 @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A5 @ F2 ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_547_image__inv__into__cancel,axiom,
! [F2: nat > nat,A5: set_nat,A7: set_nat,B7: set_nat] :
( ( ( image_nat_nat @ F2 @ A5 )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_nat_nat @ F2 @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ F2 ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_548_total__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
( ( total_on_nat @ top_top_set_nat @ R_A )
=> ( ( total_on_nat @ top_top_set_nat @ R_B )
=> ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_549_total__lex__prod,axiom,
! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8928021450653196913at_rat] :
( ( total_on_nat @ top_top_set_nat @ R_A )
=> ( ( total_on_rat @ top_top_set_rat @ R_B )
=> ( total_83873191137376397at_rat @ top_to7513191607651882425at_rat @ ( lex_prod_nat_rat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_550_total__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr1261947904930325089at_nat] :
( ( total_on_rat @ top_top_set_rat @ R_A )
=> ( ( total_on_nat @ top_top_set_nat @ R_B )
=> ( total_3275547105595907213at_nat @ top_to269121717765781945at_nat @ ( lex_prod_rat_nat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_551_total__lex__prod,axiom,
! [R_A: set_Pr8928021450653196913at_rat,R_B: set_Pr8928021450653196913at_rat] :
( ( total_on_rat @ top_top_set_rat @ R_A )
=> ( ( total_on_rat @ top_top_set_rat @ R_B )
=> ( total_8990690584057286293at_rat @ top_to3112507417142880193at_rat @ ( lex_prod_rat_rat @ R_A @ R_B ) ) ) ) ).
% total_lex_prod
thf(fact_552_totalI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( X != Y3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
| ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ X ) @ R ) ) )
=> ( total_2093374903485932954ctxt_a @ top_to3933008015346724740ctxt_a @ R ) ) ).
% totalI
thf(fact_553_totalI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R ) ) )
=> ( total_on_nat @ top_top_set_nat @ R ) ) ).
% totalI
thf(fact_554_totalI,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ! [X: rat,Y3: rat] :
( ( X != Y3 )
=> ( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X @ Y3 ) @ R )
| ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Y3 @ X ) @ R ) ) )
=> ( total_on_rat @ top_top_set_rat @ R ) ) ).
% totalI
thf(fact_555_vimage__subsetD,axiom,
! [F2: nat > nat,B5: set_nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F2 @ B5 ) @ A5 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_556_vimage__subsetD,axiom,
! [F2: nat > rat,B5: set_rat,A5: set_nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F2 @ B5 ) @ A5 )
=> ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_557_vimage__subsetD,axiom,
! [F2: rat > nat,B5: set_nat,A5: set_rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_nat @ F2 @ B5 ) @ A5 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_rat_nat @ F2 @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_558_vimage__subsetD,axiom,
! [F2: rat > rat,B5: set_rat,A5: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_rat @ F2 @ B5 ) @ A5 )
=> ( ord_less_eq_set_rat @ B5 @ ( image_rat_rat @ F2 @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_559_inj__vimage__image__eq,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( vimage_nat_rat @ F2 @ ( image_nat_rat @ F2 @ A5 ) )
= A5 ) ) ).
% inj_vimage_image_eq
thf(fact_560_inj__vimage__image__eq,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( vimage_nat_nat @ F2 @ ( image_nat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% inj_vimage_image_eq
thf(fact_561_inj__transfer,axiom,
! [F2: nat > rat,P2: nat > $o,X3: nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ! [Y3: rat] :
( ( member_rat @ Y3 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ Y3 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_562_inj__transfer,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,P2: nat > $o,X3: nat] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ top_top_set_nat )
=> ( ! [Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Y3 @ ( image_43530617296285190ctxt_a @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber1942506781820365823ctxt_a @ top_top_set_nat @ F2 @ Y3 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_563_inj__transfer,axiom,
! [F2: nat > nat,P2: nat > $o,X3: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y3 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_564_inj__transfer,axiom,
! [F2: rat > produc8175205341597284509ctxt_a,P2: rat > $o,X3: rat] :
( ( inj_on8318268736959861138ctxt_a @ F2 @ top_top_set_rat )
=> ( ! [Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Y3 @ ( image_9112142474756165118ctxt_a @ F2 @ top_top_set_rat ) )
=> ( P2 @ ( hilber1787746602425469943ctxt_a @ top_top_set_rat @ F2 @ Y3 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_565_inj__transfer,axiom,
! [F2: rat > nat,P2: rat > $o,X3: rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
=> ( P2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ Y3 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_566_image__inv__f__f,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ ( image_nat_rat @ F2 @ A5 ) )
= A5 ) ) ).
% image_inv_f_f
thf(fact_567_image__inv__f__f,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ ( image_nat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% image_inv_f_f
thf(fact_568_image__inv__f__f,axiom,
! [F2: rat > nat,A5: set_rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ ( image_rat_nat @ F2 @ A5 ) )
= A5 ) ) ).
% image_inv_f_f
thf(fact_569_inj__imp__surj__inv,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_570_inj__imp__surj__inv,axiom,
! [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ top_top_set_rat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_571_inj__imp__surj__inv,axiom,
! [F2: rat > nat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ top_top_set_nat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_572_inj__imp__surj__inv,axiom,
! [F2: rat > rat] :
( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_573_surj__imp__inj__inv,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_574_surj__imp__inj__inv,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_575_surj__imp__inj__inv,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( inj_on_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_576_surj__imp__inj__inv,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_577_inj__on__inv__into,axiom,
! [B5: set_rat,F2: nat > rat,A5: set_nat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ A5 @ F2 ) @ B5 ) ) ).
% inj_on_inv_into
thf(fact_578_inj__on__inv__into,axiom,
! [B5: set_nat,F2: nat > nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ A5 @ F2 ) @ B5 ) ) ).
% inj_on_inv_into
thf(fact_579_card__vimage__inj,axiom,
! [F2: nat > rat,A5: set_rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ A5 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_rat @ F2 @ A5 ) )
= ( finite_card_rat @ A5 ) ) ) ) ).
% card_vimage_inj
thf(fact_580_card__vimage__inj,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% card_vimage_inj
thf(fact_581_bijection_Oinj__inv,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat ) ) ).
% bijection.inj_inv
thf(fact_582_bijection_Oinj__inv,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat ) ) ).
% bijection.inj_inv
thf(fact_583_Sup__SUP__eq,axiom,
( comple8887908400527752307xt_a_o
= ( ^ [S4: set_Pr7428573071411626710xt_a_o,X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ ( comple1483637506433994474ctxt_a @ ( image_7560654748595339286ctxt_a @ collec5218855173738699400ctxt_a @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_584_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_585_relcompEpair,axiom,
! [A: ground_gmctxt_a,C2: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( relcom6868745818274832221ctxt_a @ R @ S ) )
=> ~ ! [B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_586_relcompE,axiom,
! [Xz: produc8175205341597284509ctxt_a,R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ Xz @ ( relcom6868745818274832221ctxt_a @ R @ S ) )
=> ~ ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( Xz
= ( produc6347872285926497493ctxt_a @ X @ Z3 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_587_relcomp_OrelcompI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ S )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( relcom6868745818274832221ctxt_a @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_588_relcomp_Osimps,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( relcom6868745818274832221ctxt_a @ R @ S ) )
= ( ? [A4: ground_gmctxt_a,B4: ground_gmctxt_a,C4: ground_gmctxt_a] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A4 @ B4 ) @ R )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_589_relcomp_Ocases,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,S: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( relcom6868745818274832221ctxt_a @ R @ S ) )
=> ~ ! [B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ B3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_590_card__image,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ).
% card_image
thf(fact_591_card__image,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ).
% card_image
thf(fact_592_bijection_Osurj,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_593_bijection_Osurj,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj
thf(fact_594_bijection_Oinj,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_595_bijection_Oinj,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( inj_on_rat_rat @ F2 @ top_top_set_rat ) ) ).
% bijection.inj
thf(fact_596_bijection_Oinv__right__eq__iff,axiom,
! [F2: nat > nat,B: nat,A: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( B
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ A ) )
= ( ( F2 @ B )
= A ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_597_bijection_Oinv__right__eq__iff,axiom,
! [F2: rat > rat,B: rat,A: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( B
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ A ) )
= ( ( F2 @ B )
= A ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_598_bijection_Oinv__left__eq__iff,axiom,
! [F2: nat > nat,A: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ A )
= B )
= ( ( F2 @ B )
= A ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_599_bijection_Oinv__left__eq__iff,axiom,
! [F2: rat > rat,A: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ A )
= B )
= ( ( F2 @ B )
= A ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_600_bijection_Oeq__inv__iff,axiom,
! [F2: nat > nat,A: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ A )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ B ) )
= ( A = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_601_bijection_Oeq__inv__iff,axiom,
! [F2: rat > rat,A: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ A )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ B ) )
= ( A = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_602_bijection_Oinv__right,axiom,
! [F2: nat > nat,A: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ A ) )
= A ) ) ).
% bijection.inv_right
thf(fact_603_bijection_Oinv__right,axiom,
! [F2: rat > rat,A: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ A ) )
= A ) ) ).
% bijection.inv_right
thf(fact_604_bijection_Oinv__left,axiom,
! [F2: nat > nat,A: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ ( F2 @ A ) )
= A ) ) ).
% bijection.inv_left
thf(fact_605_bijection_Oinv__left,axiom,
! [F2: rat > rat,A: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ ( F2 @ A ) )
= A ) ) ).
% bijection.inv_left
thf(fact_606_bijection_Oeq__invI,axiom,
! [F2: nat > nat,A: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ A )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ B ) )
=> ( A = B ) ) ) ).
% bijection.eq_invI
thf(fact_607_bijection_Oeq__invI,axiom,
! [F2: rat > rat,A: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ A )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ B ) )
=> ( A = B ) ) ) ).
% bijection.eq_invI
thf(fact_608_bijection_Osurj__inv,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_609_bijection_Osurj__inv,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj_inv
thf(fact_610_relcomp3__I,axiom,
! [T: ground_gmctxt_a,U2: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a,S: ground_gmctxt_a,B5: set_Pr8705490787052945405ctxt_a,V2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T @ U2 ) @ A5 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ B5 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ U2 @ V2 ) @ B5 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ V2 ) @ ( relcom6868745818274832221ctxt_a @ B5 @ ( relcom6868745818274832221ctxt_a @ A5 @ B5 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_611_finite__vimageD_H,axiom,
! [F2: nat > rat,A5: set_rat] :
( ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_rat @ A5 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( finite_finite_rat @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_612_finite__vimageD_H,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_613_finite__vimageD_H,axiom,
! [F2: rat > nat,A5: set_nat] :
( ( finite_finite_rat @ ( vimage_rat_nat @ F2 @ A5 ) )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_614_surjective__iff__injective__gen,axiom,
! [S3: set_nat,T3: set_rat,F2: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_rat @ T3 )
=> ( ( ( finite_card_nat @ S3 )
= ( finite_card_rat @ T3 ) )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ S3 ) @ T3 )
=> ( ( ! [X4: rat] :
( ( member_rat @ X4 @ T3 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
& ( ( F2 @ Y4 )
= X4 ) ) ) )
= ( inj_on_nat_rat @ F2 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_615_surjective__iff__injective__gen,axiom,
! [S3: set_nat,T3: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S3 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S3 ) @ T3 )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T3 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
& ( ( F2 @ Y4 )
= X4 ) ) ) )
= ( inj_on_nat_nat @ F2 @ S3 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_616_inj__on__iff__card__le,axiom,
! [A5: set_nat,B5: set_rat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( ? [F: nat > rat] :
( ( inj_on_nat_rat @ F @ A5 )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A5 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_rat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_617_inj__on__iff__card__le,axiom,
! [A5: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F: nat > nat] :
( ( inj_on_nat_nat @ F @ A5 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_618_finite__imageI,axiom,
! [F4: set_nat,H: nat > rat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_rat @ ( image_nat_rat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_619_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_620_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_621_finite__Plus__UNIV__iff,axiom,
( ( finite2679478125380364318at_rat @ top_to281834657035230061at_rat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_622_finite__Plus__UNIV__iff,axiom,
( ( finite5871152039838895134at_nat @ top_to2261136804003905389at_nat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_623_finite__Plus__UNIV__iff,axiom,
( ( finite2362923481445498406at_rat @ top_to5104522503381003637at_rat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_624_finite__UN,axiom,
! [A5: set_nat,B5: nat > set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A5 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( finite_finite_nat @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_625_finite__compl,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A5 ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_626_finite__compl,axiom,
! [A5: set_rat] :
( ( finite_finite_rat @ A5 )
=> ( ( finite_finite_rat @ ( uminus2201863774496077783et_rat @ A5 ) )
= ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_compl
thf(fact_627_ex__new__if__finite,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ~ ( finite7161443020207210054ctxt_a @ top_to2898181658028262733ctxt_a )
=> ( ( finite7161443020207210054ctxt_a @ A5 )
=> ? [A3: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ A3 @ A5 ) ) ) ).
% ex_new_if_finite
thf(fact_628_ex__new__if__finite,axiom,
! [A5: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A5 )
=> ? [A3: nat] :
~ ( member_nat @ A3 @ A5 ) ) ) ).
% ex_new_if_finite
thf(fact_629_ex__new__if__finite,axiom,
! [A5: set_rat] :
( ~ ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_rat @ A5 )
=> ? [A3: rat] :
~ ( member_rat @ A3 @ A5 ) ) ) ).
% ex_new_if_finite
thf(fact_630_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_631_infinite__UNIV__char__0,axiom,
~ ( finite_finite_rat @ top_top_set_rat ) ).
% infinite_UNIV_char_0
thf(fact_632_infinite__countable__subset,axiom,
! [S3: set_rat] :
( ~ ( finite_finite_rat @ S3 )
=> ? [F3: nat > rat] :
( ( inj_on_nat_rat @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F3 @ top_top_set_nat ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_633_infinite__countable__subset,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_634_infinite__iff__countable__subset,axiom,
! [S3: set_rat] :
( ( ~ ( finite_finite_rat @ S3 ) )
= ( ? [F: nat > rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_635_infinite__iff__countable__subset,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ? [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_636_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_637_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_rat @ top_top_set_rat )
=> ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat ) ) ) ).
% finite_Prod_UNIV
thf(fact_638_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_639_finite__Prod__UNIV,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( finite_finite_rat @ top_top_set_rat )
=> ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat ) ) ) ).
% finite_Prod_UNIV
thf(fact_640_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_641_finite__prod,axiom,
( ( finite2668982390342448306at_rat @ top_to7513191607651882425at_rat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_prod
thf(fact_642_finite__prod,axiom,
( ( finite5860656304800979122at_nat @ top_to269121717765781945at_nat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_643_finite__prod,axiom,
( ( finite2352427746407582394at_rat @ top_to3112507417142880193at_rat )
= ( ( finite_finite_rat @ top_top_set_rat )
& ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_prod
thf(fact_644_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_645_Finite__Set_Ofinite__set,axiom,
( ( finite6867581373910428453et_rat @ top_top_set_set_rat )
= ( finite_finite_rat @ top_top_set_rat ) ) ).
% Finite_Set.finite_set
thf(fact_646_all__finite__subset__image,axiom,
! [F2: nat > rat,A5: set_nat,P2: set_rat > $o] :
( ( ! [B6: set_rat] :
( ( ( finite_finite_rat @ B6 )
& ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F2 @ A5 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) )
=> ( P2 @ ( image_nat_rat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_647_all__finite__subset__image,axiom,
! [F2: nat > nat,A5: set_nat,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A5 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) )
=> ( P2 @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_648_ex__finite__subset__image,axiom,
! [F2: nat > rat,A5: set_nat,P2: set_rat > $o] :
( ( ? [B6: set_rat] :
( ( finite_finite_rat @ B6 )
& ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F2 @ A5 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 )
& ( P2 @ ( image_nat_rat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_649_ex__finite__subset__image,axiom,
! [F2: nat > nat,A5: set_nat,P2: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A5 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 )
& ( P2 @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_650_finite__subset__image,axiom,
! [B5: set_rat,F2: nat > rat,A5: set_nat] :
( ( finite_finite_rat @ B5 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A5 )
& ( finite_finite_nat @ C3 )
& ( B5
= ( image_nat_rat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_651_finite__subset__image,axiom,
! [B5: set_nat,F2: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A5 )
& ( finite_finite_nat @ C3 )
& ( B5
= ( image_nat_nat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_652_finite__surj,axiom,
! [A5: set_nat,B5: set_rat,F2: nat > rat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( finite_finite_rat @ B5 ) ) ) ).
% finite_surj
thf(fact_653_finite__surj,axiom,
! [A5: set_nat,B5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_654_finite__imageD,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ( inj_on_nat_rat @ F2 @ A5 )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_imageD
thf(fact_655_finite__imageD,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ( inj_on_nat_nat @ F2 @ A5 )
=> ( finite_finite_nat @ A5 ) ) ) ).
% finite_imageD
thf(fact_656_finite__image__iff,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A5 ) )
= ( finite_finite_nat @ A5 ) ) ) ).
% finite_image_iff
thf(fact_657_finite__image__iff,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A5 ) )
= ( finite_finite_nat @ A5 ) ) ) ).
% finite_image_iff
thf(fact_658_card__image__le,axiom,
! [A5: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ A5 )
=> ( ord_less_eq_nat @ ( finite_card_rat @ ( image_nat_rat @ F2 @ A5 ) ) @ ( finite_card_nat @ A5 ) ) ) ).
% card_image_le
thf(fact_659_card__image__le,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ A5 ) ) @ ( finite_card_nat @ A5 ) ) ) ).
% card_image_le
thf(fact_660_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A5 )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A5 = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_661_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A5: set_rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( ( finite_card_rat @ A5 )
= ( finite_card_rat @ top_top_set_rat ) )
=> ( A5 = top_top_set_rat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_662_finite__UNIV__surj__inj,axiom,
! [F2: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_663_finite__UNIV__surj__inj,axiom,
! [F2: rat > rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ F2 @ top_top_set_rat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_664_finite__UNIV__inj__surj,axiom,
! [F2: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_665_finite__UNIV__inj__surj,axiom,
! [F2: rat > rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
=> ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_666_finite__surj__inj,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ A5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( inj_on_nat_nat @ F2 @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_667_inj__on__finite,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
=> ( ( finite_finite_rat @ B5 )
=> ( finite_finite_nat @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_668_inj__on__finite,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_669_endo__inj__surj,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ A5 )
=> ( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( image_nat_nat @ F2 @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_670_surj__card__le,axiom,
! [A5: set_nat,B5: set_rat,F2: nat > rat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ord_less_eq_nat @ ( finite_card_rat @ B5 ) @ ( finite_card_nat @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_671_surj__card__le,axiom,
! [A5: set_nat,B5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_672_eq__card__imp__inj__on,axiom,
! [A5: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) )
=> ( inj_on_nat_rat @ F2 @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_673_eq__card__imp__inj__on,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) )
=> ( inj_on_nat_nat @ F2 @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_674_inj__on__iff__eq__card,axiom,
! [A5: set_nat,F2: nat > rat] :
( ( finite_finite_nat @ A5 )
=> ( ( inj_on_nat_rat @ F2 @ A5 )
= ( ( finite_card_rat @ ( image_nat_rat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_675_inj__on__iff__eq__card,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A5 )
=> ( ( inj_on_nat_nat @ F2 @ A5 )
= ( ( finite_card_nat @ ( image_nat_nat @ F2 @ A5 ) )
= ( finite_card_nat @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_676_finite__vimageD,axiom,
! [H: nat > nat,F4: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F4 ) )
=> ( ( ( image_nat_nat @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_677_finite__vimageD,axiom,
! [H: nat > rat,F4: set_rat] :
( ( finite_finite_nat @ ( vimage_nat_rat @ H @ F4 ) )
=> ( ( ( image_nat_rat @ H @ top_top_set_nat )
= top_top_set_rat )
=> ( finite_finite_rat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_678_finite__vimageD,axiom,
! [H: rat > nat,F4: set_nat] :
( ( finite_finite_rat @ ( vimage_rat_nat @ H @ F4 ) )
=> ( ( ( image_rat_nat @ H @ top_top_set_rat )
= top_top_set_nat )
=> ( finite_finite_nat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_679_finite__vimageD,axiom,
! [H: rat > rat,F4: set_rat] :
( ( finite_finite_rat @ ( vimage_rat_rat @ H @ F4 ) )
=> ( ( ( image_rat_rat @ H @ top_top_set_rat )
= top_top_set_rat )
=> ( finite_finite_rat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_680_finite__vimageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( ( inj_on_nat_nat @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage_nat_nat @ H @ F4 ) ) ) ) ).
% finite_vimageI
thf(fact_681_finite__vimageI,axiom,
! [F4: set_nat,H: rat > nat] :
( ( finite_finite_nat @ F4 )
=> ( ( inj_on_rat_nat @ H @ top_top_set_rat )
=> ( finite_finite_rat @ ( vimage_rat_nat @ H @ F4 ) ) ) ) ).
% finite_vimageI
thf(fact_682_card__bij__eq,axiom,
! [F2: rat > nat,A5: set_rat,B5: set_nat,G2: nat > rat] :
( ( inj_on_rat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F2 @ A5 ) @ B5 )
=> ( ( inj_on_nat_rat @ G2 @ B5 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ G2 @ B5 ) @ A5 )
=> ( ( finite_finite_rat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_rat @ A5 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_683_card__bij__eq,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat,G2: rat > nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
=> ( ( inj_on_rat_nat @ G2 @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ G2 @ B5 ) @ A5 )
=> ( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( finite_card_nat @ A5 )
= ( finite_card_rat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_684_card__bij__eq,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat,G2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
=> ( ( inj_on_nat_nat @ G2 @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B5 ) @ A5 )
=> ( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_nat @ A5 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_685_card__le__inj,axiom,
! [A5: set_nat,B5: set_rat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_rat @ B5 ) )
=> ? [F3: nat > rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F3 @ A5 ) @ B5 )
& ( inj_on_nat_rat @ F3 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_686_card__le__inj,axiom,
! [A5: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) )
=> ? [F3: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A5 ) @ B5 )
& ( inj_on_nat_nat @ F3 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_687_card__inj__on__le,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_rat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ B5 )
=> ( ( finite_finite_rat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_rat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_688_card__inj__on__le,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_689_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_690_finite__option__UNIV,axiom,
( ( finite2014924581280026175on_rat @ top_to2540212048668676366on_rat )
= ( finite_finite_rat @ top_top_set_rat ) ) ).
% finite_option_UNIV
thf(fact_691_comp__rtrancl__trancl,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a,S: ground_gmctxt_a,T: ground_gmctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( relcom6868745818274832221ctxt_a @ R2 @ S3 ) @ S3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ ( sup_su1394892571904902097ctxt_a @ R2 @ S3 ) ) @ S3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ ( transi3225167780237802813ctxt_a @ S3 ) ) ) ) ).
% comp_rtrancl_trancl
thf(fact_692_mlex__leq,axiom,
! [F2: ground_gmctxt_a > nat,X3: ground_gmctxt_a,Y: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ R2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( mlex_p7507609278379928015ctxt_a @ F2 @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_693_wf__bounded__set,axiom,
! [R: set_Pr8705490787052945405ctxt_a,Ub: ground_gmctxt_a > set_nat,F2: ground_gmctxt_a > set_nat] :
( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ A3 ) @ R )
=> ( ( finite_finite_nat @ ( Ub @ A3 ) )
& ( ord_less_eq_set_nat @ ( Ub @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_eq_set_nat @ ( F2 @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_set_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) )
=> ( wf_Ground_gmctxt_a @ R ) ) ).
% wf_bounded_set
thf(fact_694_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_695_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_696_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_697_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_698_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_699_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_700_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_701_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_702_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_703_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_704_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_705_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_706_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_707_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_708_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_709_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_710_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_711_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_712_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_713_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_714_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_715_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_716_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_717_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_718_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_719_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_720_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_721_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_722_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_723_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_724_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_725_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_726_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_727_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_728_converse__rtrancl__into__rtrancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_729_converse__rtrancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( P2 @ B )
=> ( ! [Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_730_converse__rtranclE,axiom,
! [X3: ground_gmctxt_a,Z: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( X3 != Z )
=> ~ ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y3 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_731_rtrancl__induct,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( P2 @ A )
=> ( ! [Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y3 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ Z3 ) @ R )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_732_rtrancl__trans,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,Z: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ Z ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_733_rtranclE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( A != B )
=> ~ ! [Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ Y3 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_734_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_735_rtrancl_Ortrancl__refl,axiom,
! [A: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A ) @ ( transi2468631686036622171ctxt_a @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_736_rtrancl_Osimps,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( transi2468631686036622171ctxt_a @ R ) )
= ( ? [A4: ground_gmctxt_a] :
( ( A1 = A4 )
& ( A22 = A4 ) )
| ? [A4: ground_gmctxt_a,B4: ground_gmctxt_a,C4: ground_gmctxt_a] :
( ( A1 = A4 )
& ( A22 = C4 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A4 @ B4 ) @ ( transi2468631686036622171ctxt_a @ R ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B4 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_737_rtrancl_Ocases,axiom,
! [A1: ground_gmctxt_a,A22: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ A22 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A1 @ B3 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_738_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_739_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_740_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_741_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_742_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_743_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_744_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_745_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_746_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_747_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_748_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_749_exists__least__iff,axiom,
( ( ^ [P6: nat > $o] :
? [X9: nat] : ( P6 @ X9 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_750_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_751_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_752_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_753_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_754_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_755_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_756_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_757_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_758_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_759_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_760_order__less__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_761_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_762_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_763_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_764_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_765_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_766_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_767_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_768_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_769_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_770_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_771_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_772_psubsetD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,C2: produc8175205341597284509ctxt_a] :
( ( ord_le254475249434039465ctxt_a @ A5 @ B5 )
=> ( ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_773_psubsetD,axiom,
! [A5: set_nat,B5: set_nat,C2: nat] :
( ( ord_less_set_nat @ A5 @ B5 )
=> ( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% psubsetD
thf(fact_774_transp__on__less,axiom,
! [A5: set_nat] : ( transp_on_nat @ A5 @ ord_less_nat ) ).
% transp_on_less
thf(fact_775_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_776_top_Oextremum__strict,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A ) ).
% top.extremum_strict
thf(fact_777_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_778_top_Onot__eq__extremum,axiom,
! [A: set_rat] :
( ( A != top_top_set_rat )
= ( ord_less_set_rat @ A @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_779_steps__map,axiom,
! [P2: ground_gmctxt_a > $o,Q: set_Pr8705490787052945405ctxt_a > $o,F2: ground_gmctxt_a > ground_gmctxt_a,G2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a,T: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a,U2: ground_gmctxt_a] :
( ! [T4: ground_gmctxt_a,U3: ground_gmctxt_a,R4: set_Pr8705490787052945405ctxt_a] :
( ( P2 @ T4 )
=> ( ( Q @ R4 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T4 @ U3 ) @ R4 )
=> ( ( P2 @ U3 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ T4 ) @ ( F2 @ U3 ) ) @ ( G2 @ R4 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R2 )
=> ( ( Q @ S3 )
=> ( ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T @ U2 ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ T ) @ ( F2 @ U2 ) ) @ ( transi2468631686036622171ctxt_a @ ( G2 @ R2 ) ) ) )
& ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T @ U2 ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ R2 ) @ ( relcom6868745818274832221ctxt_a @ S3 @ ( transi2468631686036622171ctxt_a @ R2 ) ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ T ) @ ( F2 @ U2 ) ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ ( G2 @ R2 ) ) @ ( relcom6868745818274832221ctxt_a @ ( G2 @ S3 ) @ ( transi2468631686036622171ctxt_a @ ( G2 @ R2 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_780_tranclD,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R2 ) )
=> ? [Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z3 ) @ R2 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R2 ) ) ) ) ).
% tranclD
thf(fact_781_rtranclD,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R2 ) ) ) ) ) ).
% rtranclD
thf(fact_782_tranclD2,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R2 ) )
=> ? [Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z3 ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ Y ) @ R2 ) ) ) ).
% tranclD2
thf(fact_783_trancl__into__rtrancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_784_rtrancl__eq__or__trancl,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi3225167780237802813ctxt_a @ R2 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_785_rtrancl__into__trancl1,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ R )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_786_rtrancl__into__trancl2,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_787_rtrancl__trancl__trancl,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,Z: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Z ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_788_trancl__rtrancl__trancl,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi3225167780237802813ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_789_rtrancl__Un__separator__converseE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,P2: set_Pr8705490787052945405ctxt_a,Q: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ ( sup_su1394892571904902097ctxt_a @ P2 @ Q ) ) )
=> ( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ B ) @ ( transi2468631686036622171ctxt_a @ P2 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ X ) @ Q )
=> ( Y3 = X ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_790_rtrancl__Un__separatorE,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,P2: set_Pr8705490787052945405ctxt_a,Q: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ ( sup_su1394892571904902097ctxt_a @ P2 @ Q ) ) )
=> ( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ X ) @ ( transi2468631686036622171ctxt_a @ P2 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ Q )
=> ( X = Y3 ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_791_image__strict__mono,axiom,
! [F2: nat > rat,B5: set_nat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ B5 )
=> ( ( ord_less_set_nat @ A5 @ B5 )
=> ( ord_less_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ) ).
% image_strict_mono
thf(fact_792_image__strict__mono,axiom,
! [F2: nat > nat,B5: set_nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ B5 )
=> ( ( ord_less_set_nat @ A5 @ B5 )
=> ( ord_less_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ) ).
% image_strict_mono
thf(fact_793_converse__rtrancl__induct2,axiom,
! [Ax: ground_gmctxt_a,Ay: ground_gmctxt_a,Bx: ground_gmctxt_a,By: ground_gmctxt_a,R: set_Pr490164551564945991ctxt_a,P2: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ Bx @ By ) ) @ ( transi2118924707453815072ctxt_a @ R ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a,Aa: ground_gmctxt_a,Ba: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) @ ( produc6347872285926497493ctxt_a @ Aa @ Ba ) ) @ R )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Aa @ Ba ) @ ( produc6347872285926497493ctxt_a @ Bx @ By ) ) @ ( transi2118924707453815072ctxt_a @ R ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_794_converse__rtranclE2,axiom,
! [Xa2: ground_gmctxt_a,Xb: ground_gmctxt_a,Za: ground_gmctxt_a,Zb: ground_gmctxt_a,R: set_Pr490164551564945991ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Xa2 @ Xb ) @ ( produc6347872285926497493ctxt_a @ Za @ Zb ) ) @ ( transi2118924707453815072ctxt_a @ R ) )
=> ( ( ( produc6347872285926497493ctxt_a @ Xa2 @ Xb )
!= ( produc6347872285926497493ctxt_a @ Za @ Zb ) )
=> ~ ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Xa2 @ Xb ) @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) ) @ R )
=> ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) @ ( produc6347872285926497493ctxt_a @ Za @ Zb ) ) @ ( transi2118924707453815072ctxt_a @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_795_rtrancl__induct2,axiom,
! [Ax: ground_gmctxt_a,Ay: ground_gmctxt_a,Bx: ground_gmctxt_a,By: ground_gmctxt_a,R: set_Pr490164551564945991ctxt_a,P2: ground_gmctxt_a > ground_gmctxt_a > $o] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ Bx @ By ) ) @ ( transi2118924707453815072ctxt_a @ R ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a,Aa: ground_gmctxt_a,Ba: ground_gmctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ Ax @ Ay ) @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) ) @ ( transi2118924707453815072ctxt_a @ R ) )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) @ ( produc6347872285926497493ctxt_a @ Aa @ Ba ) ) @ R )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_796_acyclicI__order,axiom,
! [R: set_Pr8705490787052945405ctxt_a,F2: ground_gmctxt_a > nat] :
( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B3 ) @ R )
=> ( ord_less_nat @ ( F2 @ B3 ) @ ( F2 @ A3 ) ) )
=> ( transi8399219972279719055ctxt_a @ R ) ) ).
% acyclicI_order
thf(fact_797_non__strict__into__strict,axiom,
! [NS: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a,S: ground_gmctxt_a,T: ground_gmctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( relcom6868745818274832221ctxt_a @ NS @ S3 ) @ S3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ NS ) @ S3 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ S3 ) ) ) ).
% non_strict_into_strict
thf(fact_798_right__comp__S,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,S3: set_Pr8705490787052945405ctxt_a,NS: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( relcom6868745818274832221ctxt_a @ S3 @ ( sup_su1394892571904902097ctxt_a @ ( relcom6868745818274832221ctxt_a @ S3 @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ S3 ) @ ( transi2468631686036622171ctxt_a @ NS ) ) ) @ ( transi2468631686036622171ctxt_a @ NS ) ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( relcom6868745818274832221ctxt_a @ S3 @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ S3 ) @ ( transi2468631686036622171ctxt_a @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_799_first__step__O,axiom,
! [C: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( C
= ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ C ) @ B5 ) )
=> ? [Y3: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y3 ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ A5 ) @ B5 ) ) ) ) ).
% first_step_O
thf(fact_800_first__step,axiom,
! [C: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a,Z: ground_gmctxt_a] :
( ( C
= ( sup_su1394892571904902097ctxt_a @ A5 @ B5 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ C ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ Z ) @ B5 )
=> ? [Y3: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y3 ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ A5 ) @ B5 ) ) ) ) ) ).
% first_step
thf(fact_801_firstStep,axiom,
! [L: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a,R2: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( L
= ( sup_su1394892571904902097ctxt_a @ S3 @ R2 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ L ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
| ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ R2 ) @ ( relcom6868745818274832221ctxt_a @ S3 @ ( transi2468631686036622171ctxt_a @ L ) ) ) ) ) ) ) ).
% firstStep
thf(fact_802_compatible__rtrancl__split,axiom,
! [NS: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a,Y: ground_gmctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( relcom6868745818274832221ctxt_a @ NS @ S3 ) @ S3 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ ( sup_su1394892571904902097ctxt_a @ NS @ S3 ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( sup_su1394892571904902097ctxt_a @ ( relcom6868745818274832221ctxt_a @ S3 @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ S3 ) @ ( transi2468631686036622171ctxt_a @ NS ) ) ) @ ( transi2468631686036622171ctxt_a @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_803_range__inj__infinite,axiom,
! [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ~ ( finite_finite_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_804_range__inj__infinite,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ~ ( finite_finite_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_inj_infinite
thf(fact_805_meetI,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a,C2: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ A5 ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ C2 ) @ ( transi2468631686036622171ctxt_a @ A5 ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( abstra1819221482964979745ctxt_a @ A5 ) ) ) ) ).
% meetI
thf(fact_806_surj__from__nat,axiom,
( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_from_nat
thf(fact_807_surj__from__nat,axiom,
( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_from_nat
thf(fact_808_nat__descend__induct,axiom,
! [N3: nat,P2: nat > $o,M3: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N3 @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N3 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P2 @ I3 ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_809_mlex__iff,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,F2: ground_gmctxt_a > nat,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( mlex_p7507609278379928015ctxt_a @ F2 @ R2 ) )
= ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_810_mlex__less,axiom,
! [F2: ground_gmctxt_a > nat,X3: ground_gmctxt_a,Y: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( mlex_p7507609278379928015ctxt_a @ F2 @ R2 ) ) ) ).
% mlex_less
thf(fact_811_wf__bounded__measure,axiom,
! [R: set_Pr8705490787052945405ctxt_a,Ub: ground_gmctxt_a > nat,F2: ground_gmctxt_a > nat] :
( ! [A3: ground_gmctxt_a,B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B3 @ A3 ) @ R )
=> ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_eq_nat @ ( F2 @ B3 ) @ ( Ub @ A3 ) )
& ( ord_less_nat @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) )
=> ( wf_Ground_gmctxt_a @ R ) ) ).
% wf_bounded_measure
thf(fact_812_pigeonhole,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( ord_less_nat @ ( finite_card_rat @ ( image_nat_rat @ F2 @ A5 ) ) @ ( finite_card_nat @ A5 ) )
=> ~ ( inj_on_nat_rat @ F2 @ A5 ) ) ).
% pigeonhole
thf(fact_813_pigeonhole,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ A5 ) ) @ ( finite_card_nat @ A5 ) )
=> ~ ( inj_on_nat_nat @ F2 @ A5 ) ) ).
% pigeonhole
thf(fact_814_ex__inj,axiom,
? [To_nat: nat > nat] : ( inj_on_nat_nat @ To_nat @ top_top_set_nat ) ).
% ex_inj
thf(fact_815_ex__inj,axiom,
? [To_nat: rat > nat] : ( inj_on_rat_nat @ To_nat @ top_top_set_rat ) ).
% ex_inj
thf(fact_816_rat__denum,axiom,
? [F3: nat > rat] :
( ( image_nat_rat @ F3 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_817_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_818_meetE,axiom,
! [B: ground_gmctxt_a,C2: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( abstra1819221482964979745ctxt_a @ A5 ) )
=> ~ ! [A3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B ) @ ( transi2468631686036622171ctxt_a @ A5 ) )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ C2 ) @ ( transi2468631686036622171ctxt_a @ A5 ) ) ) ) ).
% meetE
thf(fact_819_meetD,axiom,
! [B: ground_gmctxt_a,C2: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ C2 ) @ ( abstra1819221482964979745ctxt_a @ A5 ) )
=> ? [A3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B ) @ ( transi2468631686036622171ctxt_a @ A5 ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ C2 ) @ ( transi2468631686036622171ctxt_a @ A5 ) ) ) ) ).
% meetD
thf(fact_820_in__measure,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,F2: ground_gmctxt_a > nat] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( measur2076251321048797014ctxt_a @ F2 ) )
= ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_821_from__nat__def,axiom,
( from_nat_nat
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ to_nat_nat ) ) ).
% from_nat_def
thf(fact_822_from__nat__def,axiom,
( from_nat_rat
= ( hilber3317322552863949046at_nat @ top_top_set_rat @ to_nat_rat ) ) ).
% from_nat_def
thf(fact_823_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_824_top_Oordering__top__axioms,axiom,
ordering_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat @ top_top_set_rat ).
% top.ordering_top_axioms
thf(fact_825_inj__to__nat,axiom,
inj_on_nat_nat @ to_nat_nat @ top_top_set_nat ).
% inj_to_nat
thf(fact_826_inj__to__nat,axiom,
inj_on_rat_nat @ to_nat_rat @ top_top_set_rat ).
% inj_to_nat
thf(fact_827_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_828_card__range__greater__zero,axiom,
! [F2: nat > rat] :
( ( finite_finite_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_829_card__range__greater__zero,axiom,
! [F2: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ) ).
% card_range_greater_zero
thf(fact_830_card__range__greater__zero,axiom,
! [F2: rat > nat] :
( ( finite_finite_nat @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_rat_nat @ F2 @ top_top_set_rat ) ) ) ) ).
% card_range_greater_zero
thf(fact_831_qc__SN__relto__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R @ S ) @ ( relcomp_nat_nat_nat @ S @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ S @ R ) ) ) )
=> ( ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( relcomp_nat_nat_nat @ S @ ( transi2905341329935302413cl_nat @ R ) ) ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ S @ top_top_set_nat ) ) ) ).
% qc_SN_relto_iff
thf(fact_832_qc__SN__relto__iff,axiom,
! [R: set_Pr8928021450653196913at_rat,S: set_Pr8928021450653196913at_rat] :
( ( ord_le1589215037752994321at_rat @ ( relcomp_rat_rat_rat @ R @ S ) @ ( relcomp_rat_rat_rat @ S @ ( transi2270211269848806677cl_rat @ ( sup_su4770203945505871429at_rat @ S @ R ) ) ) )
=> ( ( abstract_SN_on_rat @ ( relcomp_rat_rat_rat @ ( transi2270211269848806677cl_rat @ R ) @ ( relcomp_rat_rat_rat @ S @ ( transi2270211269848806677cl_rat @ R ) ) ) @ top_top_set_rat )
= ( abstract_SN_on_rat @ S @ top_top_set_rat ) ) ) ).
% qc_SN_relto_iff
thf(fact_833_refl__not__SN,axiom,
! [X3: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ X3 ) @ R2 )
=> ~ ( abstra4731155269209573716ctxt_a @ R2 @ top_to3933008015346724740ctxt_a ) ) ).
% refl_not_SN
thf(fact_834_refl__not__SN,axiom,
! [X3: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ~ ( abstract_SN_on_nat @ R2 @ top_top_set_nat ) ) ).
% refl_not_SN
thf(fact_835_refl__not__SN,axiom,
! [X3: rat,R2: set_Pr8928021450653196913at_rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X3 @ X3 ) @ R2 )
=> ~ ( abstract_SN_on_rat @ R2 @ top_top_set_rat ) ) ).
% refl_not_SN
thf(fact_836_SN__imp__minimal,axiom,
! [A5: set_Pr490164551564945991ctxt_a] :
( ( abstra7720173292291745255ctxt_a @ A5 @ top_to2898181658028262733ctxt_a )
=> ! [Q5: set_Pr8705490787052945405ctxt_a] :
( ? [X5: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X5 @ Q5 )
=> ? [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ Q5 )
& ! [Y6: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Y6 ) @ A5 )
=> ~ ( member461722496487281478ctxt_a @ Y6 @ Q5 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_837_SN__imp__minimal,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ( abstra4731155269209573716ctxt_a @ A5 @ top_to3933008015346724740ctxt_a )
=> ! [Q5: set_Ground_gmctxt_a] :
( ? [X5: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X5 @ Q5 )
=> ? [X: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ Q5 )
& ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y6 ) @ A5 )
=> ~ ( member3306386364193653941ctxt_a @ Y6 @ Q5 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_838_SN__imp__minimal,axiom,
! [A5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ A5 @ top_top_set_nat )
=> ! [Q5: set_nat] :
( ? [X5: nat] : ( member_nat @ X5 @ Q5 )
=> ? [X: nat] :
( ( member_nat @ X @ Q5 )
& ! [Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ A5 )
=> ~ ( member_nat @ Y6 @ Q5 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_839_SN__imp__minimal,axiom,
! [A5: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ A5 @ top_top_set_rat )
=> ! [Q5: set_rat] :
( ? [X5: rat] : ( member_rat @ X5 @ Q5 )
=> ? [X: rat] :
( ( member_rat @ X @ Q5 )
& ! [Y6: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X @ Y6 ) @ A5 )
=> ~ ( member_rat @ Y6 @ Q5 ) ) ) ) ) ).
% SN_imp_minimal
thf(fact_840_SN__induct__rule,axiom,
! [R: set_Pr8705490787052945405ctxt_a,P2: ground_gmctxt_a > $o,A: ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R @ top_to3933008015346724740ctxt_a )
=> ( ! [A3: ground_gmctxt_a] :
( ! [B8: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ B8 ) @ R )
=> ( P2 @ B8 ) )
=> ( P2 @ A3 ) )
=> ( P2 @ A ) ) ) ).
% SN_induct_rule
thf(fact_841_SN__induct__rule,axiom,
! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( ! [A3: nat] :
( ! [B8: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B8 ) @ R )
=> ( P2 @ B8 ) )
=> ( P2 @ A3 ) )
=> ( P2 @ A ) ) ) ).
% SN_induct_rule
thf(fact_842_SN__induct__rule,axiom,
! [R: set_Pr8928021450653196913at_rat,P2: rat > $o,A: rat] :
( ( abstract_SN_on_rat @ R @ top_top_set_rat )
=> ( ! [A3: rat] :
( ! [B8: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ A3 @ B8 ) @ R )
=> ( P2 @ B8 ) )
=> ( P2 @ A3 ) )
=> ( P2 @ A ) ) ) ).
% SN_induct_rule
thf(fact_843_SN__subset,axiom,
! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R2 @ top_top_set_nat )
=> ( ( ord_le3146513528884898305at_nat @ R5 @ R2 )
=> ( abstract_SN_on_nat @ R5 @ top_top_set_nat ) ) ) ).
% SN_subset
thf(fact_844_SN__subset,axiom,
! [R2: set_Pr8928021450653196913at_rat,R5: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ R2 @ top_top_set_rat )
=> ( ( ord_le1589215037752994321at_rat @ R5 @ R2 )
=> ( abstract_SN_on_rat @ R5 @ top_top_set_rat ) ) ) ).
% SN_subset
thf(fact_845_SN__on__induct,axiom,
! [R2: set_Pr490164551564945991ctxt_a,A5: set_Pr8705490787052945405ctxt_a,S: produc8175205341597284509ctxt_a,P2: produc8175205341597284509ctxt_a > $o] :
( ( abstra7720173292291745255ctxt_a @ R2 @ A5 )
=> ( ( member461722496487281478ctxt_a @ S @ A5 )
=> ( ! [T4: produc8175205341597284509ctxt_a] :
( ! [U4: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ T4 @ U4 ) @ R2 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_846_SN__on__induct,axiom,
! [R2: set_Pr1261947904930325089at_nat,A5: set_nat,S: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R2 @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ! [T4: nat] :
( ! [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U4 ) @ R2 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_847_SN__on__induct,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,A5: set_Ground_gmctxt_a,S: ground_gmctxt_a,P2: ground_gmctxt_a > $o] :
( ( abstra4731155269209573716ctxt_a @ R2 @ A5 )
=> ( ( member3306386364193653941ctxt_a @ S @ A5 )
=> ( ! [T4: ground_gmctxt_a] :
( ! [U4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T4 @ U4 ) @ R2 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ) ).
% SN_on_induct
thf(fact_848_SN__on__irrefl,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A5: set_Ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R @ A5 )
=> ! [X5: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X5 @ A5 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ X5 ) @ R ) ) ) ).
% SN_on_irrefl
thf(fact_849_SN__O__comm,axiom,
! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ R2 @ S3 ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ S3 @ R2 ) @ top_top_set_nat ) ) ).
% SN_O_comm
thf(fact_850_SN__O__comm,axiom,
! [R2: set_Pr4105333604307423337at_rat,S3: set_Pr6084635751276098665at_nat] :
( ( abstract_SN_on_nat @ ( relcomp_nat_rat_nat @ R2 @ S3 ) @ top_top_set_nat )
= ( abstract_SN_on_rat @ ( relcomp_rat_nat_rat @ S3 @ R2 ) @ top_top_set_rat ) ) ).
% SN_O_comm
thf(fact_851_SN__O__comm,axiom,
! [R2: set_Pr6084635751276098665at_nat,S3: set_Pr4105333604307423337at_rat] :
( ( abstract_SN_on_rat @ ( relcomp_rat_nat_rat @ R2 @ S3 ) @ top_top_set_rat )
= ( abstract_SN_on_nat @ ( relcomp_nat_rat_nat @ S3 @ R2 ) @ top_top_set_nat ) ) ).
% SN_O_comm
thf(fact_852_SN__O__comm,axiom,
! [R2: set_Pr8928021450653196913at_rat,S3: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ ( relcomp_rat_rat_rat @ R2 @ S3 ) @ top_top_set_rat )
= ( abstract_SN_on_rat @ ( relcomp_rat_rat_rat @ S3 @ R2 ) @ top_top_set_rat ) ) ).
% SN_O_comm
thf(fact_853_SN__imp__SN__trancl,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R2 @ top_top_set_nat )
=> ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R2 ) @ top_top_set_nat ) ) ).
% SN_imp_SN_trancl
thf(fact_854_SN__imp__SN__trancl,axiom,
! [R2: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ R2 @ top_top_set_rat )
=> ( abstract_SN_on_rat @ ( transi5628869978870870775cl_rat @ R2 ) @ top_top_set_rat ) ) ).
% SN_imp_SN_trancl
thf(fact_855_SN__trancl__imp__SN,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R2 ) @ top_top_set_nat )
=> ( abstract_SN_on_nat @ R2 @ top_top_set_nat ) ) ).
% SN_trancl_imp_SN
thf(fact_856_SN__trancl__imp__SN,axiom,
! [R2: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ ( transi5628869978870870775cl_rat @ R2 ) @ top_top_set_rat )
=> ( abstract_SN_on_rat @ R2 @ top_top_set_rat ) ) ).
% SN_trancl_imp_SN
thf(fact_857_SN__trancl__SN__conv,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ ( transi6264000038957366511cl_nat @ R2 ) @ top_top_set_nat )
= ( abstract_SN_on_nat @ R2 @ top_top_set_nat ) ) ).
% SN_trancl_SN_conv
thf(fact_858_SN__trancl__SN__conv,axiom,
! [R2: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ ( transi5628869978870870775cl_rat @ R2 ) @ top_top_set_rat )
= ( abstract_SN_on_rat @ R2 @ top_top_set_rat ) ) ).
% SN_trancl_SN_conv
thf(fact_859_SN__imp__acyclic,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R2 @ top_top_set_nat )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% SN_imp_acyclic
thf(fact_860_SN__imp__acyclic,axiom,
! [R2: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ R2 @ top_top_set_rat )
=> ( transi2503928277806102089ic_rat @ R2 ) ) ).
% SN_imp_acyclic
thf(fact_861_SN__inv__image,axiom,
! [R2: set_Pr1261947904930325089at_nat,F2: nat > nat] :
( ( abstract_SN_on_nat @ R2 @ top_top_set_nat )
=> ( abstract_SN_on_nat @ ( inv_image_nat_nat @ R2 @ F2 ) @ top_top_set_nat ) ) ).
% SN_inv_image
thf(fact_862_SN__inv__image,axiom,
! [R2: set_Pr1261947904930325089at_nat,F2: rat > nat] :
( ( abstract_SN_on_nat @ R2 @ top_top_set_nat )
=> ( abstract_SN_on_rat @ ( inv_image_nat_rat @ R2 @ F2 ) @ top_top_set_rat ) ) ).
% SN_inv_image
thf(fact_863_SN__inv__image,axiom,
! [R2: set_Pr8928021450653196913at_rat,F2: nat > rat] :
( ( abstract_SN_on_rat @ R2 @ top_top_set_rat )
=> ( abstract_SN_on_nat @ ( inv_image_rat_nat @ R2 @ F2 ) @ top_top_set_nat ) ) ).
% SN_inv_image
thf(fact_864_SN__inv__image,axiom,
! [R2: set_Pr8928021450653196913at_rat,F2: rat > rat] :
( ( abstract_SN_on_rat @ R2 @ top_top_set_rat )
=> ( abstract_SN_on_rat @ ( inv_image_rat_rat @ R2 @ F2 ) @ top_top_set_rat ) ) ).
% SN_inv_image
thf(fact_865_compatible__SN,axiom,
! [S3: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ S3 @ top_top_set_nat )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S3 ) @ S3 )
=> ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ S3 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S3 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) @ top_top_set_nat ) ) ) ).
% compatible_SN
thf(fact_866_compatible__SN,axiom,
! [S3: set_Pr8928021450653196913at_rat,NS: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ S3 @ top_top_set_rat )
=> ( ( ord_le1589215037752994321at_rat @ ( relcomp_rat_rat_rat @ NS @ S3 ) @ S3 )
=> ( abstract_SN_on_rat @ ( relcomp_rat_rat_rat @ S3 @ ( relcomp_rat_rat_rat @ ( transi2270211269848806677cl_rat @ S3 ) @ ( transi2270211269848806677cl_rat @ NS ) ) ) @ top_top_set_rat ) ) ) ).
% compatible_SN
thf(fact_867_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_868_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_rat @ top_top_set_rat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_rat @ top_top_set_rat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_869_compatible__SN_H,axiom,
! [NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S3 ) @ S3 )
=> ( ( abstract_SN_on_nat @ S3 @ top_top_set_nat )
=> ( abstract_SN_on_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S3 ) ) @ ( relcomp_nat_nat_nat @ S3 @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S3 ) ) ) ) @ top_top_set_nat ) ) ) ).
% compatible_SN'
thf(fact_870_compatible__SN_H,axiom,
! [NS: set_Pr8928021450653196913at_rat,S3: set_Pr8928021450653196913at_rat] :
( ( ord_le1589215037752994321at_rat @ ( relcomp_rat_rat_rat @ NS @ S3 ) @ S3 )
=> ( ( abstract_SN_on_rat @ S3 @ top_top_set_rat )
=> ( abstract_SN_on_rat @ ( relcomp_rat_rat_rat @ ( transi2270211269848806677cl_rat @ ( sup_su4770203945505871429at_rat @ NS @ S3 ) ) @ ( relcomp_rat_rat_rat @ S3 @ ( transi2270211269848806677cl_rat @ ( sup_su4770203945505871429at_rat @ NS @ S3 ) ) ) ) @ top_top_set_rat ) ) ) ).
% compatible_SN'
thf(fact_871_quasi__commute__imp__SN,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
=> ( ( abstract_SN_on_nat @ S @ top_top_set_nat )
=> ( ( abstra4838182543893801299te_nat @ R @ S )
=> ( abstract_SN_on_nat @ ( sup_su6327502436637775413at_nat @ R @ S ) @ top_top_set_nat ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_872_quasi__commute__imp__SN,axiom,
! [R: set_Pr8928021450653196913at_rat,S: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ R @ top_top_set_rat )
=> ( ( abstract_SN_on_rat @ S @ top_top_set_rat )
=> ( ( abstra4203052483807305563te_rat @ R @ S )
=> ( abstract_SN_on_rat @ ( sup_su4770203945505871429at_rat @ R @ S ) @ top_top_set_rat ) ) ) ) ).
% quasi_commute_imp_SN
thf(fact_873_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_874_finite__fun__UNIVD1,axiom,
( ( finite1799139810636553822at_nat @ top_top_set_rat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_875_finite__fun__UNIVD1,axiom,
( ( finite7830837933032798814at_rat @ top_top_set_nat_rat )
=> ( ( ( finite_card_rat @ top_top_set_rat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_876_finite__fun__UNIVD1,axiom,
( ( finite7514283289097932902at_rat @ top_top_set_rat_rat )
=> ( ( ( finite_card_rat @ top_top_set_rat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_rat @ top_top_set_rat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_877_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_878_SN__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,R: set_Pr490164551564945991ctxt_a] :
( ! [F3: nat > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) )
=> ( abstra7720173292291745255ctxt_a @ R @ A5 ) ) ).
% SN_onI
thf(fact_879_SN__onI,axiom,
! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [F3: nat > nat] :
( ( member_nat @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) )
=> ( abstract_SN_on_nat @ R @ A5 ) ) ).
% SN_onI
thf(fact_880_SN__onI,axiom,
! [A5: set_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ! [F3: nat > ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) )
=> ( abstra4731155269209573716ctxt_a @ R @ A5 ) ) ).
% SN_onI
thf(fact_881_not__SN__onI,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F2 @ zero_zero_nat ) @ X7 )
=> ( ! [I4: nat] : ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R2 )
=> ~ ( abstra7720173292291745255ctxt_a @ R2 @ X7 ) ) ) ).
% not_SN_onI
thf(fact_882_not__SN__onI,axiom,
! [F2: nat > nat,X7: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ ( F2 @ zero_zero_nat ) @ X7 )
=> ( ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R2 )
=> ~ ( abstract_SN_on_nat @ R2 @ X7 ) ) ) ).
% not_SN_onI
thf(fact_883_not__SN__onI,axiom,
! [F2: nat > ground_gmctxt_a,X7: set_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member3306386364193653941ctxt_a @ ( F2 @ zero_zero_nat ) @ X7 )
=> ( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R2 )
=> ~ ( abstra4731155269209573716ctxt_a @ R2 @ X7 ) ) ) ).
% not_SN_onI
thf(fact_884_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr8705490787052945405ctxt_a,R2: set_Pr8705490787052945405ctxt_a,Seq: nat > ground_gmctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ R5 @ R2 )
=> ( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R5 )
=> ! [I3: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_885_wf__iff__no__infinite__down__chain,axiom,
( wf_Ground_gmctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a] :
~ ? [F: nat > ground_gmctxt_a] :
! [I2: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) @ R3 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_886_wf__no__infinite__down__chainE,axiom,
! [R: set_Pr8705490787052945405ctxt_a,F2: nat > ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R )
=> ~ ! [K: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ ( suc @ K ) ) @ ( F2 @ K ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_887_SN__onE,axiom,
! [R: set_Pr490164551564945991ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( abstra7720173292291745255ctxt_a @ R @ A5 )
=> ~ ? [F5: nat > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F5 @ zero_zero_nat ) @ A5 )
& ! [I4: nat] : ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_888_SN__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A5: set_nat] :
( ( abstract_SN_on_nat @ R @ A5 )
=> ~ ? [F5: nat > nat] :
( ( member_nat @ ( F5 @ zero_zero_nat ) @ A5 )
& ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_889_SN__onE,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A5: set_Ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R @ A5 )
=> ~ ? [F5: nat > ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ ( F5 @ zero_zero_nat ) @ A5 )
& ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R ) ) ) ).
% SN_onE
thf(fact_890_SN__on__def,axiom,
( abstra7720173292291745255ctxt_a
= ( ^ [R3: set_Pr490164551564945991ctxt_a,A6: set_Pr8705490787052945405ctxt_a] :
~ ? [F: nat > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F @ zero_zero_nat ) @ A6 )
& ! [I2: nat] : ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_891_SN__on__def,axiom,
( abstract_SN_on_nat
= ( ^ [R3: set_Pr1261947904930325089at_nat,A6: set_nat] :
~ ? [F: nat > nat] :
( ( member_nat @ ( F @ zero_zero_nat ) @ A6 )
& ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_892_SN__on__def,axiom,
( abstra4731155269209573716ctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a,A6: set_Ground_gmctxt_a] :
~ ? [F: nat > ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ ( F @ zero_zero_nat ) @ A6 )
& ! [I2: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R3 ) ) ) ) ).
% SN_on_def
thf(fact_893_not__SN__onE,axiom,
! [R: set_Pr490164551564945991ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ~ ( abstra7720173292291745255ctxt_a @ R @ A5 )
=> ~ ! [F3: nat > produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_894_not__SN__onE,axiom,
! [R: set_Pr1261947904930325089at_nat,A5: set_nat] :
( ~ ( abstract_SN_on_nat @ R @ A5 )
=> ~ ! [F3: nat > nat] :
( ( member_nat @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_895_not__SN__onE,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A5: set_Ground_gmctxt_a] :
( ~ ( abstra4731155269209573716ctxt_a @ R @ A5 )
=> ~ ! [F3: nat > ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ ( F3 @ zero_zero_nat ) @ A5 )
=> ~ ! [I3: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ).
% not_SN_onE
thf(fact_896_rtrancl__fun__conv,axiom,
! [S: ground_gmctxt_a,T: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ S @ T ) @ ( transi2468631686036622171ctxt_a @ R2 ) )
= ( ? [F: nat > ground_gmctxt_a,N2: nat] :
( ( ( F @ zero_zero_nat )
= S )
& ( ( F @ N2 )
= T )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) ) @ R2 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_897_rtrancl__imp__seq,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ? [F3: nat > ground_gmctxt_a,N4: nat] :
( ( ( F3 @ zero_zero_nat )
= X3 )
& ( ( F3 @ N4 )
= Y )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N4 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_898_chain__imp__trancl,axiom,
! [S3: nat > ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I: nat,J2: nat] :
( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S3 @ I4 ) @ ( S3 @ ( suc @ I4 ) ) ) @ R )
=> ( ( ord_less_nat @ I @ J2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S3 @ I ) @ ( S3 @ J2 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ).
% chain_imp_trancl
thf(fact_899_chain__imp__rtrancl,axiom,
! [S3: nat > ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I: nat,J2: nat] :
( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S3 @ I4 ) @ ( S3 @ ( suc @ I4 ) ) ) @ R )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( S3 @ I ) @ ( S3 @ J2 ) ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_900_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F2: nat > rat] :
( ( finite_finite_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F2 )
=> ( ! [N4: nat] :
( ( ( F2 @ N4 )
= ( F2 @ ( suc @ N4 ) ) )
=> ( ( F2 @ ( suc @ N4 ) )
= ( F2 @ ( suc @ ( suc @ N4 ) ) ) ) )
=> ? [N5: nat] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ N6 @ N5 )
=> ! [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( ord_less_nat @ M4 @ N6 )
=> ( ord_less_rat @ ( F2 @ M4 ) @ ( F2 @ N6 ) ) ) ) )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ( F2 @ N5 )
= ( F2 @ N6 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_901_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F2: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ! [N4: nat] :
( ( ( F2 @ N4 )
= ( F2 @ ( suc @ N4 ) ) )
=> ( ( F2 @ ( suc @ N4 ) )
= ( F2 @ ( suc @ ( suc @ N4 ) ) ) ) )
=> ? [N5: nat] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ N6 @ N5 )
=> ! [M4: nat] :
( ( ord_less_eq_nat @ M4 @ N5 )
=> ( ( ord_less_nat @ M4 @ N6 )
=> ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N6 ) ) ) ) )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ( F2 @ N5 )
= ( F2 @ N6 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_902_ord_Ostrict__mono__onD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat,R: produc8175205341597284509ctxt_a,S: produc8175205341597284509ctxt_a] :
( ( monoto2101069531372151366_a_nat @ A5 @ Less @ ord_less_nat @ F2 )
=> ( ( member461722496487281478ctxt_a @ R @ A5 )
=> ( ( member461722496487281478ctxt_a @ S @ A5 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_903_ord_Ostrict__mono__onD,axiom,
! [A5: set_nat,Less: nat > nat > $o,F2: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ Less @ ord_less_nat @ F2 )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( Less @ R @ S )
=> ( ord_less_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_904_ord_Ostrict__mono__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat] :
( ! [R6: produc8175205341597284509ctxt_a,S5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ R6 @ A5 )
=> ( ( member461722496487281478ctxt_a @ S5 @ A5 )
=> ( ( Less @ R6 @ S5 )
=> ( ord_less_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monoto2101069531372151366_a_nat @ A5 @ Less @ ord_less_nat @ F2 ) ) ).
% ord.strict_mono_onI
thf(fact_905_ord_Ostrict__mono__onI,axiom,
! [A5: set_nat,Less: nat > nat > $o,F2: nat > nat] :
( ! [R6: nat,S5: nat] :
( ( member_nat @ R6 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( Less @ R6 @ S5 )
=> ( ord_less_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ Less @ ord_less_nat @ F2 ) ) ).
% ord.strict_mono_onI
thf(fact_906_ord_Ostrict__mono__on__def,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat] :
( ( monoto2101069531372151366_a_nat @ A5 @ Less @ ord_less_nat @ F2 )
= ( ! [R3: produc8175205341597284509ctxt_a,S6: produc8175205341597284509ctxt_a] :
( ( ( member461722496487281478ctxt_a @ R3 @ A5 )
& ( member461722496487281478ctxt_a @ S6 @ A5 )
& ( Less @ R3 @ S6 ) )
=> ( ord_less_nat @ ( F2 @ R3 ) @ ( F2 @ S6 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_907_ord_Ostrict__mono__on__def,axiom,
! [A5: set_nat,Less: nat > nat > $o,F2: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ Less @ ord_less_nat @ F2 )
= ( ! [R3: nat,S6: nat] :
( ( ( member_nat @ R3 @ A5 )
& ( member_nat @ S6 @ A5 )
& ( Less @ R3 @ S6 ) )
=> ( ord_less_nat @ ( F2 @ R3 ) @ ( F2 @ S6 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_908_strict__mono__onD,axiom,
! [A5: set_nat,F2: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( ord_less_nat @ R @ S )
=> ( ord_less_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_909_strict__mono__onI,axiom,
! [A5: set_nat,F2: nat > nat] :
( ! [R6: nat,S5: nat] :
( ( member_nat @ R6 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( ord_less_nat @ R6 @ S5 )
=> ( ord_less_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 ) ) ).
% strict_mono_onI
thf(fact_910_strict__mono__on__eqD,axiom,
! [A5: set_nat,F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( Y = X3 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_911_mono__on__greaterD,axiom,
! [A5: set_nat,G2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ G2 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( ( ord_less_nat @ ( G2 @ Y ) @ ( G2 @ X3 ) )
=> ( ord_less_nat @ Y @ X3 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_912_strict__mono__on__leD,axiom,
! [A5: set_nat,F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_913_strict__mono__on__imp__mono__on,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_914_mono__imp__mono__on,axiom,
! [F2: rat > nat,A5: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( monotone_on_rat_nat @ A5 @ ord_less_eq_rat @ ord_less_eq_nat @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_915_mono__imp__mono__on,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_916_monoI,axiom,
! [F2: rat > nat] :
( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 ) ) ).
% monoI
thf(fact_917_monoI,axiom,
! [F2: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% monoI
thf(fact_918_monoE,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_rat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% monoE
thf(fact_919_monoE,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% monoE
thf(fact_920_monoD,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_rat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% monoD
thf(fact_921_monoD,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% monoD
thf(fact_922_strict__monoD,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( ( ord_less_rat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% strict_monoD
thf(fact_923_strict__monoD,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% strict_monoD
thf(fact_924_strict__monoI,axiom,
! [F2: rat > nat] :
( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 ) ) ).
% strict_monoI
thf(fact_925_strict__monoI,axiom,
! [F2: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 ) ) ).
% strict_monoI
thf(fact_926_strict__mono__eq,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
= ( X3 = Y ) ) ) ).
% strict_mono_eq
thf(fact_927_strict__mono__eq,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y ) )
= ( X3 = Y ) ) ) ).
% strict_mono_eq
thf(fact_928_strict__mono__less,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less_rat @ X3 @ Y ) ) ) ).
% strict_mono_less
thf(fact_929_strict__mono__less,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less_nat @ X3 @ Y ) ) ) ).
% strict_mono_less
thf(fact_930_strict__mono__on__imp__inj__on,axiom,
! [A5: set_nat,F2: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_nat @ ord_less_nat @ F2 )
=> ( inj_on_nat_nat @ F2 @ A5 ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_931_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F2 )
=> ( ( Orda @ X3 @ Y )
=> ( Ordb @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ).
% monotoneD
thf(fact_932_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( Orda @ X @ Y3 )
=> ( Ordb @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F2 ) ) ).
% monotoneI
thf(fact_933_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A6: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A6 )
=> ( ( Orda2 @ X4 @ Y4 )
=> ( Ordb2 @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_934_monotone__onI,axiom,
! [A5: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( ( Orda @ X @ Y3 )
=> ( Ordb @ ( F2 @ X ) @ ( F2 @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ Orda @ Ordb @ F2 ) ) ).
% monotone_onI
thf(fact_935_monotone__onD,axiom,
! [A5: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ A5 @ Orda @ Ordb @ F2 )
=> ( ( member_nat @ X3 @ A5 )
=> ( ( member_nat @ Y @ A5 )
=> ( ( Orda @ X3 @ Y )
=> ( Ordb @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_936_mono__onI,axiom,
! [A5: set_nat,F2: nat > nat] :
( ! [R6: nat,S5: nat] :
( ( member_nat @ R6 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( ord_less_eq_nat @ R6 @ S5 )
=> ( ord_less_eq_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% mono_onI
thf(fact_937_mono__onD,axiom,
! [A5: set_nat,F2: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( ord_less_eq_nat @ R @ S )
=> ( ord_less_eq_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% mono_onD
thf(fact_938_ord_Omono__on__def,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less_eq: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat] :
( ( monoto2101069531372151366_a_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 )
= ( ! [R3: produc8175205341597284509ctxt_a,S6: produc8175205341597284509ctxt_a] :
( ( ( member461722496487281478ctxt_a @ R3 @ A5 )
& ( member461722496487281478ctxt_a @ S6 @ A5 )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F2 @ R3 ) @ ( F2 @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_939_ord_Omono__on__def,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F2: nat > nat] :
( ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 )
= ( ! [R3: nat,S6: nat] :
( ( ( member_nat @ R3 @ A5 )
& ( member_nat @ S6 @ A5 )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F2 @ R3 ) @ ( F2 @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_940_ord_Omono__onI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less_eq: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat] :
( ! [R6: produc8175205341597284509ctxt_a,S5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ R6 @ A5 )
=> ( ( member461722496487281478ctxt_a @ S5 @ A5 )
=> ( ( Less_eq @ R6 @ S5 )
=> ( ord_less_eq_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monoto2101069531372151366_a_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 ) ) ).
% ord.mono_onI
thf(fact_941_ord_Omono__onI,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F2: nat > nat] :
( ! [R6: nat,S5: nat] :
( ( member_nat @ R6 @ A5 )
=> ( ( member_nat @ S5 @ A5 )
=> ( ( Less_eq @ R6 @ S5 )
=> ( ord_less_eq_nat @ ( F2 @ R6 ) @ ( F2 @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 ) ) ).
% ord.mono_onI
thf(fact_942_ord_Omono__onD,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,Less_eq: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a > $o,F2: produc8175205341597284509ctxt_a > nat,R: produc8175205341597284509ctxt_a,S: produc8175205341597284509ctxt_a] :
( ( monoto2101069531372151366_a_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 )
=> ( ( member461722496487281478ctxt_a @ R @ A5 )
=> ( ( member461722496487281478ctxt_a @ S @ A5 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_943_ord_Omono__onD,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F2: nat > nat,R: nat,S: nat] :
( ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 )
=> ( ( member_nat @ R @ A5 )
=> ( ( member_nat @ S @ A5 )
=> ( ( Less_eq @ R @ S )
=> ( ord_less_eq_nat @ ( F2 @ R ) @ ( F2 @ S ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_944_monotone__on__subset,axiom,
! [A5: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat,B5: set_nat] :
( ( monotone_on_nat_nat @ A5 @ Orda @ Ordb @ F2 )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( monotone_on_nat_nat @ B5 @ Orda @ Ordb @ F2 ) ) ) ).
% monotone_on_subset
thf(fact_945_ord_Omono__on__subset,axiom,
! [A5: set_nat,Less_eq: nat > nat > $o,F2: nat > nat,B5: set_nat] :
( ( monotone_on_nat_nat @ A5 @ Less_eq @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( monotone_on_nat_nat @ B5 @ Less_eq @ ord_less_eq_nat @ F2 ) ) ) ).
% ord.mono_on_subset
thf(fact_946_mono__on__subset,axiom,
! [A5: set_nat,F2: nat > nat,B5: set_nat] :
( ( monotone_on_nat_nat @ A5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ B5 @ A5 )
=> ( monotone_on_nat_nat @ B5 @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ) ).
% mono_on_subset
thf(fact_947_mono__invE,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ) ).
% mono_invE
thf(fact_948_mono__invE,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% mono_invE
thf(fact_949_strict__mono__mono,axiom,
! [F2: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 ) ) ).
% strict_mono_mono
thf(fact_950_strict__mono__mono,axiom,
! [F2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% strict_mono_mono
thf(fact_951_mono__strict__invE,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less_rat @ X3 @ Y ) ) ) ).
% mono_strict_invE
thf(fact_952_mono__strict__invE,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
=> ( ord_less_nat @ X3 @ Y ) ) ) ).
% mono_strict_invE
thf(fact_953_strict__mono__less__eq,axiom,
! [F2: rat > nat,X3: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less_eq_rat @ X3 @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_954_strict__mono__less__eq,axiom,
! [F2: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_955_strict__mono__inv,axiom,
! [F2: rat > rat,G2: rat > rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_rat @ ord_less_rat @ F2 )
=> ( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_rat @ ord_less_rat @ G2 ) ) ) ) ).
% strict_mono_inv
thf(fact_956_strict__mono__inv,axiom,
! [F2: rat > nat,G2: nat > rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X: rat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ G2 ) ) ) ) ).
% strict_mono_inv
thf(fact_957_strict__mono__inv,axiom,
! [F2: nat > rat,G2: rat > nat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ F2 )
=> ( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ G2 ) ) ) ) ).
% strict_mono_inv
thf(fact_958_strict__mono__inv,axiom,
! [F2: nat > nat,G2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G2 @ ( F2 @ X ) )
= X )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G2 ) ) ) ) ).
% strict_mono_inv
thf(fact_959_mono__sup,axiom,
! [F2: rat > nat,A5: rat,B5: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup_rat @ A5 @ B5 ) ) ) ) ).
% mono_sup
thf(fact_960_mono__sup,axiom,
! [F2: nat > nat,A5: nat,B5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% mono_sup
thf(fact_961_strict__mono__imp__inj__on,axiom,
! [F2: rat > nat,A5: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( inj_on_rat_nat @ F2 @ A5 ) ) ).
% strict_mono_imp_inj_on
thf(fact_962_strict__mono__imp__inj__on,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( inj_on_nat_nat @ F2 @ A5 ) ) ).
% strict_mono_imp_inj_on
thf(fact_963_infinite__enumerate,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ? [R6: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R6 )
& ! [N6: nat] : ( member_nat @ ( R6 @ N6 ) @ S3 ) ) ) ).
% infinite_enumerate
thf(fact_964_strict__mono__inv__on__range,axiom,
! [F2: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F2 )
=> ( monotone_on_nat_rat @ ( image_rat_nat @ F2 @ top_top_set_rat ) @ ord_less_nat @ ord_less_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) ) ) ).
% strict_mono_inv_on_range
thf(fact_965_strict__mono__inv__on__range,axiom,
! [F2: nat > rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ F2 )
=> ( monotone_on_rat_nat @ ( image_nat_rat @ F2 @ top_top_set_nat ) @ ord_less_rat @ ord_less_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) ) ) ).
% strict_mono_inv_on_range
thf(fact_966_strict__mono__inv__on__range,axiom,
! [F2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( monotone_on_nat_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ ord_less_nat @ ord_less_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) ) ) ).
% strict_mono_inv_on_range
thf(fact_967_mono__iff__le__Suc,axiom,
! [F2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
= ( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_968_strict__mono__imp__increasing,axiom,
! [F2: nat > nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F2 )
=> ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_969_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_970_zero__notin__Suc__image,axiom,
! [A5: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_971_mono__times__nat,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N3 ) ) ) ).
% mono_times_nat
thf(fact_972_inj__mult__left,axiom,
! [A: rat] :
( ( inj_on_rat_rat @ ( times_times_rat @ A ) @ top_top_set_rat )
= ( A != zero_zero_rat ) ) ).
% inj_mult_left
thf(fact_973_mono__mult,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ ( times_times_rat @ A ) ) ) ).
% mono_mult
thf(fact_974_mono__mult,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ A ) ) ) ).
% mono_mult
thf(fact_975_strict__mono__enumerate,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S3 ) ) ) ).
% strict_mono_enumerate
thf(fact_976_SN__restrict__SN__idemp,axiom,
! [A5: set_Pr1261947904930325089at_nat] : ( abstract_SN_on_nat @ ( abstra2540617779794743107SN_nat @ A5 @ A5 ) @ top_top_set_nat ) ).
% SN_restrict_SN_idemp
thf(fact_977_SN__restrict__SN__idemp,axiom,
! [A5: set_Pr8928021450653196913at_rat] : ( abstract_SN_on_rat @ ( abstra1905487719708247371SN_rat @ A5 @ A5 ) @ top_top_set_rat ) ).
% SN_restrict_SN_idemp
thf(fact_978_inj__enumerate,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( inj_on_nat_nat @ ( infini8530281810654367211te_nat @ S3 ) @ top_top_set_nat ) ) ).
% inj_enumerate
thf(fact_979_range__enumerate,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S3 ) @ top_top_set_nat )
= S3 ) ) ).
% range_enumerate
thf(fact_980_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_981_Sup__greaterThanAtLeast,axiom,
! [X3: set_nat] :
( ( ord_less_set_nat @ X3 @ top_top_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( set_or458868116921152288et_nat @ X3 ) )
= top_top_set_nat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_982_Sup__greaterThanAtLeast,axiom,
! [X3: set_rat] :
( ( ord_less_set_rat @ X3 @ top_top_set_rat )
=> ( ( comple3890839924845867745et_rat @ ( set_or6174011595382531368et_rat @ X3 ) )
= top_top_set_rat ) ) ).
% Sup_greaterThanAtLeast
thf(fact_983_mono__image__least,axiom,
! [F2: rat > nat,M3: rat,N3: rat,M5: nat,N7: nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F2 )
=> ( ( ( image_rat_nat @ F2 @ ( set_or4029947393144176647an_rat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M5 @ N7 ) )
=> ( ( ord_less_rat @ M3 @ N3 )
=> ( ( F2 @ M3 )
= M5 ) ) ) ) ).
% mono_image_least
thf(fact_984_mono__image__least,axiom,
! [F2: nat > rat,M3: nat,N3: nat,M5: rat,N7: rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F2 )
=> ( ( ( image_nat_rat @ F2 @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( set_or4029947393144176647an_rat @ M5 @ N7 ) )
=> ( ( ord_less_nat @ M3 @ N3 )
=> ( ( F2 @ M3 )
= M5 ) ) ) ) ).
% mono_image_least
thf(fact_985_mono__image__least,axiom,
! [F2: nat > nat,M3: nat,N3: nat,M5: nat,N7: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ( image_nat_nat @ F2 @ ( set_or4665077453230672383an_nat @ M3 @ N3 ) )
= ( set_or4665077453230672383an_nat @ M5 @ N7 ) )
=> ( ( ord_less_nat @ M3 @ N3 )
=> ( ( F2 @ M3 )
= M5 ) ) ) ) ).
% mono_image_least
thf(fact_986_funpow__increasing,axiom,
! [M3: nat,N3: nat,F2: set_nat > set_nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N3 @ F2 @ top_top_set_nat ) @ ( compow8708494347934031032et_nat @ M3 @ F2 @ top_top_set_nat ) ) ) ) ).
% funpow_increasing
thf(fact_987_funpow__increasing,axiom,
! [M3: nat,N3: nat,F2: set_rat > set_rat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( monoto6994633993926870149et_rat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_set_rat @ F2 )
=> ( ord_less_eq_set_rat @ ( compow1859248436694711496et_rat @ N3 @ F2 @ top_top_set_rat ) @ ( compow1859248436694711496et_rat @ M3 @ F2 @ top_top_set_rat ) ) ) ) ).
% funpow_increasing
thf(fact_988_image__Suc__atLeastLessThan,axiom,
! [I: nat,J2: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_989_surj__fn,axiom,
! [F2: nat > nat,N3: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( compow_nat_nat @ N3 @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surj_fn
thf(fact_990_surj__fn,axiom,
! [F2: rat > rat,N3: nat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ ( compow_rat_rat @ N3 @ F2 ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surj_fn
thf(fact_991_inj__fn,axiom,
! [F2: nat > nat,N3: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( compow_nat_nat @ N3 @ F2 ) @ top_top_set_nat ) ) ).
% inj_fn
thf(fact_992_inj__fn,axiom,
! [F2: rat > rat,N3: nat] :
( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
=> ( inj_on_rat_rat @ ( compow_rat_rat @ N3 @ F2 ) @ top_top_set_rat ) ) ).
% inj_fn
thf(fact_993_funpow__mono,axiom,
! [F2: rat > rat,A5: rat,B5: rat,N3: nat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ F2 )
=> ( ( ord_less_eq_rat @ A5 @ B5 )
=> ( ord_less_eq_rat @ ( compow_rat_rat @ N3 @ F2 @ A5 ) @ ( compow_rat_rat @ N3 @ F2 @ B5 ) ) ) ) ).
% funpow_mono
thf(fact_994_funpow__mono,axiom,
! [F2: nat > nat,A5: nat,B5: nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ A5 @ B5 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ N3 @ F2 @ A5 ) @ ( compow_nat_nat @ N3 @ F2 @ B5 ) ) ) ) ).
% funpow_mono
thf(fact_995_Kleene__iter__gpfp,axiom,
! [F2: set_nat > set_nat,P: set_nat,K2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ P @ ( F2 @ P ) )
=> ( ord_less_eq_set_nat @ P @ ( compow8708494347934031032et_nat @ K2 @ F2 @ top_top_set_nat ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_996_Kleene__iter__gpfp,axiom,
! [F2: set_rat > set_rat,P: set_rat,K2: nat] :
( ( monoto6994633993926870149et_rat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_set_rat @ F2 )
=> ( ( ord_less_eq_set_rat @ P @ ( F2 @ P ) )
=> ( ord_less_eq_set_rat @ P @ ( compow1859248436694711496et_rat @ K2 @ F2 @ top_top_set_rat ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_997_funpow__mono2,axiom,
! [F2: rat > rat,I: nat,J2: nat,X3: rat,Y: rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ F2 )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_eq_rat @ X3 @ ( F2 @ X3 ) )
=> ( ord_less_eq_rat @ ( compow_rat_rat @ I @ F2 @ X3 ) @ ( compow_rat_rat @ J2 @ F2 @ Y ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_998_funpow__mono2,axiom,
! [F2: nat > nat,I: nat,J2: nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ ( F2 @ X3 ) )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ I @ F2 @ X3 ) @ ( compow_nat_nat @ J2 @ F2 @ Y ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_999_gfp__Kleene__iter,axiom,
! [F2: set_nat > set_nat,K2: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K2 ) @ F2 @ top_top_set_nat )
= ( compow8708494347934031032et_nat @ K2 @ F2 @ top_top_set_nat ) )
=> ( ( comple1596078789208929544et_nat @ F2 )
= ( compow8708494347934031032et_nat @ K2 @ F2 @ top_top_set_nat ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_1000_gfp__Kleene__iter,axiom,
! [F2: set_rat > set_rat,K2: nat] :
( ( monoto6994633993926870149et_rat @ top_top_set_set_rat @ ord_less_eq_set_rat @ ord_less_eq_set_rat @ F2 )
=> ( ( ( compow1859248436694711496et_rat @ ( suc @ K2 ) @ F2 @ top_top_set_rat )
= ( compow1859248436694711496et_rat @ K2 @ F2 @ top_top_set_rat ) )
=> ( ( comple7311222267670308624et_rat @ F2 )
= ( compow1859248436694711496et_rat @ K2 @ F2 @ top_top_set_rat ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_1001_weak__coinduct__image,axiom,
! [A: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a,G2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ X7 )
=> ( ( ord_le2690539916662146973ctxt_a @ ( image_4216501698144419397ctxt_a @ G2 @ X7 ) @ ( F2 @ ( image_4216501698144419397ctxt_a @ G2 @ X7 ) ) )
=> ( member461722496487281478ctxt_a @ ( G2 @ A ) @ ( comple5045702773340321787ctxt_a @ F2 ) ) ) ) ).
% weak_coinduct_image
thf(fact_1002_weak__coinduct__image,axiom,
! [A: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a,G2: produc8175205341597284509ctxt_a > nat,F2: set_nat > set_nat] :
( ( member461722496487281478ctxt_a @ A @ X7 )
=> ( ( ord_less_eq_set_nat @ ( image_5981101931084406824_a_nat @ G2 @ X7 ) @ ( F2 @ ( image_5981101931084406824_a_nat @ G2 @ X7 ) ) )
=> ( member_nat @ ( G2 @ A ) @ ( comple1596078789208929544et_nat @ F2 ) ) ) ) ).
% weak_coinduct_image
thf(fact_1003_weak__coinduct__image,axiom,
! [A: nat,X7: set_nat,G2: nat > rat,F2: set_rat > set_rat] :
( ( member_nat @ A @ X7 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ G2 @ X7 ) @ ( F2 @ ( image_nat_rat @ G2 @ X7 ) ) )
=> ( member_rat @ ( G2 @ A ) @ ( comple7311222267670308624et_rat @ F2 ) ) ) ) ).
% weak_coinduct_image
thf(fact_1004_weak__coinduct__image,axiom,
! [A: nat,X7: set_nat,G2: nat > produc8175205341597284509ctxt_a,F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a] :
( ( member_nat @ A @ X7 )
=> ( ( ord_le2690539916662146973ctxt_a @ ( image_43530617296285190ctxt_a @ G2 @ X7 ) @ ( F2 @ ( image_43530617296285190ctxt_a @ G2 @ X7 ) ) )
=> ( member461722496487281478ctxt_a @ ( G2 @ A ) @ ( comple5045702773340321787ctxt_a @ F2 ) ) ) ) ).
% weak_coinduct_image
thf(fact_1005_weak__coinduct__image,axiom,
! [A: nat,X7: set_nat,G2: nat > nat,F2: set_nat > set_nat] :
( ( member_nat @ A @ X7 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ X7 ) @ ( F2 @ ( image_nat_nat @ G2 @ X7 ) ) )
=> ( member_nat @ ( G2 @ A ) @ ( comple1596078789208929544et_nat @ F2 ) ) ) ) ).
% weak_coinduct_image
thf(fact_1006_coinduct__set,axiom,
! [F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a] :
( ( monoto6151637465316607655ctxt_a @ top_to8078199433501558957ctxt_a @ ord_le2690539916662146973ctxt_a @ ord_le2690539916662146973ctxt_a @ F2 )
=> ( ( member461722496487281478ctxt_a @ A @ X7 )
=> ( ( ord_le2690539916662146973ctxt_a @ X7 @ ( F2 @ ( sup_su1394892571904902097ctxt_a @ X7 @ ( comple5045702773340321787ctxt_a @ F2 ) ) ) )
=> ( member461722496487281478ctxt_a @ A @ ( comple5045702773340321787ctxt_a @ F2 ) ) ) ) ) ).
% coinduct_set
thf(fact_1007_coinduct__set,axiom,
! [F2: set_nat > set_nat,A: nat,X7: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ( member_nat @ A @ X7 )
=> ( ( ord_less_eq_set_nat @ X7 @ ( F2 @ ( sup_sup_set_nat @ X7 @ ( comple1596078789208929544et_nat @ F2 ) ) ) )
=> ( member_nat @ A @ ( comple1596078789208929544et_nat @ F2 ) ) ) ) ) ).
% coinduct_set
thf(fact_1008_gfp__fun__UnI2,axiom,
! [F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a] :
( ( monoto6151637465316607655ctxt_a @ top_to8078199433501558957ctxt_a @ ord_le2690539916662146973ctxt_a @ ord_le2690539916662146973ctxt_a @ F2 )
=> ( ( member461722496487281478ctxt_a @ A @ ( comple5045702773340321787ctxt_a @ F2 ) )
=> ( member461722496487281478ctxt_a @ A @ ( F2 @ ( sup_su1394892571904902097ctxt_a @ X7 @ ( comple5045702773340321787ctxt_a @ F2 ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_1009_gfp__fun__UnI2,axiom,
! [F2: set_nat > set_nat,A: nat,X7: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ( member_nat @ A @ ( comple1596078789208929544et_nat @ F2 ) )
=> ( member_nat @ A @ ( F2 @ ( sup_sup_set_nat @ X7 @ ( comple1596078789208929544et_nat @ F2 ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_1010_def__coinduct__set,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,F2: set_Pr8705490787052945405ctxt_a > set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a] :
( ( A5
= ( comple5045702773340321787ctxt_a @ F2 ) )
=> ( ( monoto6151637465316607655ctxt_a @ top_to8078199433501558957ctxt_a @ ord_le2690539916662146973ctxt_a @ ord_le2690539916662146973ctxt_a @ F2 )
=> ( ( member461722496487281478ctxt_a @ A @ X7 )
=> ( ( ord_le2690539916662146973ctxt_a @ X7 @ ( F2 @ ( sup_su1394892571904902097ctxt_a @ X7 @ A5 ) ) )
=> ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_1011_def__coinduct__set,axiom,
! [A5: set_nat,F2: set_nat > set_nat,A: nat,X7: set_nat] :
( ( A5
= ( comple1596078789208929544et_nat @ F2 ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F2 )
=> ( ( member_nat @ A @ X7 )
=> ( ( ord_less_eq_set_nat @ X7 @ ( F2 @ ( sup_sup_set_nat @ X7 @ A5 ) ) )
=> ( member_nat @ A @ A5 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_1012_funpow__decreasing,axiom,
! [M3: nat,N3: nat,F2: nat > nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ M3 @ F2 @ bot_bot_nat ) @ ( compow_nat_nat @ N3 @ F2 @ bot_bot_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_1013_image__empty,axiom,
! [F2: nat > rat] :
( ( image_nat_rat @ F2 @ bot_bot_set_nat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_1014_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1015_empty__is__image,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( bot_bot_set_rat
= ( image_nat_rat @ F2 @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1016_empty__is__image,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F2 @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1017_image__is__empty,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( ( image_nat_rat @ F2 @ A5 )
= bot_bot_set_rat )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1018_image__is__empty,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1019_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_1020_boolean__algebra_Ocompl__one,axiom,
( ( uminus2201863774496077783et_rat @ top_top_set_rat )
= bot_bot_set_rat ) ).
% boolean_algebra.compl_one
thf(fact_1021_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_1022_boolean__algebra_Ocompl__zero,axiom,
( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
= top_top_set_rat ) ).
% boolean_algebra.compl_zero
thf(fact_1023_monotone__on__empty,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat] : ( monotone_on_nat_nat @ bot_bot_set_nat @ Orda @ Ordb @ F2 ) ).
% monotone_on_empty
thf(fact_1024_subset__emptyI,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ! [X: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ord_le2690539916662146973ctxt_a @ A5 @ bot_bo8437124304389434473ctxt_a ) ) ).
% subset_emptyI
thf(fact_1025_subset__emptyI,axiom,
! [A5: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1026_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1027_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1028_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1029_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1030_empty__not__UNIV,axiom,
bot_bot_set_rat != top_top_set_rat ).
% empty_not_UNIV
thf(fact_1031_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1032_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1033_wfE__min_H,axiom,
! [R2: set_Pr490164551564945991ctxt_a,Q: set_Pr8705490787052945405ctxt_a] :
( ( wf_Pro4728511896666056454ctxt_a @ R2 )
=> ( ( Q != bot_bo8437124304389434473ctxt_a )
=> ~ ! [Z3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Z3 @ Q )
=> ~ ! [Y6: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ Y6 @ Z3 ) @ R2 )
=> ~ ( member461722496487281478ctxt_a @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1034_wfE__min_H,axiom,
! [R2: set_Pr1261947904930325089at_nat,Q: set_nat] :
( ( wf_nat @ R2 )
=> ( ( Q != bot_bot_set_nat )
=> ~ ! [Z3: nat] :
( ( member_nat @ Z3 @ Q )
=> ~ ! [Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Z3 ) @ R2 )
=> ~ ( member_nat @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1035_wfE__min_H,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,Q: set_Ground_gmctxt_a] :
( ( wf_Ground_gmctxt_a @ R2 )
=> ( ( Q != bot_bo3757104637556896616ctxt_a )
=> ~ ! [Z3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Z3 @ Q )
=> ~ ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y6 @ Z3 ) @ R2 )
=> ~ ( member3306386364193653941ctxt_a @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1036_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1037_Compl__UNIV__eq,axiom,
( ( uminus2201863774496077783et_rat @ top_top_set_rat )
= bot_bot_set_rat ) ).
% Compl_UNIV_eq
thf(fact_1038_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1039_Compl__empty__eq,axiom,
( ( uminus2201863774496077783et_rat @ bot_bot_set_rat )
= top_top_set_rat ) ).
% Compl_empty_eq
thf(fact_1040_SN__empty,axiom,
abstract_SN_on_nat @ bot_bo2099793752762293965at_nat @ top_top_set_nat ).
% SN_empty
thf(fact_1041_SN__empty,axiom,
abstract_SN_on_rat @ bot_bo542495261630389981at_rat @ top_top_set_rat ).
% SN_empty
thf(fact_1042_inj__on__iff__surj,axiom,
! [A5: set_rat,A7: set_nat] :
( ( A5 != bot_bot_set_rat )
=> ( ( ? [F: rat > nat] :
( ( inj_on_rat_nat @ F @ A5 )
& ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A5 ) @ A7 ) ) )
= ( ? [G3: nat > rat] :
( ( image_nat_rat @ G3 @ A7 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1043_inj__on__iff__surj,axiom,
! [A5: set_nat,A7: set_rat] :
( ( A5 != bot_bot_set_nat )
=> ( ( ? [F: nat > rat] :
( ( inj_on_nat_rat @ F @ A5 )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A5 ) @ A7 ) ) )
= ( ? [G3: rat > nat] :
( ( image_rat_nat @ G3 @ A7 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1044_inj__on__iff__surj,axiom,
! [A5: set_nat,A7: set_nat] :
( ( A5 != bot_bot_set_nat )
=> ( ( ? [F: nat > nat] :
( ( inj_on_nat_nat @ F @ A5 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ A7 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat @ G3 @ A7 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1045_surj__vimage__empty,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( vimage_nat_nat @ F2 @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1046_surj__vimage__empty,axiom,
! [F2: nat > rat,A5: set_rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ( vimage_nat_rat @ F2 @ A5 )
= bot_bot_set_nat )
= ( A5 = bot_bot_set_rat ) ) ) ).
% surj_vimage_empty
thf(fact_1047_surj__vimage__empty,axiom,
! [F2: rat > nat,A5: set_nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( ( vimage_rat_nat @ F2 @ A5 )
= bot_bot_set_rat )
= ( A5 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_1048_surj__vimage__empty,axiom,
! [F2: rat > rat,A5: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( ( vimage_rat_rat @ F2 @ A5 )
= bot_bot_set_rat )
= ( A5 = bot_bot_set_rat ) ) ) ).
% surj_vimage_empty
thf(fact_1049_Kleene__iter__lpfp,axiom,
! [F2: nat > nat,P: nat,K2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ ( F2 @ P ) @ P )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ K2 @ F2 @ bot_bot_nat ) @ P ) ) ) ).
% Kleene_iter_lpfp
thf(fact_1050_cSUP__least,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > nat,M: nat] :
( ( A5 != bot_bo8437124304389434473ctxt_a )
=> ( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1051_cSUP__least,axiom,
! [A5: set_nat,F2: nat > nat,M: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A5 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1052_empty__iff,axiom,
! [C2: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ C2 @ bot_bo8437124304389434473ctxt_a ) ).
% empty_iff
thf(fact_1053_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1054_all__not__in__conv,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ( ! [X4: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ X4 @ A5 ) )
= ( A5 = bot_bo8437124304389434473ctxt_a ) ) ).
% all_not_in_conv
thf(fact_1055_all__not__in__conv,axiom,
! [A5: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A5 ) )
= ( A5 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1056_bot__empty__eq,axiom,
( bot_bo5290081426263203764xt_a_o
= ( ^ [X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ bot_bo8437124304389434473ctxt_a ) ) ) ).
% bot_empty_eq
thf(fact_1057_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1058_emptyE,axiom,
! [A: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ A @ bot_bo8437124304389434473ctxt_a ) ).
% emptyE
thf(fact_1059_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1060_equals0D,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,A: produc8175205341597284509ctxt_a] :
( ( A5 = bot_bo8437124304389434473ctxt_a )
=> ~ ( member461722496487281478ctxt_a @ A @ A5 ) ) ).
% equals0D
thf(fact_1061_equals0D,axiom,
! [A5: set_nat,A: nat] :
( ( A5 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A5 ) ) ).
% equals0D
thf(fact_1062_equals0I,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ! [Y3: produc8175205341597284509ctxt_a] :
~ ( member461722496487281478ctxt_a @ Y3 @ A5 )
=> ( A5 = bot_bo8437124304389434473ctxt_a ) ) ).
% equals0I
thf(fact_1063_equals0I,axiom,
! [A5: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A5 )
=> ( A5 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1064_ex__in__conv,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ( ? [X4: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ X4 @ A5 ) )
= ( A5 != bot_bo8437124304389434473ctxt_a ) ) ).
% ex_in_conv
thf(fact_1065_ex__in__conv,axiom,
! [A5: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A5 ) )
= ( A5 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1066_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) )
=> ( ( abstract_SN_on_nat @ S3 @ ( insert_nat @ ( T @ zero_zero_nat ) @ bot_bot_set_nat ) )
=> ? [I4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ R2 )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ S3 )
& ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ R2 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1067_chain__Un__SN__on__imp__first__step,axiom,
! [T: nat > ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a] :
( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( sup_su1394892571904902097ctxt_a @ R2 @ S3 ) )
=> ( ( abstra4731155269209573716ctxt_a @ S3 @ ( insert2292343091172363342ctxt_a @ ( T @ zero_zero_nat ) @ bot_bo3757104637556896616ctxt_a ) )
=> ? [I4: nat] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ R2 )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I4 )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ S3 )
& ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ J3 ) @ ( T @ ( suc @ J3 ) ) ) @ R2 ) ) ) ) ) ) ).
% chain_Un_SN_on_imp_first_step
thf(fact_1068_max__ext_Ocases,axiom,
! [A1: set_Pr8705490787052945405ctxt_a,A22: set_Pr8705490787052945405ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member2401363563938284688ctxt_a @ ( produc795262113445460951ctxt_a @ A1 @ A22 ) @ ( max_ex7231900002760029912ctxt_a @ R2 ) )
=> ~ ( ( finite7161443020207210054ctxt_a @ A1 )
=> ( ( finite7161443020207210054ctxt_a @ A22 )
=> ( ( A22 != bot_bo8437124304389434473ctxt_a )
=> ~ ! [X5: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X5 @ A1 )
=> ? [Xa3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Xa3 @ A22 )
& ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1069_max__ext_Ocases,axiom,
! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
=> ~ ( ( finite_finite_nat @ A1 )
=> ( ( finite_finite_nat @ A22 )
=> ( ( A22 != bot_bot_set_nat )
=> ~ ! [X5: nat] :
( ( member_nat @ X5 @ A1 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1070_max__ext_Ocases,axiom,
! [A1: set_Ground_gmctxt_a,A22: set_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member5333943563032178098ctxt_a @ ( produc3704910457525973569ctxt_a @ A1 @ A22 ) @ ( max_ex5522983650612646819ctxt_a @ R2 ) )
=> ~ ( ( finite8279891950448712117ctxt_a @ A1 )
=> ( ( finite8279891950448712117ctxt_a @ A22 )
=> ( ( A22 != bot_bo3757104637556896616ctxt_a )
=> ~ ! [X5: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X5 @ A1 )
=> ? [Xa3: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa3 @ A22 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X5 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_1071_insertCI,axiom,
! [A: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( ~ ( member461722496487281478ctxt_a @ A @ B5 )
=> ( A = B ) )
=> ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ B @ B5 ) ) ) ).
% insertCI
thf(fact_1072_insertCI,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B5 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertCI
thf(fact_1073_insert__iff,axiom,
! [A: produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ B @ A5 ) )
= ( ( A = B )
| ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_1074_insert__iff,axiom,
! [A: nat,B: nat,A5: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A5 ) )
= ( ( A = B )
| ( member_nat @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_1075_insert__absorb2,axiom,
! [X3: nat,A5: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A5 ) )
= ( insert_nat @ X3 @ A5 ) ) ).
% insert_absorb2
thf(fact_1076_image__insert,axiom,
! [F2: nat > rat,A: nat,B5: set_nat] :
( ( image_nat_rat @ F2 @ ( insert_nat @ A @ B5 ) )
= ( insert_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_1077_image__insert,axiom,
! [F2: nat > nat,A: nat,B5: set_nat] :
( ( image_nat_nat @ F2 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_1078_insert__image,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > nat] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( insert_nat @ ( F2 @ X3 ) @ ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
= ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) ) ).
% insert_image
thf(fact_1079_insert__image,axiom,
! [X3: nat,A5: set_nat,F2: nat > rat] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_rat @ ( F2 @ X3 ) @ ( image_nat_rat @ F2 @ A5 ) )
= ( image_nat_rat @ F2 @ A5 ) ) ) ).
% insert_image
thf(fact_1080_insert__image,axiom,
! [X3: nat,A5: set_nat,F2: nat > nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( insert_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A5 ) )
= ( image_nat_nat @ F2 @ A5 ) ) ) ).
% insert_image
thf(fact_1081_singletonI,axiom,
! [A: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ A @ bot_bo8437124304389434473ctxt_a ) ) ).
% singletonI
thf(fact_1082_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1083_insert__subset,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ A5 ) @ B5 )
= ( ( member461722496487281478ctxt_a @ X3 @ B5 )
& ( ord_le2690539916662146973ctxt_a @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_1084_insert__subset,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( ( member_nat @ X3 @ B5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_1085_Un__insert__right,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A5 @ B5 ) ) ) ).
% Un_insert_right
thf(fact_1086_Un__insert__left,axiom,
! [A: nat,B5: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C ) ) ) ).
% Un_insert_left
thf(fact_1087_singleton__insert__inj__eq_H,axiom,
! [A: nat,A5: set_nat,B: nat] :
( ( ( insert_nat @ A @ A5 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1088_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A5: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A5 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1089_wf__insert,axiom,
! [Y: ground_gmctxt_a,X3: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( wf_Ground_gmctxt_a @ ( insert6238942492121947757ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ X3 ) @ R ) )
= ( ( wf_Ground_gmctxt_a @ R )
& ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% wf_insert
thf(fact_1090_acyclic__insert,axiom,
! [Y: ground_gmctxt_a,X3: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( transi8399219972279719055ctxt_a @ ( insert6238942492121947757ctxt_a @ ( produc6347872285926497493ctxt_a @ Y @ X3 ) @ R ) )
= ( ( transi8399219972279719055ctxt_a @ R )
& ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( transi2468631686036622171ctxt_a @ R ) ) ) ) ).
% acyclic_insert
thf(fact_1091_subset__Compl__singleton,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A5 @ ( uminus5163801920556549748ctxt_a @ ( insert6238942492121947757ctxt_a @ B @ bot_bo8437124304389434473ctxt_a ) ) )
= ( ~ ( member461722496487281478ctxt_a @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_1092_subset__Compl__singleton,axiom,
! [A5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A5 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_1093_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1094_subset__singletonD,axiom,
! [A5: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A5 = bot_bot_set_nat )
| ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1095_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1096_Un__singleton__iff,axiom,
! [A5: set_nat,B5: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A5 @ B5 )
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( ( ( A5 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1097_singleton__Un__iff,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ( ( insert_nat @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A5 @ B5 ) )
= ( ( ( A5 = bot_bot_set_nat )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5 = bot_bot_set_nat ) )
| ( ( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1098_vimage__singleton__eq,axiom,
! [A: produc8175205341597284509ctxt_a,F2: produc8175205341597284509ctxt_a > nat,B: nat] :
( ( member461722496487281478ctxt_a @ A @ ( vimage6110315667926561778_a_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1099_vimage__singleton__eq,axiom,
! [A: nat,F2: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F2 @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_1100_singletonD,axiom,
! [B: produc8175205341597284509ctxt_a,A: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ B @ ( insert6238942492121947757ctxt_a @ A @ bot_bo8437124304389434473ctxt_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1101_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1102_singleton__iff,axiom,
! [B: produc8175205341597284509ctxt_a,A: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ B @ ( insert6238942492121947757ctxt_a @ A @ bot_bo8437124304389434473ctxt_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1103_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1104_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D3 @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B = D3 ) )
| ( ( A = D3 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1105_insert__not__empty,axiom,
! [A: nat,A5: set_nat] :
( ( insert_nat @ A @ A5 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1106_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1107_total__on__singleton,axiom,
! [X3: nat,R: set_Pr1261947904930325089at_nat] : ( total_on_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ R ) ).
% total_on_singleton
thf(fact_1108_inj__img__insertE,axiom,
! [F2: nat > rat,A5: set_nat,X3: rat,B5: set_rat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ~ ( member_rat @ X3 @ B5 )
=> ( ( ( insert_rat @ X3 @ B5 )
= ( image_nat_rat @ F2 @ A5 ) )
=> ~ ! [X10: nat,A8: set_nat] :
( ~ ( member_nat @ X10 @ A8 )
=> ( ( A5
= ( insert_nat @ X10 @ A8 ) )
=> ( ( X3
= ( F2 @ X10 ) )
=> ( B5
!= ( image_nat_rat @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1109_inj__img__insertE,axiom,
! [F2: produc8175205341597284509ctxt_a > produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on4839580843290501553ctxt_a @ F2 @ A5 )
=> ( ~ ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( ( insert6238942492121947757ctxt_a @ X3 @ B5 )
= ( image_4216501698144419397ctxt_a @ F2 @ A5 ) )
=> ~ ! [X10: produc8175205341597284509ctxt_a,A8: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X10 @ A8 )
=> ( ( A5
= ( insert6238942492121947757ctxt_a @ X10 @ A8 ) )
=> ( ( X3
= ( F2 @ X10 ) )
=> ( B5
!= ( image_4216501698144419397ctxt_a @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1110_inj__img__insertE,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A5: set_nat,X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ A5 )
=> ( ~ ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( ( insert6238942492121947757ctxt_a @ X3 @ B5 )
= ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
=> ~ ! [X10: nat,A8: set_nat] :
( ~ ( member_nat @ X10 @ A8 )
=> ( ( A5
= ( insert_nat @ X10 @ A8 ) )
=> ( ( X3
= ( F2 @ X10 ) )
=> ( B5
!= ( image_43530617296285190ctxt_a @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1111_inj__img__insertE,axiom,
! [F2: produc8175205341597284509ctxt_a > nat,A5: set_Pr8705490787052945405ctxt_a,X3: nat,B5: set_nat] :
( ( inj_on5187228193288102844_a_nat @ F2 @ A5 )
=> ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( insert_nat @ X3 @ B5 )
= ( image_5981101931084406824_a_nat @ F2 @ A5 ) )
=> ~ ! [X10: produc8175205341597284509ctxt_a,A8: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X10 @ A8 )
=> ( ( A5
= ( insert6238942492121947757ctxt_a @ X10 @ A8 ) )
=> ( ( X3
= ( F2 @ X10 ) )
=> ( B5
!= ( image_5981101931084406824_a_nat @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1112_inj__img__insertE,axiom,
! [F2: nat > nat,A5: set_nat,X3: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( insert_nat @ X3 @ B5 )
= ( image_nat_nat @ F2 @ A5 ) )
=> ~ ! [X10: nat,A8: set_nat] :
( ~ ( member_nat @ X10 @ A8 )
=> ( ( A5
= ( insert_nat @ X10 @ A8 ) )
=> ( ( X3
= ( F2 @ X10 ) )
=> ( B5
!= ( image_nat_nat @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1113_insert__subsetI,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,X7: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( ord_le2690539916662146973ctxt_a @ X7 @ A5 )
=> ( ord_le2690539916662146973ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ X7 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_1114_insert__subsetI,axiom,
! [X3: nat,A5: set_nat,X7: set_nat] :
( ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_set_nat @ X7 @ A5 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X7 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_1115_insert__mono,axiom,
! [C: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_1116_subset__insert,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( ord_le2690539916662146973ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ B5 ) )
= ( ord_le2690539916662146973ctxt_a @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_1117_subset__insert,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_1118_subset__insertI,axiom,
! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_1119_subset__insertI2,axiom,
! [A5: set_nat,B5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
=> ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_1120_insertE,axiom,
! [A: produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member461722496487281478ctxt_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_1121_insertE,axiom,
! [A: nat,B: nat,A5: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A5 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A5 ) ) ) ).
% insertE
thf(fact_1122_insertI1,axiom,
! [A: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] : ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ A @ B5 ) ) ).
% insertI1
thf(fact_1123_insertI1,axiom,
! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).
% insertI1
thf(fact_1124_insertI2,axiom,
! [A: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ B5 )
=> ( member461722496487281478ctxt_a @ A @ ( insert6238942492121947757ctxt_a @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1125_insertI2,axiom,
! [A: nat,B5: set_nat,B: nat] :
( ( member_nat @ A @ B5 )
=> ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).
% insertI2
thf(fact_1126_Set_Oset__insert,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ~ ! [B9: set_Pr8705490787052945405ctxt_a] :
( ( A5
= ( insert6238942492121947757ctxt_a @ X3 @ B9 ) )
=> ( member461722496487281478ctxt_a @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_1127_Set_Oset__insert,axiom,
! [X3: nat,A5: set_nat] :
( ( member_nat @ X3 @ A5 )
=> ~ ! [B9: set_nat] :
( ( A5
= ( insert_nat @ X3 @ B9 ) )
=> ( member_nat @ X3 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_1128_insert__ident,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ~ ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( ( insert6238942492121947757ctxt_a @ X3 @ A5 )
= ( insert6238942492121947757ctxt_a @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1129_insert__ident,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( insert_nat @ X3 @ A5 )
= ( insert_nat @ X3 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_1130_insert__absorb,axiom,
! [A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ A5 )
=> ( ( insert6238942492121947757ctxt_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_1131_insert__absorb,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( insert_nat @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_1132_insert__eq__iff,axiom,
! [A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ A @ A5 )
=> ( ~ ( member461722496487281478ctxt_a @ B @ B5 )
=> ( ( ( insert6238942492121947757ctxt_a @ A @ A5 )
= ( insert6238942492121947757ctxt_a @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C6: set_Pr8705490787052945405ctxt_a] :
( ( A5
= ( insert6238942492121947757ctxt_a @ B @ C6 ) )
& ~ ( member461722496487281478ctxt_a @ B @ C6 )
& ( B5
= ( insert6238942492121947757ctxt_a @ A @ C6 ) )
& ~ ( member461722496487281478ctxt_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1133_insert__eq__iff,axiom,
! [A: nat,A5: set_nat,B: nat,B5: set_nat] :
( ~ ( member_nat @ A @ A5 )
=> ( ~ ( member_nat @ B @ B5 )
=> ( ( ( insert_nat @ A @ A5 )
= ( insert_nat @ B @ B5 ) )
= ( ( ( A = B )
=> ( A5 = B5 ) )
& ( ( A != B )
=> ? [C6: set_nat] :
( ( A5
= ( insert_nat @ B @ C6 ) )
& ~ ( member_nat @ B @ C6 )
& ( B5
= ( insert_nat @ A @ C6 ) )
& ~ ( member_nat @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1134_insert__commute,axiom,
! [X3: nat,Y: nat,A5: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y @ A5 ) )
= ( insert_nat @ Y @ ( insert_nat @ X3 @ A5 ) ) ) ).
% insert_commute
thf(fact_1135_mk__disjoint__insert,axiom,
! [A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ A5 )
=> ? [B9: set_Pr8705490787052945405ctxt_a] :
( ( A5
= ( insert6238942492121947757ctxt_a @ A @ B9 ) )
& ~ ( member461722496487281478ctxt_a @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_1136_mk__disjoint__insert,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ? [B9: set_nat] :
( ( A5
= ( insert_nat @ A @ B9 ) )
& ~ ( member_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_1137_insert__UNIV,axiom,
! [X3: nat] :
( ( insert_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_1138_insert__UNIV,axiom,
! [X3: rat] :
( ( insert_rat @ X3 @ top_top_set_rat )
= top_top_set_rat ) ).
% insert_UNIV
thf(fact_1139_range__eq__singletonD,axiom,
! [F2: nat > rat,A: rat,X3: nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= ( insert_rat @ A @ bot_bot_set_rat ) )
=> ( ( F2 @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1140_range__eq__singletonD,axiom,
! [F2: nat > nat,A: nat,X3: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F2 @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1141_range__eq__singletonD,axiom,
! [F2: rat > nat,A: nat,X3: rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F2 @ X3 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1142_SN__on__induct_H,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R2 @ ( insert_nat @ S @ bot_bot_set_nat ) )
=> ( ! [T4: nat] :
( ! [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U4 ) @ R2 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_1143_SN__on__induct_H,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,S: ground_gmctxt_a,P2: ground_gmctxt_a > $o] :
( ( abstra4731155269209573716ctxt_a @ R2 @ ( insert2292343091172363342ctxt_a @ S @ bot_bo3757104637556896616ctxt_a ) )
=> ( ! [T4: ground_gmctxt_a] :
( ! [U4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ T4 @ U4 ) @ R2 )
=> ( P2 @ U4 ) )
=> ( P2 @ T4 ) )
=> ( P2 @ S ) ) ) ).
% SN_on_induct'
thf(fact_1144_step__reflects__SN__on,axiom,
! [A: nat,R: set_Pr1261947904930325089at_nat] :
( ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% step_reflects_SN_on
thf(fact_1145_step__reflects__SN__on,axiom,
! [A: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ! [B3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B3 ) @ R )
=> ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ B3 @ bot_bo3757104637556896616ctxt_a ) ) )
=> ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) ) ) ).
% step_reflects_SN_on
thf(fact_1146_SN__on__imp__on__minimal,axiom,
! [R: set_Pr490164551564945991ctxt_a,X3: produc8175205341597284509ctxt_a] :
( ( abstra7720173292291745255ctxt_a @ R @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) )
=> ! [Q5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ Q5 )
=> ? [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ Q5 )
& ! [Y6: produc8175205341597284509ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Y6 ) @ R )
=> ~ ( member461722496487281478ctxt_a @ Y6 @ Q5 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1147_SN__on__imp__on__minimal,axiom,
! [R: set_Pr1261947904930325089at_nat,X3: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ! [Q5: set_nat] :
( ( member_nat @ X3 @ Q5 )
=> ? [X: nat] :
( ( member_nat @ X @ Q5 )
& ! [Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R )
=> ~ ( member_nat @ Y6 @ Q5 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1148_SN__on__imp__on__minimal,axiom,
! [R: set_Pr8705490787052945405ctxt_a,X3: ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ X3 @ bot_bo3757104637556896616ctxt_a ) )
=> ! [Q5: set_Ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X3 @ Q5 )
=> ? [X: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ Q5 )
& ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y6 ) @ R )
=> ~ ( member3306386364193653941ctxt_a @ Y6 @ Q5 ) ) ) ) ) ).
% SN_on_imp_on_minimal
thf(fact_1149_step__preserves__SN__on,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( abstract_SN_on_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1150_step__preserves__SN__on,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
=> ( ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) )
=> ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ B @ bot_bo3757104637556896616ctxt_a ) ) ) ) ).
% step_preserves_SN_on
thf(fact_1151_SN__on__induct__acc__style,axiom,
! [R2: set_Pr1261947904930325089at_nat,A: nat,P2: nat > $o] :
( ( abstract_SN_on_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ! [X: nat] :
( ( abstract_SN_on_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [Y6: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R2 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1152_SN__on__induct__acc__style,axiom,
! [R2: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a,P2: ground_gmctxt_a > $o] :
( ( abstra4731155269209573716ctxt_a @ R2 @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) )
=> ( ! [X: ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R2 @ ( insert2292343091172363342ctxt_a @ X @ bot_bo3757104637556896616ctxt_a ) )
=> ( ! [Y6: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y6 ) @ R2 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ A ) ) ) ).
% SN_on_induct_acc_style
thf(fact_1153_lower__set__imp__not__SN__on,axiom,
! [S: produc8175205341597284509ctxt_a,X7: set_Pr8705490787052945405ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member461722496487281478ctxt_a @ S @ X7 )
=> ( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ X7 )
=> ? [Xa: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Xa @ X7 )
& ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Xa ) @ R2 ) ) )
=> ~ ( abstra7720173292291745255ctxt_a @ R2 @ ( insert6238942492121947757ctxt_a @ S @ bot_bo8437124304389434473ctxt_a ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1154_lower__set__imp__not__SN__on,axiom,
! [S: nat,X7: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ S @ X7 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Xa ) @ R2 ) ) )
=> ~ ( abstract_SN_on_nat @ R2 @ ( insert_nat @ S @ bot_bot_set_nat ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1155_lower__set__imp__not__SN__on,axiom,
! [S: ground_gmctxt_a,X7: set_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member3306386364193653941ctxt_a @ S @ X7 )
=> ( ! [X: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ X7 )
=> ? [Xa: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa @ X7 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Xa ) @ R2 ) ) )
=> ~ ( abstra4731155269209573716ctxt_a @ R2 @ ( insert2292343091172363342ctxt_a @ S @ bot_bo3757104637556896616ctxt_a ) ) ) ) ).
% lower_set_imp_not_SN_on
thf(fact_1156_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat] :
( ( abstract_SN_on_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ! [B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R )
=> ( abstract_SN_on_nat @ R @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1157_SN__on__all__reducts__SN__on__conv,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a] :
( ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) )
= ( ! [B4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B4 ) @ R )
=> ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ B4 @ bot_bo3757104637556896616ctxt_a ) ) ) ) ) ).
% SN_on_all_reducts_SN_on_conv
thf(fact_1158_SN__I,axiom,
! [A5: set_Pr1261947904930325089at_nat] :
( ! [A3: nat] : ( abstract_SN_on_nat @ A5 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A5 @ top_top_set_nat ) ) ).
% SN_I
thf(fact_1159_SN__I,axiom,
! [A5: set_Pr8928021450653196913at_rat] :
( ! [A3: rat] : ( abstract_SN_on_rat @ A5 @ ( insert_rat @ A3 @ bot_bot_set_rat ) )
=> ( abstract_SN_on_rat @ A5 @ top_top_set_rat ) ) ).
% SN_I
thf(fact_1160_SN__def,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( abstract_SN_on_nat @ R @ top_top_set_nat )
= ( ! [X4: nat] : ( abstract_SN_on_nat @ R @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% SN_def
thf(fact_1161_SN__def,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ( abstract_SN_on_rat @ R @ top_top_set_rat )
= ( ! [X4: rat] : ( abstract_SN_on_rat @ R @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) ) ) ).
% SN_def
thf(fact_1162_Pow__insert,axiom,
! [A: nat,A5: set_nat] :
( ( pow_nat @ ( insert_nat @ A @ A5 ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A5 ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A ) @ ( pow_nat @ A5 ) ) ) ) ).
% Pow_insert
thf(fact_1163_chain__imp__not__SN__on,axiom,
! [F2: nat > nat,R: set_Pr1261947904930325089at_nat,I: nat] :
( ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R )
=> ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ ( F2 @ I ) @ bot_bot_set_nat ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1164_chain__imp__not__SN__on,axiom,
! [F2: nat > ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I: nat] :
( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) ) @ R )
=> ~ ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ ( F2 @ I ) @ bot_bo3757104637556896616ctxt_a ) ) ) ).
% chain_imp_not_SN_on
thf(fact_1165_inf__img__fin__dom,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ? [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F2 @ A5 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ X @ bot_bot_set_rat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1166_inf__img__fin__dom,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F2 @ A5 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_1167_inf__img__fin__domE,axiom,
! [F2: nat > rat,A5: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F2 @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ~ ! [Y3: rat] :
( ( member_rat @ Y3 @ ( image_nat_rat @ F2 @ A5 ) )
=> ( finite_finite_nat @ ( vimage_nat_rat @ F2 @ ( insert_rat @ Y3 @ bot_bot_set_rat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1168_inf__img__fin__domE,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A5: set_nat] :
( ( finite7161443020207210054ctxt_a @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ~ ! [Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Y3 @ ( image_43530617296285190ctxt_a @ F2 @ A5 ) )
=> ( finite_finite_nat @ ( vimage172744354138440144ctxt_a @ F2 @ ( insert6238942492121947757ctxt_a @ Y3 @ bot_bo8437124304389434473ctxt_a ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1169_inf__img__fin__domE,axiom,
! [F2: nat > nat,A5: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A5 ) )
=> ( ~ ( finite_finite_nat @ A5 )
=> ~ ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_nat_nat @ F2 @ A5 ) )
=> ( finite_finite_nat @ ( vimage_nat_nat @ F2 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_1170_steps__preserve__SN__on,axiom,
! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A5 ) )
=> ( ( abstract_SN_on_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( abstract_SN_on_nat @ A5 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1171_steps__preserve__SN__on,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ A5 ) )
=> ( ( abstra4731155269209573716ctxt_a @ A5 @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) )
=> ( abstra4731155269209573716ctxt_a @ A5 @ ( insert2292343091172363342ctxt_a @ B @ bot_bo3757104637556896616ctxt_a ) ) ) ) ).
% steps_preserve_SN_on
thf(fact_1172_steps__reflect__SN__on,axiom,
! [R: set_Pr1261947904930325089at_nat,B: nat,A: nat] :
( ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( abstract_SN_on_nat @ R @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1173_steps__reflect__SN__on,axiom,
! [R: set_Pr8705490787052945405ctxt_a,B: ground_gmctxt_a,A: ground_gmctxt_a] :
( ~ ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ B @ bot_bo3757104637556896616ctxt_a ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( transi2468631686036622171ctxt_a @ R ) )
=> ~ ( abstra4731155269209573716ctxt_a @ R @ ( insert2292343091172363342ctxt_a @ A @ bot_bo3757104637556896616ctxt_a ) ) ) ) ).
% steps_reflect_SN_on
thf(fact_1174_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_1175_max__ext_Omax__extI,axiom,
! [X7: set_Pr8705490787052945405ctxt_a,Y8: set_Pr8705490787052945405ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( finite7161443020207210054ctxt_a @ X7 )
=> ( ( finite7161443020207210054ctxt_a @ Y8 )
=> ( ( Y8 != bot_bo8437124304389434473ctxt_a )
=> ( ! [X: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ X7 )
=> ? [Xa: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Xa @ Y8 )
& ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X @ Xa ) @ R2 ) ) )
=> ( member2401363563938284688ctxt_a @ ( produc795262113445460951ctxt_a @ X7 @ Y8 ) @ ( max_ex7231900002760029912ctxt_a @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1176_max__ext_Omax__extI,axiom,
! [X7: set_nat,Y8: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ X7 )
=> ( ( finite_finite_nat @ Y8 )
=> ( ( Y8 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ Y8 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Xa ) @ R2 ) ) )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X7 @ Y8 ) @ ( max_ext_nat @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1177_max__ext_Omax__extI,axiom,
! [X7: set_Ground_gmctxt_a,Y8: set_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( finite8279891950448712117ctxt_a @ X7 )
=> ( ( finite8279891950448712117ctxt_a @ Y8 )
=> ( ( Y8 != bot_bo3757104637556896616ctxt_a )
=> ( ! [X: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X @ X7 )
=> ? [Xa: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Xa @ Y8 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Xa ) @ R2 ) ) )
=> ( member5333943563032178098ctxt_a @ ( produc3704910457525973569ctxt_a @ X7 @ Y8 ) @ ( max_ex5522983650612646819ctxt_a @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1178_max__ext_Osimps,axiom,
! [A1: set_Pr8705490787052945405ctxt_a,A22: set_Pr8705490787052945405ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member2401363563938284688ctxt_a @ ( produc795262113445460951ctxt_a @ A1 @ A22 ) @ ( max_ex7231900002760029912ctxt_a @ R2 ) )
= ( ( finite7161443020207210054ctxt_a @ A1 )
& ( finite7161443020207210054ctxt_a @ A22 )
& ( A22 != bot_bo8437124304389434473ctxt_a )
& ! [X4: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X4 @ A1 )
=> ? [Y4: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ Y4 @ A22 )
& ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1179_max__ext_Osimps,axiom,
! [A1: set_nat,A22: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R2 ) )
= ( ( finite_finite_nat @ A1 )
& ( finite_finite_nat @ A22 )
& ( A22 != bot_bot_set_nat )
& ! [X4: nat] :
( ( member_nat @ X4 @ A1 )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1180_max__ext_Osimps,axiom,
! [A1: set_Ground_gmctxt_a,A22: set_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member5333943563032178098ctxt_a @ ( produc3704910457525973569ctxt_a @ A1 @ A22 ) @ ( max_ex5522983650612646819ctxt_a @ R2 ) )
= ( ( finite8279891950448712117ctxt_a @ A1 )
& ( finite8279891950448712117ctxt_a @ A22 )
& ( A22 != bot_bo3757104637556896616ctxt_a )
& ! [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A1 )
=> ? [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A22 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_1181_non__strict__ending,axiom,
! [T: nat > nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S3 ) @ S3 )
=> ( ( abstract_SN_on_nat @ S3 @ ( insert_nat @ ( T @ zero_zero_nat ) @ bot_bot_set_nat ) )
=> ? [J4: nat] :
! [I3: nat] :
( ( ord_less_eq_nat @ J4 @ I3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_1356011639430497352at_nat @ R2 @ S3 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1182_non__strict__ending,axiom,
! [T: nat > ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a,S3: set_Pr8705490787052945405ctxt_a] :
( ! [I4: nat] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ I4 ) @ ( T @ ( suc @ I4 ) ) ) @ ( sup_su1394892571904902097ctxt_a @ R2 @ S3 ) )
=> ( ( ord_le2690539916662146973ctxt_a @ ( relcom6868745818274832221ctxt_a @ R2 @ S3 ) @ S3 )
=> ( ( abstra4731155269209573716ctxt_a @ S3 @ ( insert2292343091172363342ctxt_a @ ( T @ zero_zero_nat ) @ bot_bo3757104637556896616ctxt_a ) )
=> ? [J4: nat] :
! [I3: nat] :
( ( ord_less_eq_nat @ J4 @ I3 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( T @ I3 ) @ ( T @ ( suc @ I3 ) ) ) @ ( minus_3912115830687649252ctxt_a @ R2 @ S3 ) ) ) ) ) ) ).
% non_strict_ending
thf(fact_1183_Diff__iff,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) )
= ( ( member461722496487281478ctxt_a @ C2 @ A5 )
& ~ ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_1184_Diff__iff,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( ( member_nat @ C2 @ A5 )
& ~ ( member_nat @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_1185_DiffI,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( ~ ( member461722496487281478ctxt_a @ C2 @ B5 )
=> ( member461722496487281478ctxt_a @ C2 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1186_DiffI,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ A5 )
=> ( ~ ( member_nat @ C2 @ B5 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1187_Diff__insert0,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ B5 ) )
= ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1188_Diff__insert0,axiom,
! [X3: nat,A5: set_nat,B5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_1189_insert__Diff1,axiom,
! [X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( minus_3912115830687649252ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ A5 ) @ B5 )
= ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1190_insert__Diff1,axiom,
! [X3: nat,B5: set_nat,A5: set_nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_1191_surj__diff,axiom,
! [A: rat] :
( ( image_rat_rat @ ( minus_minus_rat @ A ) @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_diff
thf(fact_1192_Diff__UNIV,axiom,
! [A5: set_nat] :
( ( minus_minus_set_nat @ A5 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1193_Diff__UNIV,axiom,
! [A5: set_rat] :
( ( minus_minus_set_rat @ A5 @ top_top_set_rat )
= bot_bot_set_rat ) ).
% Diff_UNIV
thf(fact_1194_insert__Diff__single,axiom,
! [A: nat,A5: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_1195_inj__on__insert,axiom,
! [F2: nat > rat,A: nat,A5: set_nat] :
( ( inj_on_nat_rat @ F2 @ ( insert_nat @ A @ A5 ) )
= ( ( inj_on_nat_rat @ F2 @ A5 )
& ~ ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1196_inj__on__insert,axiom,
! [F2: nat > produc8175205341597284509ctxt_a,A: nat,A5: set_nat] :
( ( inj_on8473028916354757018ctxt_a @ F2 @ ( insert_nat @ A @ A5 ) )
= ( ( inj_on8473028916354757018ctxt_a @ F2 @ A5 )
& ~ ( member461722496487281478ctxt_a @ ( F2 @ A ) @ ( image_43530617296285190ctxt_a @ F2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1197_inj__on__insert,axiom,
! [F2: nat > nat,A: nat,A5: set_nat] :
( ( inj_on_nat_nat @ F2 @ ( insert_nat @ A @ A5 ) )
= ( ( inj_on_nat_nat @ F2 @ A5 )
& ~ ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1198_Diff__insert,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A5 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1199_insert__Diff,axiom,
! [A: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ A @ A5 )
=> ( ( insert6238942492121947757ctxt_a @ A @ ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ A @ bot_bo8437124304389434473ctxt_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1200_insert__Diff,axiom,
! [A: nat,A5: set_nat] :
( ( member_nat @ A @ A5 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_1201_Diff__insert2,axiom,
! [A5: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_1202_Diff__insert__absorb,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( minus_3912115830687649252ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ A5 ) @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1203_Diff__insert__absorb,axiom,
! [X3: nat,A5: set_nat] :
( ~ ( member_nat @ X3 @ A5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_1204_subset__Diff__insert,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a,C: set_Pr8705490787052945405ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A5 @ ( minus_3912115830687649252ctxt_a @ B5 @ ( insert6238942492121947757ctxt_a @ X3 @ C ) ) )
= ( ( ord_le2690539916662146973ctxt_a @ A5 @ ( minus_3912115830687649252ctxt_a @ B5 @ C ) )
& ~ ( member461722496487281478ctxt_a @ X3 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1205_subset__Diff__insert,axiom,
! [A5: set_nat,B5: set_nat,X3: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X3 @ C ) ) )
= ( ( ord_less_eq_set_nat @ A5 @ ( minus_minus_set_nat @ B5 @ C ) )
& ~ ( member_nat @ X3 @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_1206_insert__Diff__if,axiom,
! [X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ( ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( minus_3912115830687649252ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ A5 ) @ B5 )
= ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) ) )
& ( ~ ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( minus_3912115830687649252ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 @ A5 ) @ B5 )
= ( insert6238942492121947757ctxt_a @ X3 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1207_insert__Diff__if,axiom,
! [X3: nat,B5: set_nat,A5: set_nat] :
( ( ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( minus_minus_set_nat @ A5 @ B5 ) ) )
& ( ~ ( member_nat @ X3 @ B5 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A5 ) @ B5 )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1208_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1209_Compl__eq__Diff__UNIV,axiom,
( uminus2201863774496077783et_rat
= ( minus_minus_set_rat @ top_top_set_rat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1210_psubset__imp__ex__mem,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le254475249434039465ctxt_a @ A5 @ B5 )
=> ? [B3: produc8175205341597284509ctxt_a] : ( member461722496487281478ctxt_a @ B3 @ ( minus_3912115830687649252ctxt_a @ B5 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1211_psubset__imp__ex__mem,axiom,
! [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1212_DiffD2,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) )
=> ~ ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_1213_DiffD2,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
=> ~ ( member_nat @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_1214_DiffD1,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) )
=> ( member461722496487281478ctxt_a @ C2 @ A5 ) ) ).
% DiffD1
thf(fact_1215_DiffD1,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
=> ( member_nat @ C2 @ A5 ) ) ).
% DiffD1
thf(fact_1216_DiffE,axiom,
! [C2: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ C2 @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) )
=> ~ ( ( member461722496487281478ctxt_a @ C2 @ A5 )
=> ( member461722496487281478ctxt_a @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_1217_DiffE,axiom,
! [C2: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
=> ~ ( ( member_nat @ C2 @ A5 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_1218_image__diff__subset,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] : ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_1219_image__diff__subset,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_1220_subset__insert__iff,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le2690539916662146973ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ B5 ) )
= ( ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ord_le2690539916662146973ctxt_a @ ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) ) @ B5 ) )
& ( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ord_le2690539916662146973ctxt_a @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1221_subset__insert__iff,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ( ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_1222_Diff__single__insert,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_1223_image__set__diff,axiom,
! [F2: nat > rat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_1224_image__set__diff,axiom,
! [F2: nat > nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_1225_inj__on__image__set__diff,axiom,
! [F2: nat > rat,C: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F2 @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ B5 ) @ C )
=> ( ( ord_less_eq_set_nat @ B5 @ C )
=> ( ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F2 @ A5 ) @ ( image_nat_rat @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1226_inj__on__image__set__diff,axiom,
! [F2: nat > nat,C: set_nat,A5: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F2 @ C )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A5 @ B5 ) @ C )
=> ( ( ord_less_eq_set_nat @ B5 @ C )
=> ( ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A5 ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1227_rtrancl__diff__decomp,axiom,
! [X3: ground_gmctxt_a,Y: ground_gmctxt_a,A5: set_Pr8705490787052945405ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( minus_3912115830687649252ctxt_a @ ( transi2468631686036622171ctxt_a @ A5 ) @ ( transi2468631686036622171ctxt_a @ B5 ) ) )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ Y ) @ ( relcom6868745818274832221ctxt_a @ ( transi2468631686036622171ctxt_a @ A5 ) @ ( relcom6868745818274832221ctxt_a @ ( minus_3912115830687649252ctxt_a @ A5 @ B5 ) @ ( transi2468631686036622171ctxt_a @ A5 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1228_Compl__insert,axiom,
! [X3: nat,A5: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A5 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A5 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_1229_in__image__insert__iff,axiom,
! [B5: set_se6366015629438403421ctxt_a,X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a] :
( ! [C3: set_Pr8705490787052945405ctxt_a] :
( ( member6521329796374918822ctxt_a @ C3 @ B5 )
=> ~ ( member461722496487281478ctxt_a @ X3 @ C3 ) )
=> ( ( member6521329796374918822ctxt_a @ A5 @ ( image_5735731932157988421ctxt_a @ ( insert6238942492121947757ctxt_a @ X3 ) @ B5 ) )
= ( ( member461722496487281478ctxt_a @ X3 @ A5 )
& ( member6521329796374918822ctxt_a @ ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1230_in__image__insert__iff,axiom,
! [B5: set_set_nat,X3: nat,A5: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B5 )
=> ~ ( member_nat @ X3 @ C3 ) )
=> ( ( member_set_nat @ A5 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B5 ) )
= ( ( member_nat @ X3 @ A5 )
& ( member_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1231_psubset__insert__iff,axiom,
! [A5: set_Pr8705490787052945405ctxt_a,X3: produc8175205341597284509ctxt_a,B5: set_Pr8705490787052945405ctxt_a] :
( ( ord_le254475249434039465ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ B5 ) )
= ( ( ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ord_le254475249434039465ctxt_a @ A5 @ B5 ) )
& ( ~ ( member461722496487281478ctxt_a @ X3 @ B5 )
=> ( ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ord_le254475249434039465ctxt_a @ ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) ) @ B5 ) )
& ( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ord_le2690539916662146973ctxt_a @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1232_psubset__insert__iff,axiom,
! [A5: set_nat,X3: nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ ( insert_nat @ X3 @ B5 ) )
= ( ( ( member_nat @ X3 @ B5 )
=> ( ord_less_set_nat @ A5 @ B5 ) )
& ( ~ ( member_nat @ X3 @ B5 )
=> ( ( ( member_nat @ X3 @ A5 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1233_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1234_is__singletonI_H,axiom,
! [A5: set_Pr8705490787052945405ctxt_a] :
( ( A5 != bot_bo8437124304389434473ctxt_a )
=> ( ! [X: produc8175205341597284509ctxt_a,Y3: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ X @ A5 )
=> ( ( member461722496487281478ctxt_a @ Y3 @ A5 )
=> ( X = Y3 ) ) )
=> ( is_sin4775672710296068497ctxt_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1235_is__singletonI_H,axiom,
! [A5: set_nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A5 )
=> ( ( member_nat @ Y3 @ A5 )
=> ( X = Y3 ) ) )
=> ( is_singleton_nat @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1236_inj__on__fun__updI,axiom,
! [F2: nat > rat,A5: set_nat,Y: rat,X3: nat] :
( ( inj_on_nat_rat @ F2 @ A5 )
=> ( ~ ( member_rat @ Y @ ( image_nat_rat @ F2 @ A5 ) )
=> ( inj_on_nat_rat @ ( fun_upd_nat_rat @ F2 @ X3 @ Y ) @ A5 ) ) ) ).
% inj_on_fun_updI
thf(fact_1237_inj__on__fun__updI,axiom,
! [F2: nat > nat,A5: set_nat,Y: nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ A5 )
=> ( ~ ( member_nat @ Y @ ( image_nat_nat @ F2 @ A5 ) )
=> ( inj_on_nat_nat @ ( fun_upd_nat_nat @ F2 @ X3 @ Y ) @ A5 ) ) ) ).
% inj_on_fun_updI
thf(fact_1238_is__singletonE,axiom,
! [A5: set_nat] :
( ( is_singleton_nat @ A5 )
=> ~ ! [X: nat] :
( A5
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1239_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X4: nat] :
( A6
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1240_fun__upd__image,axiom,
! [X3: produc8175205341597284509ctxt_a,A5: set_Pr8705490787052945405ctxt_a,F2: produc8175205341597284509ctxt_a > nat,Y: nat] :
( ( ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( image_5981101931084406824_a_nat @ ( fun_up339567317610056112_a_nat @ F2 @ X3 @ Y ) @ A5 )
= ( insert_nat @ Y @ ( image_5981101931084406824_a_nat @ F2 @ ( minus_3912115830687649252ctxt_a @ A5 @ ( insert6238942492121947757ctxt_a @ X3 @ bot_bo8437124304389434473ctxt_a ) ) ) ) ) )
& ( ~ ( member461722496487281478ctxt_a @ X3 @ A5 )
=> ( ( image_5981101931084406824_a_nat @ ( fun_up339567317610056112_a_nat @ F2 @ X3 @ Y ) @ A5 )
= ( image_5981101931084406824_a_nat @ F2 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1241_fun__upd__image,axiom,
! [X3: nat,A5: set_nat,F2: nat > rat,Y: rat] :
( ( ( member_nat @ X3 @ A5 )
=> ( ( image_nat_rat @ ( fun_upd_nat_rat @ F2 @ X3 @ Y ) @ A5 )
= ( insert_rat @ Y @ ( image_nat_rat @ F2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ( image_nat_rat @ ( fun_upd_nat_rat @ F2 @ X3 @ Y ) @ A5 )
= ( image_nat_rat @ F2 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1242_fun__upd__image,axiom,
! [X3: nat,A5: set_nat,F2: nat > nat,Y: nat] :
( ( ( member_nat @ X3 @ A5 )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F2 @ X3 @ Y ) @ A5 )
= ( insert_nat @ Y @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X3 @ A5 )
=> ( ( image_nat_nat @ ( fun_upd_nat_nat @ F2 @ X3 @ Y ) @ A5 )
= ( image_nat_nat @ F2 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1243_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1244_Field__insert,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_1245_Field__insert,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
( ( field_5946474487916882833ctxt_a @ ( insert6238942492121947757ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R ) )
= ( sup_su721339912568147456ctxt_a @ ( insert2292343091172363342ctxt_a @ A @ ( insert2292343091172363342ctxt_a @ B @ bot_bo3757104637556896616ctxt_a ) ) @ ( field_5946474487916882833ctxt_a @ R ) ) ) ).
% Field_insert
thf(fact_1246_the__elem__eq,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% the_elem_eq
thf(fact_1247_FieldI1,axiom,
! [I: produc8175205341597284509ctxt_a,J2: produc8175205341597284509ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ I @ J2 ) @ R2 )
=> ( member461722496487281478ctxt_a @ I @ ( field_8422804572985872746ctxt_a @ R2 ) ) ) ).
% FieldI1
thf(fact_1248_FieldI1,axiom,
! [I: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 )
=> ( member_nat @ I @ ( field_nat @ R2 ) ) ) ).
% FieldI1
thf(fact_1249_FieldI1,axiom,
! [I: ground_gmctxt_a,J2: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ I @ J2 ) @ R2 )
=> ( member3306386364193653941ctxt_a @ I @ ( field_5946474487916882833ctxt_a @ R2 ) ) ) ).
% FieldI1
thf(fact_1250_FieldI2,axiom,
! [I: produc8175205341597284509ctxt_a,J2: produc8175205341597284509ctxt_a,R2: set_Pr490164551564945991ctxt_a] :
( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ I @ J2 ) @ R2 )
=> ( member461722496487281478ctxt_a @ J2 @ ( field_8422804572985872746ctxt_a @ R2 ) ) ) ).
% FieldI2
thf(fact_1251_FieldI2,axiom,
! [I: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J2 ) @ R2 )
=> ( member_nat @ J2 @ ( field_nat @ R2 ) ) ) ).
% FieldI2
thf(fact_1252_FieldI2,axiom,
! [I: ground_gmctxt_a,J2: ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ I @ J2 ) @ R2 )
=> ( member3306386364193653941ctxt_a @ J2 @ ( field_5946474487916882833ctxt_a @ R2 ) ) ) ).
% FieldI2
thf(fact_1253_finite__Field,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R )
=> ( finite_finite_nat @ ( field_nat @ R ) ) ) ).
% finite_Field
thf(fact_1254_the__elem__image__unique,axiom,
! [A5: set_nat,F2: nat > rat,X3: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( ( F2 @ Y3 )
= ( F2 @ X3 ) ) )
=> ( ( the_elem_rat @ ( image_nat_rat @ F2 @ A5 ) )
= ( F2 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1255_the__elem__image__unique,axiom,
! [A5: set_nat,F2: nat > nat,X3: nat] :
( ( A5 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( ( F2 @ Y3 )
= ( F2 @ X3 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F2 @ A5 ) )
= ( F2 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1256_wf__eq__minimal2,axiom,
( wf_Ground_gmctxt_a
= ( ^ [R3: set_Pr8705490787052945405ctxt_a] :
! [A6: set_Ground_gmctxt_a] :
( ( ( ord_le4032861497522155828ctxt_a @ A6 @ ( field_5946474487916882833ctxt_a @ R3 ) )
& ( A6 != bot_bo3757104637556896616ctxt_a ) )
=> ? [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A6 )
& ! [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A6 )
=> ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1257_cofinal__def,axiom,
( bNF_Ca3356332997443478541ctxt_a
= ( ^ [A6: set_Ground_gmctxt_a,R3: set_Pr8705490787052945405ctxt_a] :
! [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ ( field_5946474487916882833ctxt_a @ R3 ) )
=> ? [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A6 )
& ( X4 != Y4 )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R3 ) ) ) ) ) ).
% cofinal_def
thf(fact_1258_linear__order__on__singleton,axiom,
! [X3: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1259_linear__order__on__singleton,axiom,
! [X3: ground_gmctxt_a] : ( order_988128385783186345ctxt_a @ ( insert2292343091172363342ctxt_a @ X3 @ bot_bo3757104637556896616ctxt_a ) @ ( insert6238942492121947757ctxt_a @ ( produc6347872285926497493ctxt_a @ X3 @ X3 ) @ bot_bo8437124304389434473ctxt_a ) ) ).
% linear_order_on_singleton
thf(fact_1260_Total__subset__Id,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ( total_2093374903485932954ctxt_a @ ( field_5946474487916882833ctxt_a @ R ) @ R )
=> ( ( ord_le2690539916662146973ctxt_a @ R @ id_Ground_gmctxt_a )
=> ( ( R = bot_bo8437124304389434473ctxt_a )
| ? [A3: ground_gmctxt_a] :
( R
= ( insert6238942492121947757ctxt_a @ ( produc6347872285926497493ctxt_a @ A3 @ A3 ) @ bot_bo8437124304389434473ctxt_a ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1261_pair__in__Id__conv,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ id_Ground_gmctxt_a )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_1262_IdI,axiom,
! [A: ground_gmctxt_a] : ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ A ) @ id_Ground_gmctxt_a ) ).
% IdI
thf(fact_1263_Linear__order__in__diff__Id,axiom,
! [R: set_Pr490164551564945991ctxt_a,A: produc8175205341597284509ctxt_a,B: produc8175205341597284509ctxt_a] :
( ( order_3248224809147042642ctxt_a @ ( field_8422804572985872746ctxt_a @ R ) @ R )
=> ( ( member461722496487281478ctxt_a @ A @ ( field_8422804572985872746ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ B @ ( field_8422804572985872746ctxt_a @ R ) )
=> ( ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ A @ B ) @ R )
= ( ~ ( member4022424729989878928ctxt_a @ ( produc5881458463083805655ctxt_a @ B @ A ) @ ( minus_3029729196815224238ctxt_a @ R @ id_Pro2518137219852809209ctxt_a ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1264_Linear__order__in__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1265_Linear__order__in__diff__Id,axiom,
! [R: set_Pr8705490787052945405ctxt_a,A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( order_988128385783186345ctxt_a @ ( field_5946474487916882833ctxt_a @ R ) @ R )
=> ( ( member3306386364193653941ctxt_a @ A @ ( field_5946474487916882833ctxt_a @ R ) )
=> ( ( member3306386364193653941ctxt_a @ B @ ( field_5946474487916882833ctxt_a @ R ) )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ R )
= ( ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ B @ A ) @ ( minus_3912115830687649252ctxt_a @ R @ id_Ground_gmctxt_a ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1266_IdD,axiom,
! [A: ground_gmctxt_a,B: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ id_Ground_gmctxt_a )
=> ( A = B ) ) ).
% IdD
thf(fact_1267_IdE,axiom,
! [P: produc8175205341597284509ctxt_a] :
( ( member461722496487281478ctxt_a @ P @ id_Ground_gmctxt_a )
=> ~ ! [X: ground_gmctxt_a] :
( P
!= ( produc6347872285926497493ctxt_a @ X @ X ) ) ) ).
% IdE
thf(fact_1268_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ( order_988128385783186345ctxt_a @ ( field_5946474487916882833ctxt_a @ R ) @ R )
=> ( ( wf_Ground_gmctxt_a @ ( minus_3912115830687649252ctxt_a @ R @ id_Ground_gmctxt_a ) )
= ( ! [A6: set_Ground_gmctxt_a] :
( ( ord_le4032861497522155828ctxt_a @ A6 @ ( field_5946474487916882833ctxt_a @ R ) )
=> ( ( A6 != bot_bo3757104637556896616ctxt_a )
=> ? [X4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ X4 @ A6 )
& ! [Y4: ground_gmctxt_a] :
( ( member3306386364193653941ctxt_a @ Y4 @ A6 )
=> ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X4 @ Y4 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1269_irrefl__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( irrefl_on_nat @ top_top_set_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat ) ) ).
% irrefl_diff_Id
thf(fact_1270_irrefl__diff__Id,axiom,
! [R: set_Pr8928021450653196913at_rat] : ( irrefl_on_rat @ top_top_set_rat @ ( minus_9022085185153369176at_rat @ R @ id_rat ) ) ).
% irrefl_diff_Id
thf(fact_1271_strongly__confluentI,axiom,
! [R: set_Pr8705490787052945405ctxt_a] :
( ! [X: ground_gmctxt_a,Y3: ground_gmctxt_a,Z3: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ R )
=> ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Z3 ) @ R )
=> ? [U4: ground_gmctxt_a] :
( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Y3 @ U4 ) @ ( transi2468631686036622171ctxt_a @ R ) )
& ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ Z3 @ U4 ) @ ( sup_su1394892571904902097ctxt_a @ R @ id_Ground_gmctxt_a ) ) ) ) )
=> ( abstra7652130819992315389ctxt_a @ R @ top_to3933008015346724740ctxt_a ) ) ).
% strongly_confluentI
thf(fact_1272_strongly__confluentI,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R )
=> ? [U4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ U4 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ U4 ) @ ( sup_su6327502436637775413at_nat @ R @ id_nat ) ) ) ) )
=> ( abstra1637537658730489263on_nat @ R @ top_top_set_nat ) ) ).
% strongly_confluentI
thf(fact_1273_strongly__confluentI,axiom,
! [R: set_Pr8928021450653196913at_rat] :
( ! [X: rat,Y3: rat,Z3: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X @ Y3 ) @ R )
=> ( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ X @ Z3 ) @ R )
=> ? [U4: rat] :
( ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Y3 @ U4 ) @ ( transi2270211269848806677cl_rat @ R ) )
& ( member4615739369455165370at_rat @ ( product_Pair_rat_rat @ Z3 @ U4 ) @ ( sup_su4770203945505871429at_rat @ R @ id_rat ) ) ) ) )
=> ( abstra1002407598643993527on_rat @ R @ top_top_set_rat ) ) ).
% strongly_confluentI
% Conjectures (1)
thf(conj_0,conjecture,
member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ d @ c ) @ ground_comp_gmctxt_a ).
%------------------------------------------------------------------------------