TPTP Problem File: SLH0015^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc3/0010_Prover/prob_00024_001083__11960872_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1526 ( 394 unt; 255 typ; 0 def)
% Number of atoms : 3933 (1325 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12070 ( 229 ~; 43 |; 169 &;9660 @)
% ( 0 <=>;1969 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 2005 (2005 >; 0 *; 0 +; 0 <<)
% Number of symbols : 234 ( 233 usr; 13 con; 0-5 aty)
% Number of variables : 3951 ( 173 ^;3617 !; 161 ?;3951 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:38.259
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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% Explicit typings (233)
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thf(sy_c_Fun_Omonotone__on_001t__Syntax__Orule_001t__Rat__Orat,type,
monotone_on_rule_rat: set_rule > ( rule > rule > $o ) > ( rat > rat > $o ) > ( rule > 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__Rat__Orat,type,
the_inv_into_nat_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Syntax__Orule,type,
the_in5544616208814386890t_rule: set_nat > ( nat > rule ) > rule > nat ).
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_Fun_Othe__inv__into_001t__Rat__Orat_001t__Rat__Orat,type,
the_inv_into_rat_rat: set_rat > ( rat > rat ) > rat > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Syntax__Orule,type,
the_in2217467737105579218t_rule: set_rat > ( rat > rule ) > rule > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Nat__Onat,type,
the_in8781880623634246602le_nat: set_rule > ( rule > nat ) > nat > rule ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Rat__Orat,type,
the_in8146750563547750866le_rat: set_rule > ( rule > rat ) > rat > rule ).
thf(sy_c_Fun_Othe__inv__into_001t__Syntax__Orule_001t__Syntax__Orule,type,
the_in80044576880915775e_rule: set_rule > ( rule > rule ) > rule > rule ).
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__Rat__Orat_J,type,
uminus2201863774496077783et_rat: set_rat > set_rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Orule_J,type,
uminus4869265918275750596t_rule: set_rule > set_rule ).
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_Obijection_001t__Syntax__Orule,type,
hilber6733072011887318294n_rule: ( rule > rule ) > $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__Rat__Orat,type,
hilber2998747136712319222at_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Syntax__Orule,type,
hilber8541579349336805475t_rule: set_nat > ( nat > rule ) > rule > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Nat__Onat,type,
hilber3317322552863949046at_nat: set_rat > ( rat > nat ) > nat > 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_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Syntax__Orule,type,
hilber5214430877627997803t_rule: set_rat > ( rat > rule ) > rule > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Nat__Onat,type,
hilber2555471727301889379le_nat: set_rule > ( rule > nat ) > nat > rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Rat__Orat,type,
hilber1920341667215393643le_rat: set_rule > ( rule > rat ) > rat > rule ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Syntax__Orule_001t__Syntax__Orule,type,
hilber2978553400015838680e_rule: set_rule > ( rule > rule ) > rule > rule ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_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,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Syntax__Orule_J,type,
ord_less_set_rule: set_rule > set_rule > $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,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
ord_le7968974978423766289t_rule: set_set_rule > set_set_rule > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Orule_J,type,
ord_less_eq_set_rule: set_rule > set_rule > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Rat__Orat,type,
order_Greatest_rat: ( rat > $o ) > rat ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Rat__Orat_J,type,
ordering_top_set_rat: ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > set_rat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Syntax__Orule_J,type,
orderi2038897200410189450t_rule: ( set_rule > set_rule > $o ) > ( set_rule > set_rule > $o ) > set_rule > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Rat__Orat_M_Eo_J,type,
top_top_rat_o: rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Syntax__Orule_M_Eo_J,type,
top_top_rule_o: rule > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Rat__Orat_J,type,
top_top_set_rat: set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
top_top_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Syntax__Orule_J_J,type,
top_top_set_set_rule: set_set_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_It__Nat__Onat_J_J,type,
top_to7548458143485696966am_nat: set_stream_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_It__Rat__Orat_J_J,type,
top_to1168471806008019406am_rat: set_stream_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Stream__Ostream_It__Syntax__Orule_J_J,type,
top_to3705917391389534779m_rule: set_stream_rule ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Syntax__Orule_J,type,
top_top_set_rule: set_rule ).
thf(sy_c_Prover_Orules,type,
rules: stream_rule ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Rat__Orat,type,
field_6020823756834552118ts_rat: set_rat ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Rat__Orat,type,
field_2639924705303425560at_rat: rat > rat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Syntax__Orule,type,
collect_rule: ( rule > $o ) > set_rule ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Rat__Orat,type,
pow_rat: set_rat > set_set_rat ).
thf(sy_c_Set_OPow_001t__Syntax__Orule,type,
pow_rule: set_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Rat__Orat_M_Eo_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_o_set_rat: ( ( rat > $o ) > set_rat ) > set_rat_o > set_set_rat ).
thf(sy_c_Set_Oimage_001_062_It__Syntax__Orule_M_Eo_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_1281159361656534528t_rule: ( ( rule > $o ) > set_rule ) > set_rule_o > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Rat__Orat,type,
image_nat_rat: ( nat > rat ) > set_nat > set_rat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Syntax__Orule,type,
image_nat_rule: ( nat > rule ) > set_nat > set_rule ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Nat__Onat,type,
image_rat_nat: ( rat > nat ) > set_rat > set_nat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Rat__Orat,type,
image_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Syntax__Orule,type,
image_rat_rule: ( rat > rule ) > set_rat > set_rule ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_4408659257933336347et_rat: ( set_nat > set_rat ) > set_set_nat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_458447791132712456t_rule: ( set_nat > set_rule ) > set_set_nat > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7912102293542740589et_nat: ( stream_nat > set_nat ) > set_stream_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_3934614161387701885et_rat: ( stream_rat > set_rat ) > set_stream_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Stream__Ostream_It__Syntax__Orule_J_001t__Set__Oset_It__Syntax__Orule_J,type,
image_6459725099818367575t_rule: ( stream_rule > set_rule ) > set_stream_rule > set_set_rule ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Nat__Onat,type,
image_rule_nat: ( rule > nat ) > set_rule > set_nat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Rat__Orat,type,
image_rule_rat: ( rule > rat ) > set_rule > set_rat ).
thf(sy_c_Set_Oimage_001t__Syntax__Orule_001t__Syntax__Orule,type,
image_rule_rule: ( rule > rule ) > set_rule > set_rule ).
thf(sy_c_Stream_Osmember_001t__Nat__Onat,type,
smember_nat: nat > stream_nat > $o ).
thf(sy_c_Stream_Osmember_001t__Rat__Orat,type,
smember_rat: rat > stream_rat > $o ).
thf(sy_c_Stream_Osmember_001t__Syntax__Orule,type,
smember_rule: rule > stream_rule > $o ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001t__Rat__Orat,type,
smerge_rat: stream_stream_rat > stream_rat ).
thf(sy_c_Stream_Osmerge_001t__Syntax__Orule,type,
smerge_rule: stream_stream_rule > stream_rule ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Rat__Orat,type,
snth_rat: stream_rat > nat > rat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Rat__Orat_J,type,
snth_stream_rat: stream_stream_rat > nat > stream_rat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Syntax__Orule_J,type,
snth_stream_rule: stream_stream_rule > nat > stream_rule ).
thf(sy_c_Stream_Osnth_001t__Syntax__Orule,type,
snth_rule: stream_rule > nat > rule ).
thf(sy_c_Stream_Ostream_Ocorec__stream_001t__Nat__Onat_001t__Nat__Onat,type,
corec_stream_nat_nat: ( nat > nat ) > ( nat > $o ) > ( nat > stream_nat ) > ( nat > nat ) > nat > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Nat__Onat_001t__Nat__Onat,type,
smap_nat_nat: ( nat > nat ) > stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Nat__Onat_001t__Rat__Orat,type,
smap_nat_rat: ( nat > rat ) > stream_nat > stream_rat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Nat__Onat_001t__Syntax__Orule,type,
smap_nat_rule: ( nat > rule ) > stream_nat > stream_rule ).
thf(sy_c_Stream_Ostream_Osmap_001t__Rat__Orat_001t__Nat__Onat,type,
smap_rat_nat: ( rat > nat ) > stream_rat > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Rat__Orat_001t__Rat__Orat,type,
smap_rat_rat: ( rat > rat ) > stream_rat > stream_rat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Rat__Orat_001t__Syntax__Orule,type,
smap_rat_rule: ( rat > rule ) > stream_rat > stream_rule ).
thf(sy_c_Stream_Ostream_Osmap_001t__Syntax__Orule_001t__Nat__Onat,type,
smap_rule_nat: ( rule > nat ) > stream_rule > stream_nat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Syntax__Orule_001t__Rat__Orat,type,
smap_rule_rat: ( rule > rat ) > stream_rule > stream_rat ).
thf(sy_c_Stream_Ostream_Osmap_001t__Syntax__Orule_001t__Syntax__Orule,type,
smap_rule_rule: ( rule > rule ) > stream_rule > stream_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Rat__Orat,type,
sset_rat: stream_rat > set_rat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Nat__Onat_J,type,
sset_stream_nat: stream_stream_nat > set_stream_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Rat__Orat_J,type,
sset_stream_rat: stream_stream_rat > set_stream_rat ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__Syntax__Orule_J,type,
sset_stream_rule: stream_stream_rule > set_stream_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Syntax__Orule,type,
sset_rule: stream_rule > set_rule ).
thf(sy_c_Stream_Ostreams_001t__Nat__Onat,type,
streams_nat: set_nat > set_stream_nat ).
thf(sy_c_Stream_Ostreams_001t__Rat__Orat,type,
streams_rat: set_rat > set_stream_rat ).
thf(sy_c_Stream_Ostreams_001t__Syntax__Orule,type,
streams_rule: set_rule > set_stream_rule ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Syntax__Orule_J,type,
member_set_rule: set_rule > set_set_rule > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Nat__Onat_J,type,
member_stream_nat: stream_nat > set_stream_nat > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Rat__Orat_J,type,
member_stream_rat: stream_rat > set_stream_rat > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Syntax__Orule_J,type,
member_stream_rule: stream_rule > set_stream_rule > $o ).
thf(sy_c_member_001t__Syntax__Orule,type,
member_rule: rule > set_rule > $o ).
% Relevant facts (1270)
thf(fact_0_rules__def,axiom,
( rules
= ( fair_f4564919574533178778m_rule @ rule_of_nat ) ) ).
% rules_def
thf(fact_1_UNIV__I,axiom,
! [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).
% UNIV_I
thf(fact_2_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_3_UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_I
thf(fact_4_iso__tuple__UNIV__I,axiom,
! [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).
% iso_tuple_UNIV_I
thf(fact_5_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_6_iso__tuple__UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_7_UNIV__eq__I,axiom,
! [A: set_rule] :
( ! [X2: rule] : ( member_rule @ X2 @ A )
=> ( top_top_set_rule = A ) ) ).
% UNIV_eq_I
thf(fact_8_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_9_UNIV__eq__I,axiom,
! [A: set_rat] :
( ! [X2: rat] : ( member_rat @ X2 @ A )
=> ( top_top_set_rat = A ) ) ).
% UNIV_eq_I
thf(fact_10_UNIV__witness,axiom,
? [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).
% UNIV_witness
thf(fact_11_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_12_UNIV__witness,axiom,
? [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_13_UNIV__stream,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( sset_rule @ ( fair_f4564919574533178778m_rule @ F ) )
= top_top_set_rule ) ) ).
% UNIV_stream
thf(fact_14_UNIV__stream,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( sset_nat @ ( fair_fair_stream_nat @ F ) )
= top_top_set_nat ) ) ).
% UNIV_stream
thf(fact_15_UNIV__stream,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( sset_rat @ ( fair_fair_stream_rat @ F ) )
= top_top_set_rat ) ) ).
% UNIV_stream
thf(fact_16_surj__rule__of__nat,axiom,
( ( image_nat_rule @ rule_of_nat @ top_top_set_nat )
= top_top_set_rule ) ).
% surj_rule_of_nat
thf(fact_17_rule__nat,axiom,
! [R: rule] :
( ( rule_of_nat @ ( nat_of_rule @ R ) )
= R ) ).
% rule_nat
thf(fact_18_Stream_Osmember__def,axiom,
( smember_rat
= ( ^ [X3: rat,S: stream_rat] : ( member_rat @ X3 @ ( sset_rat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_19_Stream_Osmember__def,axiom,
( smember_nat
= ( ^ [X3: nat,S: stream_nat] : ( member_nat @ X3 @ ( sset_nat @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_20_Stream_Osmember__def,axiom,
( smember_rule
= ( ^ [X3: rule,S: stream_rule] : ( member_rule @ X3 @ ( sset_rule @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_21_Sup__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Sup_UNIV
thf(fact_22_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_23_Sup__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Sup_UNIV
thf(fact_24_fair__stream,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( fair_fair_rule @ ( fair_f4564919574533178778m_rule @ F ) ) ) ).
% fair_stream
thf(fact_25_fair__stream,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( fair_fair_nat @ ( fair_fair_stream_nat @ F ) ) ) ).
% fair_stream
thf(fact_26_fair__stream,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( fair_fair_rat @ ( fair_fair_stream_rat @ F ) ) ) ).
% fair_stream
thf(fact_27_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_28_image__eqI,axiom,
! [B: rat,F: nat > rat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_29_image__eqI,axiom,
! [B: rule,F: nat > rule,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_rule @ B @ ( image_nat_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_30_image__eqI,axiom,
! [B: nat,F: rat > nat,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_nat @ B @ ( image_rat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_31_image__eqI,axiom,
! [B: rat,F: rat > rat,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_32_image__eqI,axiom,
! [B: rule,F: rat > rule,X: rat,A: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A )
=> ( member_rule @ B @ ( image_rat_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_33_image__eqI,axiom,
! [B: nat,F: rule > nat,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_nat @ B @ ( image_rule_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_34_image__eqI,axiom,
! [B: rat,F: rule > rat,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_rat @ B @ ( image_rule_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_35_image__eqI,axiom,
! [B: rule,F: rule > rule,X: rule,A: set_rule] :
( ( B
= ( F @ X ) )
=> ( ( member_rule @ X @ A )
=> ( member_rule @ B @ ( image_rule_rule @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_36_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rule,D: nat > rule,Inf: set_rule > rule] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rule @ C @ A ) )
= ( Inf @ ( image_nat_rule @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_37_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rat @ C @ A ) )
= ( Inf @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_38_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_39_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rule,D: nat > rule,Sup: set_rule > rule] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rule @ C @ A ) )
= ( Sup @ ( image_nat_rule @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_40_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rat @ C @ A ) )
= ( Sup @ ( image_nat_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_41_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_42_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_43_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_44_imageI,axiom,
! [X: nat,A: set_nat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_45_imageI,axiom,
! [X: nat,A: set_nat,F: nat > rule] :
( ( member_nat @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_nat_rule @ F @ A ) ) ) ).
% imageI
thf(fact_46_imageI,axiom,
! [X: rat,A: set_rat,F: rat > nat] :
( ( member_rat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_rat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_47_imageI,axiom,
! [X: rat,A: set_rat,F: rat > rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_rat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_48_imageI,axiom,
! [X: rat,A: set_rat,F: rat > rule] :
( ( member_rat @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_rat_rule @ F @ A ) ) ) ).
% imageI
thf(fact_49_imageI,axiom,
! [X: rule,A: set_rule,F: rule > nat] :
( ( member_rule @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_rule_nat @ F @ A ) ) ) ).
% imageI
thf(fact_50_imageI,axiom,
! [X: rule,A: set_rule,F: rule > rat] :
( ( member_rule @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_rule_rat @ F @ A ) ) ) ).
% imageI
thf(fact_51_imageI,axiom,
! [X: rule,A: set_rule,F: rule > rule] :
( ( member_rule @ X @ A )
=> ( member_rule @ ( F @ X ) @ ( image_rule_rule @ F @ A ) ) ) ).
% imageI
thf(fact_52_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_53_image__iff,axiom,
! [Z: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ Z @ ( image_nat_rat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_54_image__iff,axiom,
! [Z: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ Z @ ( image_nat_rule @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_55_bex__imageD,axiom,
! [F: nat > rule,A: set_nat,P: rule > $o] :
( ? [X4: rule] :
( ( member_rule @ X4 @ ( image_nat_rule @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_56_bex__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ? [X4: rat] :
( ( member_rat @ X4 @ ( image_nat_rat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_57_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_58_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > rule,G: nat > rule] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rule @ F @ M )
= ( image_nat_rule @ G @ N ) ) ) ) ).
% image_cong
thf(fact_59_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > rat,G: nat > rat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rat @ F @ M )
= ( image_nat_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_60_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_61_ball__imageD,axiom,
! [F: nat > rule,A: set_nat,P: rule > $o] :
( ! [X2: rule] :
( ( member_rule @ X2 @ ( image_nat_rule @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_62_ball__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ! [X2: rat] :
( ( member_rat @ X2 @ ( image_nat_rat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_63_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_64_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_65_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: rat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_66_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: rule,F: nat > rule] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_nat_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_67_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: nat,F: rat > nat] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_68_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: rat,F: rat > rat] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_69_rev__image__eqI,axiom,
! [X: rat,A: set_rat,B: rule,F: rat > rule] :
( ( member_rat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rat_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_70_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: nat,F: rule > nat] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rule_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_71_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: rat,F: rule > rat] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rule_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_72_rev__image__eqI,axiom,
! [X: rule,A: set_rule,B: rule,F: rule > rule] :
( ( member_rule @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rule_rule @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_73_top__set__def,axiom,
( top_top_set_rule
= ( collect_rule @ top_top_rule_o ) ) ).
% top_set_def
thf(fact_74_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_75_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_76_range__eqI,axiom,
! [B: nat,F: rule > nat,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rule_nat @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_77_range__eqI,axiom,
! [B: rat,F: rule > rat,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rule_rat @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_78_range__eqI,axiom,
! [B: rule,F: rule > rule,X: rule] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rule_rule @ F @ top_top_set_rule ) ) ) ).
% range_eqI
thf(fact_79_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_80_range__eqI,axiom,
! [B: rat,F: nat > rat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_81_range__eqI,axiom,
! [B: rule,F: nat > rule,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_nat_rule @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_82_range__eqI,axiom,
! [B: nat,F: rat > nat,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_83_range__eqI,axiom,
! [B: rat,F: rat > rat,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rat_rat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_84_range__eqI,axiom,
! [B: rule,F: rat > rule,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_rule @ B @ ( image_rat_rule @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_85_rangeI,axiom,
! [F: rule > nat,X: rule] : ( member_nat @ ( F @ X ) @ ( image_rule_nat @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_86_rangeI,axiom,
! [F: rule > rat,X: rule] : ( member_rat @ ( F @ X ) @ ( image_rule_rat @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_87_rangeI,axiom,
! [F: rule > rule,X: rule] : ( member_rule @ ( F @ X ) @ ( image_rule_rule @ F @ top_top_set_rule ) ) ).
% rangeI
thf(fact_88_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_89_rangeI,axiom,
! [F: nat > rat,X: nat] : ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_90_rangeI,axiom,
! [F: nat > rule,X: nat] : ( member_rule @ ( F @ X ) @ ( image_nat_rule @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_91_rangeI,axiom,
! [F: rat > nat,X: rat] : ( member_nat @ ( F @ X ) @ ( image_rat_nat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_92_rangeI,axiom,
! [F: rat > rat,X: rat] : ( member_rat @ ( F @ X ) @ ( image_rat_rat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_93_rangeI,axiom,
! [F: rat > rule,X: rat] : ( member_rule @ ( F @ X ) @ ( image_rat_rule @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_94_Union__UNIV,axiom,
( ( comple2146307154184993742t_rule @ top_top_set_set_rule )
= top_top_set_rule ) ).
% Union_UNIV
thf(fact_95_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_96_Union__UNIV,axiom,
( ( comple3890839924845867745et_rat @ top_top_set_set_rat )
= top_top_set_rat ) ).
% Union_UNIV
thf(fact_97_surj__def,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ! [Y: rule] :
? [X3: rule] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_98_surj__def,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X3: rule] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_99_surj__def,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X3: rule] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_100_surj__def,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
= ( ! [Y: rule] :
? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_101_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_102_surj__def,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_103_surj__def,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
= ( ! [Y: rule] :
? [X3: rat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_104_surj__def,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X3: rat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_105_surj__def,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X3: rat] :
( Y
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_106_surjI,axiom,
! [G: rule > rule,F: rule > rule] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_rule @ G @ top_top_set_rule )
= top_top_set_rule ) ) ).
% surjI
thf(fact_107_surjI,axiom,
! [G: rule > nat,F: nat > rule] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_nat @ G @ top_top_set_rule )
= top_top_set_nat ) ) ).
% surjI
thf(fact_108_surjI,axiom,
! [G: rule > rat,F: rat > rule] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rule_rat @ G @ top_top_set_rule )
= top_top_set_rat ) ) ).
% surjI
thf(fact_109_surjI,axiom,
! [G: nat > rule,F: rule > nat] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_rule @ G @ top_top_set_nat )
= top_top_set_rule ) ) ).
% surjI
thf(fact_110_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_111_surjI,axiom,
! [G: nat > rat,F: rat > nat] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_112_surjI,axiom,
! [G: rat > rule,F: rule > rat] :
( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_rule @ G @ top_top_set_rat )
= top_top_set_rule ) ) ).
% surjI
thf(fact_113_surjI,axiom,
! [G: rat > nat,F: nat > rat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_114_surjI,axiom,
! [G: rat > rat,F: rat > rat] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_115_surjE,axiom,
! [F: rule > rule,Y2: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ~ ! [X2: rule] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_116_surjE,axiom,
! [F: rule > nat,Y2: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ~ ! [X2: rule] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_117_surjE,axiom,
! [F: rule > rat,Y2: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ~ ! [X2: rule] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_118_surjE,axiom,
! [F: nat > rule,Y2: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_119_surjE,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_120_surjE,axiom,
! [F: nat > rat,Y2: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_121_surjE,axiom,
! [F: rat > rule,Y2: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ~ ! [X2: rat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_122_surjE,axiom,
! [F: rat > nat,Y2: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X2: rat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_123_surjE,axiom,
! [F: rat > rat,Y2: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X2: rat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_124_surjD,axiom,
! [F: rule > rule,Y2: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ? [X2: rule] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_125_surjD,axiom,
! [F: rule > nat,Y2: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ? [X2: rule] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_126_surjD,axiom,
! [F: rule > rat,Y2: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ? [X2: rule] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_127_surjD,axiom,
! [F: nat > rule,Y2: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_128_surjD,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_129_surjD,axiom,
! [F: nat > rat,Y2: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_130_surjD,axiom,
! [F: rat > rule,Y2: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ? [X2: rat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_131_surjD,axiom,
! [F: rat > nat,Y2: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ? [X2: rat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_132_surjD,axiom,
! [F: rat > rat,Y2: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ? [X2: rat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_133_surj__from__nat,axiom,
( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_from_nat
thf(fact_134_surj__from__nat,axiom,
( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_from_nat
thf(fact_135_sset__range,axiom,
( sset_rule
= ( ^ [S: stream_rule] : ( image_nat_rule @ ( snth_rule @ S ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_136_sset__range,axiom,
( sset_rat
= ( ^ [S: stream_rat] : ( image_nat_rat @ ( snth_rat @ S ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_137_sset__range,axiom,
( sset_nat
= ( ^ [S: stream_nat] : ( image_nat_nat @ ( snth_nat @ S ) @ top_top_set_nat ) ) ) ).
% sset_range
thf(fact_138_fair__surj,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( fair_fair_rule @ ( smap_nat_rule @ F @ fair_fair_nats ) ) ) ).
% fair_surj
thf(fact_139_fair__surj,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( fair_fair_nat @ ( smap_nat_nat @ F @ fair_fair_nats ) ) ) ).
% fair_surj
thf(fact_140_fair__surj,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( fair_fair_rat @ ( smap_nat_rat @ F @ fair_fair_nats ) ) ) ).
% fair_surj
thf(fact_141_rat__denum,axiom,
? [F2: nat > rat] :
( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_142_top__empty__eq,axiom,
( top_top_rule_o
= ( ^ [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ) ) ).
% top_empty_eq
thf(fact_143_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_144_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_145_SUP__id__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_id_eq
thf(fact_146_UnionI,axiom,
! [X5: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_147_UnionI,axiom,
! [X5: set_rat,C: set_set_rat,A: rat] :
( ( member_set_rat @ X5 @ C )
=> ( ( member_rat @ A @ X5 )
=> ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) ) ) ) ).
% UnionI
thf(fact_148_UnionI,axiom,
! [X5: set_rule,C: set_set_rule,A: rule] :
( ( member_set_rule @ X5 @ C )
=> ( ( member_rule @ A @ X5 )
=> ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) ) ) ) ).
% UnionI
thf(fact_149_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A2: rule,P: rule > $o] :
( ( member_rule @ A2 @ ( collect_rule @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_152_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X3: rat] : ( member_rat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A: set_rule] :
( ( collect_rule
@ ^ [X3: rule] : ( member_rule @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_155_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_156_Union__iff,axiom,
! [A: rat,C: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) )
= ( ? [X3: set_rat] :
( ( member_set_rat @ X3 @ C )
& ( member_rat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_157_Union__iff,axiom,
! [A: rule,C: set_set_rule] :
( ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) )
= ( ? [X3: set_rule] :
( ( member_set_rule @ X3 @ C )
& ( member_rule @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_158_image__id,axiom,
( ( image_nat_nat @ id_nat )
= id_set_nat ) ).
% image_id
thf(fact_159_snth__smap,axiom,
! [F: nat > nat,S2: stream_nat,N2: nat] :
( ( snth_nat @ ( smap_nat_nat @ F @ S2 ) @ N2 )
= ( F @ ( snth_nat @ S2 @ N2 ) ) ) ).
% snth_smap
thf(fact_160_lt__ex,axiom,
! [X: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% lt_ex
thf(fact_161_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_162_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_163_dense,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ? [Z2: rat] :
( ( ord_less_rat @ X @ Z2 )
& ( ord_less_rat @ Z2 @ Y2 ) ) ) ).
% dense
thf(fact_164_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_165_less__imp__neq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_166_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_167_order_Oasym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order.asym
thf(fact_168_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_169_ord__eq__less__trans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( A2 = B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_170_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_171_ord__less__eq__trans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_rat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_172_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_173_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_174_antisym__conv3,axiom,
! [Y2: rat,X: rat] :
( ~ ( ord_less_rat @ Y2 @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_175_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_176_linorder__cases,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_177_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_178_dual__order_Oasym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ~ ( ord_less_rat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_179_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_180_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_181_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_182_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_183_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: rat] : ( P @ A3 @ A3 )
=> ( ! [A3: rat,B3: rat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_184_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_185_order_Ostrict__trans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_186_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_187_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( ( ord_less_rat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_188_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_189_dual__order_Ostrict__trans,axiom,
! [B: rat,A2: rat,C2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_190_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_191_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_192_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_193_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_194_smap__alt,axiom,
! [F: nat > nat,S2: stream_nat,S3: stream_nat] :
( ( ( smap_nat_nat @ F @ S2 )
= S3 )
= ( ! [N3: nat] :
( ( F @ ( snth_nat @ S2 @ N3 ) )
= ( snth_nat @ S3 @ N3 ) ) ) ) ).
% smap_alt
thf(fact_195_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_196_Sup__SUP__eq,axiom,
( comple4580332206425622756_rat_o
= ( ^ [S4: set_rat_o,X3: rat] : ( member_rat @ X3 @ ( comple3890839924845867745et_rat @ ( image_rat_o_set_rat @ collect_rat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_197_Sup__SUP__eq,axiom,
( comple1826244231481717815rule_o
= ( ^ [S4: set_rule_o,X3: rule] : ( member_rule @ X3 @ ( comple2146307154184993742t_rule @ ( image_1281159361656534528t_rule @ collect_rule @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_198_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_199_linorder__neqE,axiom,
! [X: rat,Y2: rat] :
( ( X != Y2 )
=> ( ~ ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_200_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C ) ) ) ).
% UnionE
thf(fact_201_UnionE,axiom,
! [A: rat,C: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C ) )
=> ~ ! [X7: set_rat] :
( ( member_rat @ A @ X7 )
=> ~ ( member_set_rat @ X7 @ C ) ) ) ).
% UnionE
thf(fact_202_UnionE,axiom,
! [A: rule,C: set_set_rule] :
( ( member_rule @ A @ ( comple2146307154184993742t_rule @ C ) )
=> ~ ! [X7: set_rule] :
( ( member_rule @ A @ X7 )
=> ~ ( member_set_rule @ X7 @ C ) ) ) ).
% UnionE
thf(fact_203_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_204_order__less__asym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_205_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_206_linorder__neq__iff,axiom,
! [X: rat,Y2: rat] :
( ( X != Y2 )
= ( ( ord_less_rat @ X @ Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_207_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_208_order__less__asym_H,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_209_order__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_210_order__less__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_211_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_212_ord__eq__less__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_213_ord__eq__less__subst,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_214_ord__eq__less__subst,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_215_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_216_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_217_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_218_ord__less__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_219_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_220_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_221_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_222_order__less__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_223_order__less__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_224_order__less__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_225_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_226_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_227_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_229_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_230_order__less__not__sym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_231_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_232_order__less__imp__triv,axiom,
! [X: rat,Y2: rat,P: $o] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_233_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_234_linorder__less__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_235_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_236_order__less__imp__not__eq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_237_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_238_order__less__imp__not__eq2,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_239_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_240_order__less__imp__not__less,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_241_Sup_OSUP__id__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_nat @ id_nat @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_242_Inf_OINF__id__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_nat @ id_nat @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_243_top_Oextremum__strict,axiom,
! [A2: set_rule] :
~ ( ord_less_set_rule @ top_top_set_rule @ A2 ) ).
% top.extremum_strict
thf(fact_244_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_245_top_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A2 ) ).
% top.extremum_strict
thf(fact_246_top_Onot__eq__extremum,axiom,
! [A2: set_rule] :
( ( A2 != top_top_set_rule )
= ( ord_less_set_rule @ A2 @ top_top_set_rule ) ) ).
% top.not_eq_extremum
thf(fact_247_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_248_top_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != top_top_set_rat )
= ( ord_less_set_rat @ A2 @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_249_stream_Omap__ident__strong,axiom,
! [T: stream_rule,F: rule > rule] :
( ! [Z2: rule] :
( ( member_rule @ Z2 @ ( sset_rule @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( smap_rule_rule @ F @ T )
= T ) ) ).
% stream.map_ident_strong
thf(fact_250_stream_Omap__ident__strong,axiom,
! [T: stream_rat,F: rat > rat] :
( ! [Z2: rat] :
( ( member_rat @ Z2 @ ( sset_rat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( smap_rat_rat @ F @ T )
= T ) ) ).
% stream.map_ident_strong
thf(fact_251_stream_Omap__ident__strong,axiom,
! [T: stream_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( sset_nat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( smap_nat_nat @ F @ T )
= T ) ) ).
% stream.map_ident_strong
thf(fact_252_surj__id,axiom,
( ( image_rule_rule @ id_rule @ top_top_set_rule )
= top_top_set_rule ) ).
% surj_id
thf(fact_253_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_254_surj__id,axiom,
( ( image_rat_rat @ id_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_id
thf(fact_255_fair__stream__def,axiom,
( fair_f4564919574533178778m_rule
= ( ^ [F3: nat > rule] : ( smap_nat_rule @ F3 @ fair_fair_nats ) ) ) ).
% fair_stream_def
thf(fact_256_fair__stream__def,axiom,
( fair_fair_stream_rat
= ( ^ [F3: nat > rat] : ( smap_nat_rat @ F3 @ fair_fair_nats ) ) ) ).
% fair_stream_def
thf(fact_257_fair__stream__def,axiom,
( fair_fair_stream_nat
= ( ^ [F3: nat > nat] : ( smap_nat_nat @ F3 @ fair_fair_nats ) ) ) ).
% fair_stream_def
thf(fact_258_snth__sset,axiom,
! [S2: stream_rule,N2: nat] : ( member_rule @ ( snth_rule @ S2 @ N2 ) @ ( sset_rule @ S2 ) ) ).
% snth_sset
thf(fact_259_snth__sset,axiom,
! [S2: stream_rat,N2: nat] : ( member_rat @ ( snth_rat @ S2 @ N2 ) @ ( sset_rat @ S2 ) ) ).
% snth_sset
thf(fact_260_snth__sset,axiom,
! [S2: stream_nat,N2: nat] : ( member_nat @ ( snth_nat @ S2 @ N2 ) @ ( sset_nat @ S2 ) ) ).
% snth_sset
thf(fact_261_stream_Oset__map,axiom,
! [F: rule > rule,V: stream_rule] :
( ( sset_rule @ ( smap_rule_rule @ F @ V ) )
= ( image_rule_rule @ F @ ( sset_rule @ V ) ) ) ).
% stream.set_map
thf(fact_262_stream_Oset__map,axiom,
! [F: rat > rule,V: stream_rat] :
( ( sset_rule @ ( smap_rat_rule @ F @ V ) )
= ( image_rat_rule @ F @ ( sset_rat @ V ) ) ) ).
% stream.set_map
thf(fact_263_stream_Oset__map,axiom,
! [F: nat > rule,V: stream_nat] :
( ( sset_rule @ ( smap_nat_rule @ F @ V ) )
= ( image_nat_rule @ F @ ( sset_nat @ V ) ) ) ).
% stream.set_map
thf(fact_264_stream_Oset__map,axiom,
! [F: rule > rat,V: stream_rule] :
( ( sset_rat @ ( smap_rule_rat @ F @ V ) )
= ( image_rule_rat @ F @ ( sset_rule @ V ) ) ) ).
% stream.set_map
thf(fact_265_stream_Oset__map,axiom,
! [F: rat > rat,V: stream_rat] :
( ( sset_rat @ ( smap_rat_rat @ F @ V ) )
= ( image_rat_rat @ F @ ( sset_rat @ V ) ) ) ).
% stream.set_map
thf(fact_266_stream_Oset__map,axiom,
! [F: nat > rat,V: stream_nat] :
( ( sset_rat @ ( smap_nat_rat @ F @ V ) )
= ( image_nat_rat @ F @ ( sset_nat @ V ) ) ) ).
% stream.set_map
thf(fact_267_stream_Oset__map,axiom,
! [F: rule > nat,V: stream_rule] :
( ( sset_nat @ ( smap_rule_nat @ F @ V ) )
= ( image_rule_nat @ F @ ( sset_rule @ V ) ) ) ).
% stream.set_map
thf(fact_268_stream_Oset__map,axiom,
! [F: rat > nat,V: stream_rat] :
( ( sset_nat @ ( smap_rat_nat @ F @ V ) )
= ( image_rat_nat @ F @ ( sset_rat @ V ) ) ) ).
% stream.set_map
thf(fact_269_stream_Oset__map,axiom,
! [F: nat > nat,V: stream_nat] :
( ( sset_nat @ ( smap_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( sset_nat @ V ) ) ) ).
% stream.set_map
thf(fact_270_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_271_snth__sset__smerge,axiom,
! [Ss: stream_stream_rule,N2: nat,M3: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss @ N2 ) @ M3 ) @ ( sset_rule @ ( smerge_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_272_snth__sset__smerge,axiom,
! [Ss: stream_stream_rat,N2: nat,M3: nat] : ( member_rat @ ( snth_rat @ ( snth_stream_rat @ Ss @ N2 ) @ M3 ) @ ( sset_rat @ ( smerge_rat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_273_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N2: nat,M3: nat] : ( member_nat @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M3 ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_274_sset__smerge,axiom,
! [Ss: stream_stream_rule] :
( ( sset_rule @ ( smerge_rule @ Ss ) )
= ( comple2146307154184993742t_rule @ ( image_6459725099818367575t_rule @ sset_rule @ ( sset_stream_rule @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_275_sset__smerge,axiom,
! [Ss: stream_stream_rat] :
( ( sset_rat @ ( smerge_rat @ Ss ) )
= ( comple3890839924845867745et_rat @ ( image_3934614161387701885et_rat @ sset_rat @ ( sset_stream_rat @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_276_sset__smerge,axiom,
! [Ss: stream_stream_nat] :
( ( sset_nat @ ( smerge_nat @ Ss ) )
= ( comple7399068483239264473et_nat @ ( image_7912102293542740589et_nat @ sset_nat @ ( sset_stream_nat @ Ss ) ) ) ) ).
% sset_smerge
thf(fact_277_DEADID_Oin__rel,axiom,
( ( ^ [Y5: rule,Z3: rule] : ( Y5 = Z3 ) )
= ( ^ [A4: rule,B4: rule] :
? [Z4: rule] :
( ( member_rule @ Z4 @ top_top_set_rule )
& ( ( id_rule @ Z4 )
= A4 )
& ( ( id_rule @ Z4 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_278_DEADID_Oin__rel,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
? [Z4: nat] :
( ( member_nat @ Z4 @ top_top_set_nat )
& ( ( id_nat @ Z4 )
= A4 )
& ( ( id_nat @ Z4 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_279_DEADID_Oin__rel,axiom,
( ( ^ [Y5: rat,Z3: rat] : ( Y5 = Z3 ) )
= ( ^ [A4: rat,B4: rat] :
? [Z4: rat] :
( ( member_rat @ Z4 @ top_top_set_rat )
& ( ( id_rat @ Z4 )
= A4 )
& ( ( id_rat @ Z4 )
= B4 ) ) ) ) ).
% DEADID.in_rel
thf(fact_280_fair__def,axiom,
( fair_fair_rule
= ( ^ [S: stream_rule] :
! [X3: rule] :
( ( member_rule @ X3 @ ( sset_rule @ S ) )
=> ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( ( snth_rule @ S @ N3 )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_281_fair__def,axiom,
( fair_fair_rat
= ( ^ [S: stream_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ ( sset_rat @ S ) )
=> ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( ( snth_rat @ S @ N3 )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_282_fair__def,axiom,
( fair_fair_nat
= ( ^ [S: stream_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ ( sset_nat @ S ) )
=> ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( ( snth_nat @ S @ N3 )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_283_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_284_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_285_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_286_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_287_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_288_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_289_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_290_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_291_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_292_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_293_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_294_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_295_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_296_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_297_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_298_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_299_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_300_order__antisym__conv,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ( ( ord_less_eq_rat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_301_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_302_linorder__le__cases,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_303_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_305_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_306_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_307_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_308_ord__eq__le__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_309_ord__eq__le__subst,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_310_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_311_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_312_linorder__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
| ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_313_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_314_order__eq__refl,axiom,
! [X: rat,Y2: rat] :
( ( X = Y2 )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_315_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_316_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_317_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_318_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_319_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_320_order__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_321_order__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_322_order__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_323_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_324_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z3: rat] : ( Y5 = Z3 ) )
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_325_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_326_antisym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_327_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_328_dual__order_Otrans,axiom,
! [B: rat,A2: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_329_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_330_dual__order_Oantisym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_331_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_332_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: rat,Z3: rat] : ( Y5 = Z3 ) )
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_333_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_334_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: rat,B3: rat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_335_order__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_336_order__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_337_order_Otrans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_338_order_Otrans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_339_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_340_order__antisym,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_341_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_342_ord__le__eq__trans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_343_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_344_ord__eq__le__trans,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( A2 = B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_345_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_346_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: rat,Z3: rat] : ( Y5 = Z3 ) )
= ( ^ [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
& ( ord_less_eq_rat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_347_le__cases3,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_348_le__cases3,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_349_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_350_nle__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B ) )
= ( ( ord_less_eq_rat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_351_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_352_order__le__imp__less__or__eq,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_353_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_354_linorder__le__less__linear,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
| ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_355_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_356_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_357_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_358_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_359_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_360_order__less__le__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_361_order__less__le__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_362_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_363_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_364_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_365_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_366_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_367_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_368_order__le__less__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_369_order__le__less__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_370_order__le__less__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_rat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_371_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_372_order__less__le__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_373_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_374_order__le__less__trans,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_375_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_376_order__neq__le__trans,axiom,
! [A2: rat,B: rat] :
( ( A2 != B )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_377_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_378_order__le__neq__trans,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_379_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_380_order__less__imp__le,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_381_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_382_linorder__not__less,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( ord_less_eq_rat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_383_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_384_linorder__not__le,axiom,
! [X: rat,Y2: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y2 ) )
= ( ord_less_rat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_385_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_386_order__less__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_387_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_388_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_389_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_390_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ord_less_eq_rat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_391_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_392_order_Ostrict__implies__order,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_393_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_394_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_395_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_396_dual__order_Ostrict__trans2,axiom,
! [B: rat,A2: rat,C2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_397_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_398_dual__order_Ostrict__trans1,axiom,
! [B: rat,A2: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_399_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_400_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_401_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_402_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_rat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_403_dense__le__bounded,axiom,
! [X: rat,Y2: rat,Z: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ! [W: rat] :
( ( ord_less_rat @ X @ W )
=> ( ( ord_less_rat @ W @ Y2 )
=> ( ord_less_eq_rat @ W @ Z ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_404_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y2: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z @ W )
=> ( ( ord_less_rat @ W @ X )
=> ( ord_less_eq_rat @ Y2 @ W ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_405_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_406_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_407_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_408_order_Ostrict__trans2,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_rat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_409_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_410_order_Ostrict__trans1,axiom,
! [A2: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_411_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_412_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_413_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_414_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_rat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_415_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_416_not__le__imp__less,axiom,
! [Y2: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y2 @ X )
=> ( ord_less_rat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_417_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_418_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
& ~ ( ord_less_eq_rat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_419_dense__le,axiom,
! [Y2: rat,Z: rat] :
( ! [X2: rat] :
( ( ord_less_rat @ X2 @ Y2 )
=> ( ord_less_eq_rat @ X2 @ Z ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_le
thf(fact_420_dense__ge,axiom,
! [Z: rat,Y2: rat] :
( ! [X2: rat] :
( ( ord_less_rat @ Z @ X2 )
=> ( ord_less_eq_rat @ Y2 @ X2 ) )
=> ( ord_less_eq_rat @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_421_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_422_antisym__conv2,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ~ ( ord_less_rat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_423_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_424_antisym__conv1,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_425_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_426_nless__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_rat @ A2 @ B ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_427_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_428_leI,axiom,
! [X: rat,Y2: rat] :
( ~ ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X ) ) ).
% leI
thf(fact_429_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_430_leD,axiom,
! [Y2: rat,X: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ~ ( ord_less_rat @ X @ Y2 ) ) ).
% leD
thf(fact_431_top_Oextremum__uniqueI,axiom,
! [A2: set_rule] :
( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
=> ( A2 = top_top_set_rule ) ) ).
% top.extremum_uniqueI
thf(fact_432_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_433_top_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
=> ( A2 = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_434_top_Oextremum__unique,axiom,
! [A2: set_rule] :
( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
= ( A2 = top_top_set_rule ) ) ).
% top.extremum_unique
thf(fact_435_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_436_top_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
= ( A2 = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_437_top__greatest,axiom,
! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).
% top_greatest
thf(fact_438_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_439_top__greatest,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).
% top_greatest
thf(fact_440_all__ex__fair__nats,axiom,
! [M3: nat,X: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( ( snth_nat @ fair_fair_nats @ N4 )
= X ) ) ).
% all_ex_fair_nats
thf(fact_441_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_442_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_443_minf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ~ ( ord_less_eq_rat @ T @ X4 ) ) ).
% minf(8)
thf(fact_444_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_445_minf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ord_less_eq_rat @ X4 @ T ) ) ).
% minf(6)
thf(fact_446_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_447_pinf_I8_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ord_less_eq_rat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_448_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_449_pinf_I6_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ~ ( ord_less_eq_rat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_450_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_451_verit__comp__simplify1_I3_J,axiom,
! [B5: rat,A5: rat] :
( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
= ( ord_less_rat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_452_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_453_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_454_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_455_subsetI,axiom,
! [A: set_rat,B2: set_rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ X2 @ B2 ) )
=> ( ord_less_eq_set_rat @ A @ B2 ) ) ).
% subsetI
thf(fact_456_subsetI,axiom,
! [A: set_rule,B2: set_rule] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ X2 @ B2 ) )
=> ( ord_less_eq_set_rule @ A @ B2 ) ) ).
% subsetI
thf(fact_457_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_458_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_459_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_460_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_461_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_462_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_463_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_464_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_465_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_466_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A6: set_rat,B6: set_rat] :
! [T2: rat] :
( ( member_rat @ T2 @ A6 )
=> ( member_rat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_467_subset__iff,axiom,
( ord_less_eq_set_rule
= ( ^ [A6: set_rule,B6: set_rule] :
! [T2: rule] :
( ( member_rule @ T2 @ A6 )
=> ( member_rule @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_468_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_469_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A6: set_rat,B6: set_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( member_rat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_470_subset__eq,axiom,
( ord_less_eq_set_rule
= ( ^ [A6: set_rule,B6: set_rule] :
! [X3: rule] :
( ( member_rule @ X3 @ A6 )
=> ( member_rule @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_471_subsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_472_subsetD,axiom,
! [A: set_rat,B2: set_rat,C2: rat] :
( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_473_subsetD,axiom,
! [A: set_rule,B2: set_rule,C2: rule] :
( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_474_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_475_in__mono,axiom,
! [A: set_rat,B2: set_rat,X: rat] :
( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ X @ A )
=> ( member_rat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_476_in__mono,axiom,
! [A: set_rule,B2: set_rule,X: rule] :
( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ X @ A )
=> ( member_rule @ X @ B2 ) ) ) ).
% in_mono
thf(fact_477_subset__UNIV,axiom,
! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).
% subset_UNIV
thf(fact_478_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_479_subset__UNIV,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_480_subset__image__iff,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_rule @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_481_subset__image__iff,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_482_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_483_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_484_image__subset__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_rat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_485_image__subset__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_rule @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_486_subset__imageE,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B2
!= ( image_nat_rule @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_487_subset__imageE,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B2
!= ( image_nat_rat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_488_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_489_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_490_image__subsetI,axiom,
! [A: set_nat,F: nat > rat,B2: set_rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_491_image__subsetI,axiom,
! [A: set_nat,F: nat > rule,B2: set_rule] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_492_image__subsetI,axiom,
! [A: set_rat,F: rat > nat,B2: set_nat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_493_image__subsetI,axiom,
! [A: set_rat,F: rat > rat,B2: set_rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_494_image__subsetI,axiom,
! [A: set_rat,F: rat > rule,B2: set_rule] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_rat_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_495_image__subsetI,axiom,
! [A: set_rule,F: rule > nat,B2: set_nat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_496_image__subsetI,axiom,
! [A: set_rule,F: rule > rat,B2: set_rat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rat @ ( image_rule_rat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_497_image__subsetI,axiom,
! [A: set_rule,F: rule > rule,B2: set_rule] :
( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_rule @ ( image_rule_rule @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_498_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > rule] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ).
% image_mono
thf(fact_499_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > rat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_500_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_501_range__subsetD,axiom,
! [F: rule > nat,B2: set_nat,I: rule] :
( ( ord_less_eq_set_nat @ ( image_rule_nat @ F @ top_top_set_rule ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_502_range__subsetD,axiom,
! [F: rule > rat,B2: set_rat,I: rule] :
( ( ord_less_eq_set_rat @ ( image_rule_rat @ F @ top_top_set_rule ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_503_range__subsetD,axiom,
! [F: rule > rule,B2: set_rule,I: rule] :
( ( ord_less_eq_set_rule @ ( image_rule_rule @ F @ top_top_set_rule ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_504_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_505_range__subsetD,axiom,
! [F: nat > rat,B2: set_rat,I: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_506_range__subsetD,axiom,
! [F: nat > rule,B2: set_rule,I: nat] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ top_top_set_nat ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_507_range__subsetD,axiom,
! [F: rat > nat,B2: set_nat,I: rat] :
( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ top_top_set_rat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_508_range__subsetD,axiom,
! [F: rat > rat,B2: set_rat,I: rat] :
( ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ top_top_set_rat ) @ B2 )
=> ( member_rat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_509_range__subsetD,axiom,
! [F: rat > rule,B2: set_rule,I: rat] :
( ( ord_less_eq_set_rule @ ( image_rat_rule @ F @ top_top_set_rat ) @ B2 )
=> ( member_rule @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_510_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_511_verit__la__disequality,axiom,
! [A2: rat,B: rat] :
( ( A2 = B )
| ~ ( ord_less_eq_rat @ A2 @ B )
| ~ ( ord_less_eq_rat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_512_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_513_verit__comp__simplify1_I2_J,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_514_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_515_verit__comp__simplify1_I1_J,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_516_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_517_pinf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_518_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_519_pinf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_520_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_521_pinf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_522_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_523_pinf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_524_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_525_pinf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ~ ( ord_less_rat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_526_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_527_pinf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ord_less_rat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_528_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_529_minf_I1_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_530_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_531_minf_I2_J,axiom,
! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z5: rat] :
! [X2: rat] :
( ( ord_less_rat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_532_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_533_minf_I3_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_534_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_535_minf_I4_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_536_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_537_minf_I5_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ( ord_less_rat @ X4 @ T ) ) ).
% minf(5)
thf(fact_538_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_539_minf_I7_J,axiom,
! [T: rat] :
? [Z2: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z2 )
=> ~ ( ord_less_rat @ T @ X4 ) ) ).
% minf(7)
thf(fact_540_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_541_all__subset__image,axiom,
! [F: nat > rule,A: set_nat,P: set_rule > $o] :
( ( ! [B6: set_rule] :
( ( ord_less_eq_set_rule @ B6 @ ( image_nat_rule @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A )
=> ( P @ ( image_nat_rule @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_542_all__subset__image,axiom,
! [F: nat > rat,A: set_nat,P: set_rat > $o] :
( ( ! [B6: set_rat] :
( ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A )
=> ( P @ ( image_nat_rat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_543_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_544_top_Oordering__top__axioms,axiom,
orderi2038897200410189450t_rule @ ord_less_eq_set_rule @ ord_less_set_rule @ top_top_set_rule ).
% top.ordering_top_axioms
thf(fact_545_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_546_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_547_from__nat__def,axiom,
( from_nat_nat
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ to_nat_nat ) ) ).
% from_nat_def
thf(fact_548_from__nat__def,axiom,
( from_nat_rat
= ( hilber3317322552863949046at_nat @ top_top_set_rat @ to_nat_rat ) ) ).
% from_nat_def
thf(fact_549_GreatestI2__order,axiom,
! [P: rat > $o,X: rat,Q: rat > $o] :
( ( P @ X )
=> ( ! [Y3: rat] :
( ( P @ Y3 )
=> ( ord_less_eq_rat @ Y3 @ X ) )
=> ( ! [X2: rat] :
( ( P @ X2 )
=> ( ! [Y4: rat] :
( ( P @ Y4 )
=> ( ord_less_eq_rat @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_rat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_550_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_551_Greatest__equality,axiom,
! [P: rat > $o,X: rat] :
( ( P @ X )
=> ( ! [Y3: rat] :
( ( P @ Y3 )
=> ( ord_less_eq_rat @ Y3 @ X ) )
=> ( ( order_Greatest_rat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_552_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_553_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_554_psubsetD,axiom,
! [A: set_rat,B2: set_rat,C2: rat] :
( ( ord_less_set_rat @ A @ B2 )
=> ( ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_555_psubsetD,axiom,
! [A: set_rule,B2: set_rule,C2: rule] :
( ( ord_less_set_rule @ A @ B2 )
=> ( ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_556_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_557_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_558_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_559_inv__id,axiom,
( ( hilber2978553400015838680e_rule @ top_top_set_rule @ id_rule )
= id_rule ) ).
% inv_id
thf(fact_560_inv__id,axiom,
( ( hilber3633877196798814958at_nat @ top_top_set_nat @ id_nat )
= id_nat ) ).
% inv_id
thf(fact_561_inv__id,axiom,
( ( hilber2682192492777453310at_rat @ top_top_set_rat @ id_rat )
= id_rat ) ).
% inv_id
thf(fact_562_image__inv__into__cancel,axiom,
! [F: rule > nat,A: set_rule,A7: set_nat,B7: set_nat] :
( ( ( image_rule_nat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_rule_nat @ F @ ( image_nat_rule @ ( hilber2555471727301889379le_nat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_563_image__inv__into__cancel,axiom,
! [F: rat > nat,A: set_rat,A7: set_nat,B7: set_nat] :
( ( ( image_rat_nat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_rat_nat @ F @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_564_image__inv__into__cancel,axiom,
! [F: nat > rule,A: set_nat,A7: set_rule,B7: set_rule] :
( ( ( image_nat_rule @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_rule @ B7 @ A7 )
=> ( ( image_nat_rule @ F @ ( image_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_565_image__inv__into__cancel,axiom,
! [F: nat > rat,A: set_nat,A7: set_rat,B7: set_rat] :
( ( ( image_nat_rat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_rat @ B7 @ A7 )
=> ( ( image_nat_rat @ F @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_566_image__inv__into__cancel,axiom,
! [F: nat > nat,A: set_nat,A7: set_nat,B7: set_nat] :
( ( ( image_nat_nat @ F @ A )
= A7 )
=> ( ( ord_less_eq_set_nat @ B7 @ A7 )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ B7 ) )
= B7 ) ) ) ).
% image_inv_into_cancel
thf(fact_567_surj__f__inv__f,axiom,
! [F: rule > rule,Y2: rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_568_surj__f__inv__f,axiom,
! [F: rule > nat,Y2: nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( F @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_569_surj__f__inv__f,axiom,
! [F: rule > rat,Y2: rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( F @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_570_surj__f__inv__f,axiom,
! [F: nat > rule,Y2: rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( F @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_571_surj__f__inv__f,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_572_surj__f__inv__f,axiom,
! [F: nat > rat,Y2: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( F @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_573_surj__f__inv__f,axiom,
! [F: rat > rule,Y2: rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ( F @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_574_surj__f__inv__f,axiom,
! [F: rat > nat,Y2: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( F @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_575_surj__f__inv__f,axiom,
! [F: rat > rat,Y2: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_576_surj__iff__all,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_577_surj__iff__all,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_578_surj__iff__all,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_579_surj__iff__all,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_580_surj__iff__all,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_581_surj__iff__all,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_582_surj__iff__all,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
= ( ! [X3: rule] :
( ( F @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_583_surj__iff__all,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [X3: nat] :
( ( F @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_584_surj__iff__all,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [X3: rat] :
( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ X3 ) )
= X3 ) ) ) ).
% surj_iff_all
thf(fact_585_image__f__inv__f,axiom,
! [F: rule > rule,A: set_rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( image_rule_rule @ F @ ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_586_image__f__inv__f,axiom,
! [F: rule > nat,A: set_nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( image_rule_nat @ F @ ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_587_image__f__inv__f,axiom,
! [F: rule > rat,A: set_rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( image_rule_rat @ F @ ( image_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_588_image__f__inv__f,axiom,
! [F: nat > rule,A: set_rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( image_nat_rule @ F @ ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_589_image__f__inv__f,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_590_image__f__inv__f,axiom,
! [F: nat > rat,A: set_rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_591_image__f__inv__f,axiom,
! [F: rat > rule,A: set_rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ( image_rat_rule @ F @ ( image_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_592_image__f__inv__f,axiom,
! [F: rat > nat,A: set_nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_593_image__f__inv__f,axiom,
! [F: rat > rat,A: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F @ ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_594_surj__imp__inv__eq,axiom,
! [F: rule > rule,G: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_595_surj__imp__inv__eq,axiom,
! [F: rule > nat,G: nat > rule] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2555471727301889379le_nat @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_596_surj__imp__inv__eq,axiom,
! [F: rule > rat,G: rat > rule] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ! [X2: rule] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber1920341667215393643le_rat @ top_top_set_rule @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_597_surj__imp__inv__eq,axiom,
! [F: nat > rule,G: rule > nat] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber8541579349336805475t_rule @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_598_surj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_599_surj__imp__inv__eq,axiom,
! [F: nat > rat,G: rat > nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2998747136712319222at_rat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_600_surj__imp__inv__eq,axiom,
! [F: rat > rule,G: rule > rat] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber5214430877627997803t_rule @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_601_surj__imp__inv__eq,axiom,
! [F: rat > nat,G: nat > rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber3317322552863949046at_nat @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_602_surj__imp__inv__eq,axiom,
! [F: rat > rat,G: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_603_f__inv__into__f,axiom,
! [Y2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Y2 @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( hilber3633877196798814958at_nat @ A @ F @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_604_f__inv__into__f,axiom,
! [Y2: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ Y2 @ ( image_nat_rat @ F @ A ) )
=> ( ( F @ ( hilber2998747136712319222at_rat @ A @ F @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_605_f__inv__into__f,axiom,
! [Y2: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ Y2 @ ( image_nat_rule @ F @ A ) )
=> ( ( F @ ( hilber8541579349336805475t_rule @ A @ F @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_606_inv__into__into,axiom,
! [X: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_607_inv__into__into,axiom,
! [X: nat,F: rat > nat,A: set_rat] :
( ( member_nat @ X @ ( image_rat_nat @ F @ A ) )
=> ( member_rat @ ( hilber3317322552863949046at_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_608_inv__into__into,axiom,
! [X: nat,F: rule > nat,A: set_rule] :
( ( member_nat @ X @ ( image_rule_nat @ F @ A ) )
=> ( member_rule @ ( hilber2555471727301889379le_nat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_609_inv__into__into,axiom,
! [X: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( member_nat @ ( hilber2998747136712319222at_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_610_inv__into__into,axiom,
! [X: rat,F: rat > rat,A: set_rat] :
( ( member_rat @ X @ ( image_rat_rat @ F @ A ) )
=> ( member_rat @ ( hilber2682192492777453310at_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_611_inv__into__into,axiom,
! [X: rat,F: rule > rat,A: set_rule] :
( ( member_rat @ X @ ( image_rule_rat @ F @ A ) )
=> ( member_rule @ ( hilber1920341667215393643le_rat @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_612_inv__into__into,axiom,
! [X: rule,F: nat > rule,A: set_nat] :
( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( member_nat @ ( hilber8541579349336805475t_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_613_inv__into__into,axiom,
! [X: rule,F: rat > rule,A: set_rat] :
( ( member_rule @ X @ ( image_rat_rule @ F @ A ) )
=> ( member_rat @ ( hilber5214430877627997803t_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_614_inv__into__into,axiom,
! [X: rule,F: rule > rule,A: set_rule] :
( ( member_rule @ X @ ( image_rule_rule @ F @ A ) )
=> ( member_rule @ ( hilber2978553400015838680e_rule @ A @ F @ X ) @ A ) ) ).
% inv_into_into
thf(fact_615_inv__into__injective,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y2: nat] :
( ( ( hilber3633877196798814958at_nat @ A @ F @ X )
= ( hilber3633877196798814958at_nat @ A @ F @ Y2 ) )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ Y2 @ ( image_nat_nat @ F @ A ) )
=> ( X = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_616_inv__into__injective,axiom,
! [A: set_nat,F: nat > rat,X: rat,Y2: rat] :
( ( ( hilber2998747136712319222at_rat @ A @ F @ X )
= ( hilber2998747136712319222at_rat @ A @ F @ Y2 ) )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( ( member_rat @ Y2 @ ( image_nat_rat @ F @ A ) )
=> ( X = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_617_inv__into__injective,axiom,
! [A: set_nat,F: nat > rule,X: rule,Y2: rule] :
( ( ( hilber8541579349336805475t_rule @ A @ F @ X )
= ( hilber8541579349336805475t_rule @ A @ F @ Y2 ) )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( ( member_rule @ Y2 @ ( image_nat_rule @ F @ A ) )
=> ( X = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_618_image__Fpow__mono,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
=> ( ord_le7968974978423766289t_rule @ ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_rule @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_619_image__Fpow__mono,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_rat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_620_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_621_bijection_Osurj__inv,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bijection.surj_inv
thf(fact_622_bijection_Osurj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_623_bijection_Osurj__inv,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj_inv
thf(fact_624_surj__iff,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
= ( ( comp_rule_rule_rule @ F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) )
= id_rule ) ) ).
% surj_iff
thf(fact_625_surj__iff,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
= ( ( comp_rule_nat_nat @ F @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) )
= id_nat ) ) ).
% surj_iff
thf(fact_626_surj__iff,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
= ( ( comp_rule_rat_rat @ F @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) )
= id_rat ) ) ).
% surj_iff
thf(fact_627_surj__iff,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
= ( ( comp_nat_rule_rule @ F @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) )
= id_rule ) ) ).
% surj_iff
thf(fact_628_surj__iff,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ( comp_nat_nat_nat @ F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) )
= id_nat ) ) ).
% surj_iff
thf(fact_629_surj__iff,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ( comp_nat_rat_rat @ F @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) )
= id_rat ) ) ).
% surj_iff
thf(fact_630_surj__iff,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
= ( ( comp_rat_rule_rule @ F @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) )
= id_rule ) ) ).
% surj_iff
thf(fact_631_surj__iff,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ( comp_rat_nat_nat @ F @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) )
= id_nat ) ) ).
% surj_iff
thf(fact_632_surj__iff,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ( comp_rat_rat_rat @ F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) )
= id_rat ) ) ).
% surj_iff
thf(fact_633_comp__apply,axiom,
( comp_nat_nat_nat
= ( ^ [F3: nat > nat,G2: nat > nat,X3: nat] : ( F3 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_634_comp__id,axiom,
! [F: nat > nat] :
( ( comp_nat_nat_nat @ F @ id_nat )
= F ) ).
% comp_id
thf(fact_635_id__comp,axiom,
! [G: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ G )
= G ) ).
% id_comp
thf(fact_636_comp__eq__dest__lhs,axiom,
! [A2: nat > nat,B: nat > nat,C2: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B )
= C2 )
=> ( ( A2 @ ( B @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_637_comp__eq__elim,axiom,
! [A2: nat > nat,B: nat > nat,C2: nat > nat,D3: nat > nat] :
( ( ( comp_nat_nat_nat @ A2 @ B )
= ( comp_nat_nat_nat @ C2 @ D3 ) )
=> ! [V2: nat] :
( ( A2 @ ( B @ V2 ) )
= ( C2 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_638_comp__eq__dest,axiom,
! [A2: nat > nat,B: nat > nat,C2: nat > nat,D3: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B )
= ( comp_nat_nat_nat @ C2 @ D3 ) )
=> ( ( A2 @ ( B @ V ) )
= ( C2 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_639_comp__assoc,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_640_comp__def,axiom,
( comp_nat_nat_nat
= ( ^ [F3: nat > nat,G2: nat > nat,X3: nat] : ( F3 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_641_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: nat > nat,R2: nat > nat,F: nat > nat,L: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_nat_nat_nat @ F @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_642_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: nat > nat,L1: nat > nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_643_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_644_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: nat > nat,L: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= L )
=> ( ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_645_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X: nat,N: nat > nat,H: nat > nat,F: nat > nat] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ M ) @ G @ X )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_646_comp__cong,axiom,
! [F: nat > nat,G: nat > nat,X: nat,F4: nat > nat,G3: nat > nat,X8: nat] :
( ( ( F @ ( G @ X ) )
= ( F4 @ ( G3 @ X8 ) ) )
=> ( ( comp_nat_nat_nat @ F @ G @ X )
= ( comp_nat_nat_nat @ F4 @ G3 @ X8 ) ) ) ).
% comp_cong
thf(fact_647_Inf_OINF__image,axiom,
! [Inf: set_rule > rule,G: rule > rule,F: nat > rule,A: set_nat] :
( ( Inf @ ( image_rule_rule @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Inf @ ( image_nat_rule @ ( comp_rule_rule_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_648_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: rule > rat,F: nat > rule,A: set_nat] :
( ( Inf @ ( image_rule_rat @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_rule_rat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_649_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: rule > nat,F: nat > rule,A: set_nat] :
( ( Inf @ ( image_rule_nat @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_rule_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_650_Inf_OINF__image,axiom,
! [Inf: set_rule > rule,G: rat > rule,F: nat > rat,A: set_nat] :
( ( Inf @ ( image_rat_rule @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Inf @ ( image_nat_rule @ ( comp_rat_rule_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_651_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: rat > rat,F: nat > rat,A: set_nat] :
( ( Inf @ ( image_rat_rat @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_652_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: rat > nat,F: nat > rat,A: set_nat] :
( ( Inf @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_653_Inf_OINF__image,axiom,
! [Inf: set_rule > rule,G: nat > rule,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_rule @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_rule @ ( comp_nat_rule_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_654_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: nat > rat,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_rat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_nat_rat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_655_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_656_Sup_OSUP__image,axiom,
! [Sup: set_rule > rule,G: rule > rule,F: nat > rule,A: set_nat] :
( ( Sup @ ( image_rule_rule @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Sup @ ( image_nat_rule @ ( comp_rule_rule_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_657_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: rule > rat,F: nat > rule,A: set_nat] :
( ( Sup @ ( image_rule_rat @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_rule_rat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_658_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: rule > nat,F: nat > rule,A: set_nat] :
( ( Sup @ ( image_rule_nat @ G @ ( image_nat_rule @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_rule_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_659_Sup_OSUP__image,axiom,
! [Sup: set_rule > rule,G: rat > rule,F: nat > rat,A: set_nat] :
( ( Sup @ ( image_rat_rule @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Sup @ ( image_nat_rule @ ( comp_rat_rule_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_660_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: rat > rat,F: nat > rat,A: set_nat] :
( ( Sup @ ( image_rat_rat @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_661_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: rat > nat,F: nat > rat,A: set_nat] :
( ( Sup @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_662_Sup_OSUP__image,axiom,
! [Sup: set_rule > rule,G: nat > rule,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_rule @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_rule @ ( comp_nat_rule_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_663_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: nat > rat,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_rat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_nat_rat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_664_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_665_image__comp,axiom,
! [F: rule > rule,G: nat > rule,R: set_nat] :
( ( image_rule_rule @ F @ ( image_nat_rule @ G @ R ) )
= ( image_nat_rule @ ( comp_rule_rule_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_666_image__comp,axiom,
! [F: rule > rat,G: nat > rule,R: set_nat] :
( ( image_rule_rat @ F @ ( image_nat_rule @ G @ R ) )
= ( image_nat_rat @ ( comp_rule_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_667_image__comp,axiom,
! [F: rule > nat,G: nat > rule,R: set_nat] :
( ( image_rule_nat @ F @ ( image_nat_rule @ G @ R ) )
= ( image_nat_nat @ ( comp_rule_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_668_image__comp,axiom,
! [F: rat > rule,G: nat > rat,R: set_nat] :
( ( image_rat_rule @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_rule @ ( comp_rat_rule_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_669_image__comp,axiom,
! [F: rat > rat,G: nat > rat,R: set_nat] :
( ( image_rat_rat @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_rat @ ( comp_rat_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_670_image__comp,axiom,
! [F: rat > nat,G: nat > rat,R: set_nat] :
( ( image_rat_nat @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_nat @ ( comp_rat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_671_image__comp,axiom,
! [F: nat > rule,G: nat > nat,R: set_nat] :
( ( image_nat_rule @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_rule @ ( comp_nat_rule_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_672_image__comp,axiom,
! [F: nat > rat,G: nat > nat,R: set_nat] :
( ( image_nat_rat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_rat @ ( comp_nat_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_673_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_674_image__eq__imp__comp,axiom,
! [F: nat > rule,A: set_nat,G: nat > rule,B2: set_nat,H: rule > rule] :
( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ G @ B2 ) )
=> ( ( image_nat_rule @ ( comp_rule_rule_nat @ H @ F ) @ A )
= ( image_nat_rule @ ( comp_rule_rule_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_675_image__eq__imp__comp,axiom,
! [F: nat > rule,A: set_nat,G: nat > rule,B2: set_nat,H: rule > rat] :
( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ G @ B2 ) )
=> ( ( image_nat_rat @ ( comp_rule_rat_nat @ H @ F ) @ A )
= ( image_nat_rat @ ( comp_rule_rat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_676_image__eq__imp__comp,axiom,
! [F: nat > rule,A: set_nat,G: nat > rule,B2: set_nat,H: rule > nat] :
( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_rule_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_rule_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_677_image__eq__imp__comp,axiom,
! [F: nat > rat,A: set_nat,G: nat > rat,B2: set_nat,H: rat > rule] :
( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ G @ B2 ) )
=> ( ( image_nat_rule @ ( comp_rat_rule_nat @ H @ F ) @ A )
= ( image_nat_rule @ ( comp_rat_rule_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_678_image__eq__imp__comp,axiom,
! [F: nat > rat,A: set_nat,G: nat > rat,B2: set_nat,H: rat > rat] :
( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ G @ B2 ) )
=> ( ( image_nat_rat @ ( comp_rat_rat_nat @ H @ F ) @ A )
= ( image_nat_rat @ ( comp_rat_rat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_679_image__eq__imp__comp,axiom,
! [F: nat > rat,A: set_nat,G: nat > rat,B2: set_nat,H: rat > nat] :
( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_rat_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_rat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_680_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat,H: nat > rule] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_rule @ ( comp_nat_rule_nat @ H @ F ) @ A )
= ( image_nat_rule @ ( comp_nat_rule_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_681_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat,H: nat > rat] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_rat @ ( comp_nat_rat_nat @ H @ F ) @ A )
= ( image_nat_rat @ ( comp_nat_rat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_682_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_683_pointfree__idE,axiom,
! [F: nat > nat,G: nat > nat,X: nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= id_nat )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_684_comp__eq__id__dest,axiom,
! [A2: nat > nat,B: nat > nat,C2: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B )
= ( comp_nat_nat_nat @ id_nat @ C2 ) )
=> ( ( A2 @ ( B @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_685_stream_Omap__comp,axiom,
! [G: nat > nat,F: nat > nat,V: stream_nat] :
( ( smap_nat_nat @ G @ ( smap_nat_nat @ F @ V ) )
= ( smap_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% stream.map_comp
thf(fact_686_bijection_Oinv__comp__right,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( comp_rule_rule_rule @ F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) )
= id_rule ) ) ).
% bijection.inv_comp_right
thf(fact_687_bijection_Oinv__comp__right,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( comp_nat_nat_nat @ F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) )
= id_nat ) ) ).
% bijection.inv_comp_right
thf(fact_688_bijection_Oinv__comp__right,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( comp_rat_rat_rat @ F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) )
= id_rat ) ) ).
% bijection.inv_comp_right
thf(fact_689_bijection_Oinv__comp__left,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( comp_rule_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ F )
= id_rule ) ) ).
% bijection.inv_comp_left
thf(fact_690_bijection_Oinv__comp__left,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% bijection.inv_comp_left
thf(fact_691_bijection_Oinv__comp__left,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( comp_rat_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ F )
= id_rat ) ) ).
% bijection.inv_comp_left
thf(fact_692_comp__surj,axiom,
! [F: rule > rule,G: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( ( image_rule_rule @ G @ top_top_set_rule )
= top_top_set_rule )
=> ( ( image_rule_rule @ ( comp_rule_rule_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ) ).
% comp_surj
thf(fact_693_comp__surj,axiom,
! [F: rule > rule,G: rule > nat] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( ( image_rule_nat @ G @ top_top_set_rule )
= top_top_set_nat )
=> ( ( image_rule_nat @ ( comp_rule_nat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_694_comp__surj,axiom,
! [F: rule > rule,G: rule > rat] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ( ( image_rule_rat @ G @ top_top_set_rule )
= top_top_set_rat )
=> ( ( image_rule_rat @ ( comp_rule_rat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_695_comp__surj,axiom,
! [F: rule > nat,G: nat > rule] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( ( image_nat_rule @ G @ top_top_set_nat )
= top_top_set_rule )
=> ( ( image_rule_rule @ ( comp_nat_rule_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ) ).
% comp_surj
thf(fact_696_comp__surj,axiom,
! [F: rule > nat,G: nat > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_rule_nat @ ( comp_nat_nat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_697_comp__surj,axiom,
! [F: rule > nat,G: nat > rat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_rule_rat @ ( comp_nat_rat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_698_comp__surj,axiom,
! [F: rule > rat,G: rat > rule] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( ( image_rat_rule @ G @ top_top_set_rat )
= top_top_set_rule )
=> ( ( image_rule_rule @ ( comp_rat_rule_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ) ).
% comp_surj
thf(fact_699_comp__surj,axiom,
! [F: rule > rat,G: rat > nat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rule_nat @ ( comp_rat_nat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_700_comp__surj,axiom,
! [F: rule > rat,G: rat > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rule_rat @ ( comp_rat_rat_rule @ G @ F ) @ top_top_set_rule )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_701_comp__surj,axiom,
! [F: nat > rule,G: rule > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ( ( image_rule_rule @ G @ top_top_set_rule )
= top_top_set_rule )
=> ( ( image_nat_rule @ ( comp_rule_rule_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_rule ) ) ) ).
% comp_surj
thf(fact_702_SUP__image,axiom,
! [G: rule > nat,F: nat > rule,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_rule_nat @ G @ ( image_nat_rule @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_rule_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_703_SUP__image,axiom,
! [G: rat > nat,F: nat > rat,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_704_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_705_bijection_Osurj,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule ) ) ).
% bijection.surj
thf(fact_706_bijection_Osurj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_707_bijection_Osurj,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj
thf(fact_708_inv__unique__comp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= id_nat )
=> ( ( ( comp_nat_nat_nat @ G @ F )
= id_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% inv_unique_comp
thf(fact_709_bijection_Oeq__invI,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_710_bijection_Oeq__invI,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_711_bijection_Oeq__invI,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ B ) )
=> ( A2 = B ) ) ) ).
% bijection.eq_invI
thf(fact_712_bijection_Oinv__left,axiom,
! [F: rule > rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_713_bijection_Oinv__left,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_714_bijection_Oinv__left,axiom,
! [F: rat > rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ ( F @ A2 ) )
= A2 ) ) ).
% bijection.inv_left
thf(fact_715_bijection_Oinv__right,axiom,
! [F: rule > rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( F @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_716_bijection_Oinv__right,axiom,
! [F: nat > nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_717_bijection_Oinv__right,axiom,
! [F: rat > rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( F @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 ) )
= A2 ) ) ).
% bijection.inv_right
thf(fact_718_bijection_Oeq__inv__iff,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_719_bijection_Oeq__inv__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_720_bijection_Oeq__inv__iff,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ B ) )
= ( A2 = B ) ) ) ).
% bijection.eq_inv_iff
thf(fact_721_bijection_Oinv__left__eq__iff,axiom,
! [F: rule > rule,A2: rule,B: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_722_bijection_Oinv__left__eq__iff,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_723_bijection_Oinv__left__eq__iff,axiom,
! [F: rat > rat,A2: rat,B: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 )
= B )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_724_bijection_Oinv__right__eq__iff,axiom,
! [F: rule > rule,B: rule,A2: rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( ( B
= ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_725_bijection_Oinv__right__eq__iff,axiom,
! [F: nat > nat,B: nat,A2: nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( ( B
= ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_726_bijection_Oinv__right__eq__iff,axiom,
! [F: rat > rat,B: rat,A2: rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( ( B
= ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ A2 ) )
= ( ( F @ B )
= A2 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_727_fun_Omap__id,axiom,
! [T: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ T )
= T ) ).
% fun.map_id
thf(fact_728_fun_Oset__map,axiom,
! [F: nat > rule,V: rule > nat] :
( ( image_rule_rule @ ( comp_nat_rule_rule @ F @ V ) @ top_top_set_rule )
= ( image_nat_rule @ F @ ( image_rule_nat @ V @ top_top_set_rule ) ) ) ).
% fun.set_map
thf(fact_729_fun_Oset__map,axiom,
! [F: nat > rat,V: rule > nat] :
( ( image_rule_rat @ ( comp_nat_rat_rule @ F @ V ) @ top_top_set_rule )
= ( image_nat_rat @ F @ ( image_rule_nat @ V @ top_top_set_rule ) ) ) ).
% fun.set_map
thf(fact_730_fun_Oset__map,axiom,
! [F: nat > nat,V: rule > nat] :
( ( image_rule_nat @ ( comp_nat_nat_rule @ F @ V ) @ top_top_set_rule )
= ( image_nat_nat @ F @ ( image_rule_nat @ V @ top_top_set_rule ) ) ) ).
% fun.set_map
thf(fact_731_fun_Oset__map,axiom,
! [F: rule > rule,V: nat > rule] :
( ( image_nat_rule @ ( comp_rule_rule_nat @ F @ V ) @ top_top_set_nat )
= ( image_rule_rule @ F @ ( image_nat_rule @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_732_fun_Oset__map,axiom,
! [F: rat > rule,V: nat > rat] :
( ( image_nat_rule @ ( comp_rat_rule_nat @ F @ V ) @ top_top_set_nat )
= ( image_rat_rule @ F @ ( image_nat_rat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_733_fun_Oset__map,axiom,
! [F: nat > rule,V: nat > nat] :
( ( image_nat_rule @ ( comp_nat_rule_nat @ F @ V ) @ top_top_set_nat )
= ( image_nat_rule @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_734_fun_Oset__map,axiom,
! [F: rule > rat,V: nat > rule] :
( ( image_nat_rat @ ( comp_rule_rat_nat @ F @ V ) @ top_top_set_nat )
= ( image_rule_rat @ F @ ( image_nat_rule @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_735_fun_Oset__map,axiom,
! [F: rat > rat,V: nat > rat] :
( ( image_nat_rat @ ( comp_rat_rat_nat @ F @ V ) @ top_top_set_nat )
= ( image_rat_rat @ F @ ( image_nat_rat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_736_fun_Oset__map,axiom,
! [F: nat > rat,V: nat > nat] :
( ( image_nat_rat @ ( comp_nat_rat_nat @ F @ V ) @ top_top_set_nat )
= ( image_nat_rat @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_737_fun_Oset__map,axiom,
! [F: rule > nat,V: nat > rule] :
( ( image_nat_nat @ ( comp_rule_nat_nat @ F @ V ) @ top_top_set_nat )
= ( image_rule_nat @ F @ ( image_nat_rule @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_738_fun_Omap__cong,axiom,
! [X: nat > nat,Ya: nat > nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_nat_nat_nat @ F @ X )
= ( comp_nat_nat_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_739_fun_Omap__cong0,axiom,
! [X: nat > nat,F: nat > nat,G: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ X @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( comp_nat_nat_nat @ F @ X )
= ( comp_nat_nat_nat @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_740_fun_Oinj__map__strong,axiom,
! [X: nat > nat,Xa: nat > nat,F: nat > nat,Fa: nat > nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ X @ top_top_set_nat ) )
=> ( ( member_nat @ Za @ ( image_nat_nat @ Xa @ top_top_set_nat ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( comp_nat_nat_nat @ F @ X )
= ( comp_nat_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_741_fun_Omap__ident__strong,axiom,
! [T: rule > nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_rule_nat @ T @ top_top_set_rule ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_nat_nat_rule @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_742_fun_Omap__ident__strong,axiom,
! [T: rule > rat,F: rat > rat] :
( ! [Z2: rat] :
( ( member_rat @ Z2 @ ( image_rule_rat @ T @ top_top_set_rule ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rat_rat_rule @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_743_fun_Omap__ident__strong,axiom,
! [T: rule > rule,F: rule > rule] :
( ! [Z2: rule] :
( ( member_rule @ Z2 @ ( image_rule_rule @ T @ top_top_set_rule ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rule_rule_rule @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_744_fun_Omap__ident__strong,axiom,
! [T: nat > nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ T @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_nat_nat_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_745_fun_Omap__ident__strong,axiom,
! [T: nat > rat,F: rat > rat] :
( ! [Z2: rat] :
( ( member_rat @ Z2 @ ( image_nat_rat @ T @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rat_rat_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_746_fun_Omap__ident__strong,axiom,
! [T: nat > rule,F: rule > rule] :
( ! [Z2: rule] :
( ( member_rule @ Z2 @ ( image_nat_rule @ T @ top_top_set_nat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rule_rule_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_747_fun_Omap__ident__strong,axiom,
! [T: rat > nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( image_rat_nat @ T @ top_top_set_rat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_nat_nat_rat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_748_fun_Omap__ident__strong,axiom,
! [T: rat > rat,F: rat > rat] :
( ! [Z2: rat] :
( ( member_rat @ Z2 @ ( image_rat_rat @ T @ top_top_set_rat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rat_rat_rat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_749_fun_Omap__ident__strong,axiom,
! [T: rat > rule,F: rule > rule] :
( ! [Z2: rule] :
( ( member_rule @ Z2 @ ( image_rat_rule @ T @ top_top_set_rat ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( comp_rule_rule_rat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_750_fun_Omap__id0,axiom,
( ( comp_nat_nat_nat @ id_nat )
= id_nat_nat ) ).
% fun.map_id0
thf(fact_751_surj__fun__eq,axiom,
! [F: nat > nat,X5: set_nat,G1: nat > nat,G22: nat > nat] :
( ( ( image_nat_nat @ F @ X5 )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ( comp_nat_nat_nat @ G1 @ F @ X2 )
= ( comp_nat_nat_nat @ G22 @ F @ X2 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_752_isomorphism__expand,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( ( comp_nat_nat_nat @ F @ G )
= id_nat )
& ( ( comp_nat_nat_nat @ G @ F )
= id_nat ) )
= ( ! [X3: nat] :
( ( F @ ( G @ X3 ) )
= X3 )
& ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 ) ) ) ).
% isomorphism_expand
thf(fact_753_left__right__inverse__eq,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= id_nat )
=> ( ( ( comp_nat_nat_nat @ G @ H )
= id_nat )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_754_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ ( comp_rat_rat_nat @ field_2639924705303425560at_rat @ nat_to_rat_surj ) @ top_top_set_nat ) ) ).
% Rats_eq_range_of_rat_o_nat_to_rat_surj
thf(fact_755_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_nat,F: nat > nat > nat,X: nat] :
( ( finite7982400111564556781at_nat @ S5 @ F )
=> ( ( member_nat @ X @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_756_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_rat,F: rat > nat > nat,X: rat] :
( ( finite7665845467629690869at_nat @ S5 @ F )
=> ( ( member_rat @ X @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_757_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_rule,F: rule > nat > nat,X: rule] :
( ( finite5166552848317074786le_nat @ S5 @ F )
=> ( ( member_rule @ X @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_758_surj__of__rat__nat__to__rat__surj,axiom,
! [R: rat] :
( ( member_rat @ R @ field_6020823756834552118ts_rat )
=> ? [N4: nat] :
( R
= ( field_2639924705303425560at_rat @ ( nat_to_rat_surj @ N4 ) ) ) ) ).
% surj_of_rat_nat_to_rat_surj
thf(fact_759_Rats__def,axiom,
( field_6020823756834552118ts_rat
= ( image_rat_rat @ field_2639924705303425560at_rat @ top_top_set_rat ) ) ).
% Rats_def
thf(fact_760_Rats__of__rat,axiom,
! [R: rat] : ( member_rat @ ( field_2639924705303425560at_rat @ R ) @ field_6020823756834552118ts_rat ) ).
% Rats_of_rat
thf(fact_761_Rats__induct,axiom,
! [Q3: rat,P: rat > $o] :
( ( member_rat @ Q3 @ field_6020823756834552118ts_rat )
=> ( ! [R3: rat] : ( P @ ( field_2639924705303425560at_rat @ R3 ) )
=> ( P @ Q3 ) ) ) ).
% Rats_induct
thf(fact_762_Rats__cases,axiom,
! [Q3: rat] :
( ( member_rat @ Q3 @ field_6020823756834552118ts_rat )
=> ~ ! [R3: rat] :
( Q3
!= ( field_2639924705303425560at_rat @ R3 ) ) ) ).
% Rats_cases
thf(fact_763_of__rat__less,axiom,
! [R: rat,S2: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S2 ) )
= ( ord_less_rat @ R @ S2 ) ) ).
% of_rat_less
thf(fact_764_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% less_eq_rat_def
thf(fact_765_of__rat__less__eq,axiom,
! [R: rat,S2: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S2 ) )
= ( ord_less_eq_rat @ R @ S2 ) ) ).
% of_rat_less_eq
thf(fact_766_inv__o__cancel,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% inv_o_cancel
thf(fact_767_image__Pow__mono,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ B2 )
=> ( ord_le7968974978423766289t_rule @ ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) ) @ ( pow_rule @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_768_image__Pow__mono,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B2 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) ) @ ( pow_rat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_769_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_770_inj__iff,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ F )
= id_nat ) ) ).
% inj_iff
thf(fact_771_inj__on__to__nat,axiom,
! [S5: set_nat] : ( inj_on_nat_nat @ to_nat_nat @ S5 ) ).
% inj_on_to_nat
thf(fact_772_inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( hilber3633877196798814958at_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% inv_into_f_f
thf(fact_773_Pow__UNIV,axiom,
( ( pow_rule @ top_top_set_rule )
= top_top_set_set_rule ) ).
% Pow_UNIV
thf(fact_774_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_775_Pow__UNIV,axiom,
( ( pow_rat @ top_top_set_rat )
= top_top_set_set_rat ) ).
% Pow_UNIV
thf(fact_776_inv__into__image__cancel,axiom,
! [F: nat > rule,A: set_nat,S5: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( ord_less_eq_set_nat @ S5 @ A )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ ( image_nat_rule @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_777_inv__into__image__cancel,axiom,
! [F: nat > rat,A: set_nat,S5: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( ord_less_eq_set_nat @ S5 @ A )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ ( image_nat_rat @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_778_inv__into__image__cancel,axiom,
! [F: rule > nat,A: set_rule,S5: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( ord_less_eq_set_rule @ S5 @ A )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ A @ F ) @ ( image_rule_nat @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_779_inv__into__image__cancel,axiom,
! [F: rat > nat,A: set_rat,S5: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( ord_less_eq_set_rat @ S5 @ A )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A @ F ) @ ( image_rat_nat @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_780_inv__into__image__cancel,axiom,
! [F: nat > nat,A: set_nat,S5: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ S5 @ A )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ ( image_nat_nat @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_781_o__inv__o__cancel,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ G @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) @ F )
= G ) ) ).
% o_inv_o_cancel
thf(fact_782_inj__on__image__Pow,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_783_inj__on__image__Pow,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_784_inj__on__image__Pow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_785_inj__on__image,axiom,
! [F: nat > rule,A: set_set_nat] :
( ( inj_on_nat_rule @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ A ) ) ).
% inj_on_image
thf(fact_786_inj__on__image,axiom,
! [F: nat > rat,A: set_set_nat] :
( ( inj_on_nat_rat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_787_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_788_inj__on__image__Fpow,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on4755138273128556404t_rule @ ( image_nat_rule @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_789_inj__on__image__Fpow,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_790_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_791_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_792_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_793_inj__on__id,axiom,
! [A: set_nat] : ( inj_on_nat_nat @ id_nat @ A ) ).
% inj_on_id
thf(fact_794_injD,axiom,
! [F: nat > nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% injD
thf(fact_795_injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_796_inj__eq,axiom,
! [F: nat > nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
= ( X = Y2 ) ) ) ).
% inj_eq
thf(fact_797_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X3: nat,Y: nat] :
( ( ( F @ X3 )
= ( F @ Y ) )
=> ( X3 = Y ) ) ) ) ).
% inj_def
thf(fact_798_ex__inj,axiom,
? [To_nat: nat > nat] : ( inj_on_nat_nat @ To_nat @ top_top_set_nat ) ).
% ex_inj
thf(fact_799_ex__inj,axiom,
? [To_nat: rat > nat] : ( inj_on_rat_nat @ To_nat @ top_top_set_rat ) ).
% ex_inj
thf(fact_800_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_801_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_802_stream_Oinj__map,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on1381642877210728371am_nat @ ( smap_nat_nat @ F ) @ top_to7548458143485696966am_nat ) ) ).
% stream.inj_map
thf(fact_803_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_804_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X != Y2 )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( F @ X )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_805_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_806_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_807_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F3: nat > nat,A6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ! [Y: nat] :
( ( member_nat @ Y @ A6 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_808_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_809_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_810_inj__on__imageI2,axiom,
! [F4: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F4 @ F ) @ A )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_811_fun_Oinj__map,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on2461717442902640625at_nat @ ( comp_nat_nat_nat @ F ) @ top_top_set_nat_nat ) ) ).
% fun.inj_map
thf(fact_812_inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( F @ X )
= Y2 )
=> ( ( hilber3633877196798814958at_nat @ A @ F @ Y2 )
= X ) ) ) ) ).
% inv_into_f_eq
thf(fact_813_comp__fun__commute__on__def,axiom,
( finite3582905537739598962at_nat
= ( ^ [S4: set_nat,F3: nat > nat > nat] :
! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ S4 )
=> ( ( member_nat @ Y @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ Y ) @ ( F3 @ X3 ) )
= ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_814_comp__fun__commute__on__def,axiom,
( finite3266350893804733050at_nat
= ( ^ [S4: set_rat,F3: rat > nat > nat] :
! [X3: rat,Y: rat] :
( ( member_rat @ X3 @ S4 )
=> ( ( member_rat @ Y @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ Y ) @ ( F3 @ X3 ) )
= ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_815_comp__fun__commute__on__def,axiom,
( finite1185941773068520167le_nat
= ( ^ [S4: set_rule,F3: rule > nat > nat] :
! [X3: rule,Y: rule] :
( ( member_rule @ X3 @ S4 )
=> ( ( member_rule @ Y @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ Y ) @ ( F3 @ X3 ) )
= ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_816_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_nat,F: nat > nat > nat,X: nat,Y2: nat] :
( ( finite3582905537739598962at_nat @ S5 @ F )
=> ( ( member_nat @ X @ S5 )
=> ( ( member_nat @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_817_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_rat,F: rat > nat > nat,X: rat,Y2: rat] :
( ( finite3266350893804733050at_nat @ S5 @ F )
=> ( ( member_rat @ X @ S5 )
=> ( ( member_rat @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_818_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_rule,F: rule > nat > nat,X: rule,Y2: rule] :
( ( finite1185941773068520167le_nat @ S5 @ F )
=> ( ( member_rule @ X @ S5 )
=> ( ( member_rule @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_819_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_nat,F: nat > nat > nat,X: nat,Y2: nat,G: nat > nat] :
( ( finite3582905537739598962at_nat @ S5 @ F )
=> ( ( member_nat @ X @ S5 )
=> ( ( member_nat @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( comp_nat_nat_nat @ ( F @ X ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( comp_nat_nat_nat @ ( F @ Y2 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_820_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_rat,F: rat > nat > nat,X: rat,Y2: rat,G: nat > nat] :
( ( finite3266350893804733050at_nat @ S5 @ F )
=> ( ( member_rat @ X @ S5 )
=> ( ( member_rat @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( comp_nat_nat_nat @ ( F @ X ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( comp_nat_nat_nat @ ( F @ Y2 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_821_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_rule,F: rule > nat > nat,X: rule,Y2: rule,G: nat > nat] :
( ( finite1185941773068520167le_nat @ S5 @ F )
=> ( ( member_rule @ X @ S5 )
=> ( ( member_rule @ Y2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y2 ) @ ( comp_nat_nat_nat @ ( F @ X ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( comp_nat_nat_nat @ ( F @ Y2 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_822_comp__fun__commute__on_Ointro,axiom,
! [S5: set_nat,F: nat > nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ S5 )
=> ( ( member_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( finite3582905537739598962at_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_823_comp__fun__commute__on_Ointro,axiom,
! [S5: set_rat,F: rat > nat > nat] :
( ! [X2: rat,Y3: rat] :
( ( member_rat @ X2 @ S5 )
=> ( ( member_rat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( finite3266350893804733050at_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_824_comp__fun__commute__on_Ointro,axiom,
! [S5: set_rule,F: rule > nat > nat] :
( ! [X2: rule,Y3: rule] :
( ( member_rule @ X2 @ S5 )
=> ( ( member_rule @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( finite1185941773068520167le_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_825_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_826_linorder__injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( F @ X2 )
!= ( F @ Y3 ) ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_827_range__ex1__eq,axiom,
! [F: rule > nat,B: nat] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( member_nat @ B @ ( image_rule_nat @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y: rule] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_828_range__ex1__eq,axiom,
! [F: rule > rat,B: rat] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( member_rat @ B @ ( image_rule_rat @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y: rule] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_829_range__ex1__eq,axiom,
! [F: rule > rule,B: rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ B @ ( image_rule_rule @ F @ top_top_set_rule ) )
= ( ? [X3: rule] :
( ( B
= ( F @ X3 ) )
& ! [Y: rule] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_830_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_831_range__ex1__eq,axiom,
! [F: nat > rat,B: rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_832_range__ex1__eq,axiom,
! [F: nat > rule,B: rule] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( member_rule @ B @ ( image_nat_rule @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y: nat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_833_range__ex1__eq,axiom,
! [F: rat > nat,B: nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y: rat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_834_range__ex1__eq,axiom,
! [F: rat > rat,B: rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( member_rat @ B @ ( image_rat_rat @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y: rat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_835_range__ex1__eq,axiom,
! [F: rat > rule,B: rule] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( member_rule @ B @ ( image_rat_rule @ F @ top_top_set_rat ) )
= ( ? [X3: rat] :
( ( B
= ( F @ X3 ) )
& ! [Y: rat] :
( ( B
= ( F @ Y ) )
=> ( Y = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_836_inj__image__eq__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_837_inj__image__eq__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_838_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_839_inj__image__mem__iff,axiom,
! [F: rule > nat,A2: rule,A: set_rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rule_nat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_840_inj__image__mem__iff,axiom,
! [F: rule > rat,A2: rule,A: set_rule] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rule_rat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_841_inj__image__mem__iff,axiom,
! [F: rule > rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rule_rule @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_842_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_843_inj__image__mem__iff,axiom,
! [F: nat > rat,A2: nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_nat_rat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_844_inj__image__mem__iff,axiom,
! [F: nat > rule,A2: nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_nat_rule @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_845_inj__image__mem__iff,axiom,
! [F: rat > nat,A2: rat,A: set_rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rat_nat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_846_inj__image__mem__iff,axiom,
! [F: rat > rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rat_rat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_847_inj__image__mem__iff,axiom,
! [F: rat > rule,A2: rat,A: set_rat] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rat_rule @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_848_inj__on__image__eq__iff,axiom,
! [F: nat > rule,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_rule @ F @ A )
= ( image_nat_rule @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_849_inj__on__image__eq__iff,axiom,
! [F: nat > rat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_rat @ F @ A )
= ( image_nat_rat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_850_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_851_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_852_inj__on__image__mem__iff,axiom,
! [F: nat > rat,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_nat_rat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_853_inj__on__image__mem__iff,axiom,
! [F: nat > rule,B2: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ B2 )
=> ( ( member_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_nat_rule @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_854_inj__on__image__mem__iff,axiom,
! [F: rat > nat,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_nat @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rat_nat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_855_inj__on__image__mem__iff,axiom,
! [F: rat > rat,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rat @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rat_rat @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_856_inj__on__image__mem__iff,axiom,
! [F: rat > rule,B2: set_rat,A2: rat,A: set_rat] :
( ( inj_on_rat_rule @ F @ B2 )
=> ( ( member_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rat_rule @ F @ A ) )
= ( member_rat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_857_inj__on__image__mem__iff,axiom,
! [F: rule > nat,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_nat @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_rule_nat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_858_inj__on__image__mem__iff,axiom,
! [F: rule > rat,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rat @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rat @ ( F @ A2 ) @ ( image_rule_rat @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_859_inj__on__image__mem__iff,axiom,
! [F: rule > rule,B2: set_rule,A2: rule,A: set_rule] :
( ( inj_on_rule_rule @ F @ B2 )
=> ( ( member_rule @ A2 @ B2 )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( ( member_rule @ ( F @ A2 ) @ ( image_rule_rule @ F @ A ) )
= ( member_rule @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_860_subset__image__inj,axiom,
! [S5: set_rule,F: nat > rule,T3: set_nat] :
( ( ord_less_eq_set_rule @ S5 @ ( image_nat_rule @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_rule @ F @ U )
& ( S5
= ( image_nat_rule @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_861_subset__image__inj,axiom,
! [S5: set_rat,F: nat > rat,T3: set_nat] :
( ( ord_less_eq_set_rat @ S5 @ ( image_nat_rat @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_rat @ F @ U )
& ( S5
= ( image_nat_rat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_862_subset__image__inj,axiom,
! [S5: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S5 @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_nat @ F @ U )
& ( S5
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_863_inj__compose,axiom,
! [F: rule > nat,G: nat > rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( inj_on_nat_rule @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_rule_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_864_inj__compose,axiom,
! [F: nat > nat,G: rule > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_rule_nat @ G @ top_top_set_rule )
=> ( inj_on_rule_nat @ ( comp_nat_nat_rule @ F @ G ) @ top_top_set_rule ) ) ) ).
% inj_compose
thf(fact_865_inj__compose,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_866_inj__compose,axiom,
! [F: nat > nat,G: rat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_rat_nat @ G @ top_top_set_rat )
=> ( inj_on_rat_nat @ ( comp_nat_nat_rat @ F @ G ) @ top_top_set_rat ) ) ) ).
% inj_compose
thf(fact_867_inj__compose,axiom,
! [F: rat > nat,G: nat > rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( inj_on_nat_rat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_rat_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_868_comp__inj__on__iff,axiom,
! [F: nat > rule,A: set_nat,F4: rule > nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( inj_on_rule_nat @ F4 @ ( image_nat_rule @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_rule_nat_nat @ F4 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_869_comp__inj__on__iff,axiom,
! [F: nat > rat,A: set_nat,F4: rat > nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( inj_on_rat_nat @ F4 @ ( image_nat_rat @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_rat_nat_nat @ F4 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_870_comp__inj__on__iff,axiom,
! [F: nat > nat,A: set_nat,F4: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ F4 @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F4 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_871_inj__on__imageI,axiom,
! [G: rule > nat,F: nat > rule,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_rule_nat_nat @ G @ F ) @ A )
=> ( inj_on_rule_nat @ G @ ( image_nat_rule @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_872_inj__on__imageI,axiom,
! [G: rat > nat,F: nat > rat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A )
=> ( inj_on_rat_nat @ G @ ( image_nat_rat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_873_inj__on__imageI,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A )
=> ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_874_comp__inj__on,axiom,
! [F: nat > rule,A: set_nat,G: rule > nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( inj_on_rule_nat @ G @ ( image_nat_rule @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_rule_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_875_comp__inj__on,axiom,
! [F: nat > rat,A: set_nat,G: rat > nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( inj_on_rat_nat @ G @ ( image_nat_rat @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_876_comp__inj__on,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_877_image__strict__mono,axiom,
! [F: nat > rule,B2: set_nat,A: set_nat] :
( ( inj_on_nat_rule @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_878_image__strict__mono,axiom,
! [F: nat > rat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_rat @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_879_image__strict__mono,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_880_inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_881_inv__f__eq,axiom,
! [F: nat > nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= Y2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y2 )
= X ) ) ) ).
% inv_f_eq
thf(fact_882_inj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [X2: nat] :
( ( F @ ( G @ X2 ) )
= X2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_883_bijection_Oinj,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( inj_on_rule_rule @ F @ top_top_set_rule ) ) ).
% bijection.inj
thf(fact_884_bijection_Oinj,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_885_bijection_Oinj,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( inj_on_rat_rat @ F @ top_top_set_rat ) ) ).
% bijection.inj
thf(fact_886_inj__to__nat,axiom,
inj_on_nat_nat @ to_nat_nat @ top_top_set_nat ).
% inj_to_nat
thf(fact_887_inj__to__nat,axiom,
inj_on_rat_nat @ to_nat_rat @ top_top_set_rat ).
% inj_to_nat
thf(fact_888_image__Pow__surj,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( ( image_nat_rule @ F @ A )
= B2 )
=> ( ( image_458447791132712456t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) )
= ( pow_rule @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_889_image__Pow__surj,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( ( image_nat_rat @ F @ A )
= B2 )
=> ( ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) )
= ( pow_rat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_890_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B2 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_891_inj__image__subset__iff,axiom,
! [F: nat > rule,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ A ) @ ( image_nat_rule @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_892_inj__image__subset__iff,axiom,
! [F: nat > rat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_893_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_894_surj__imp__inj__inv,axiom,
! [F: rule > rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( inj_on_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_895_surj__imp__inj__inv,axiom,
! [F: rule > nat] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( inj_on_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_896_surj__imp__inj__inv,axiom,
! [F: rule > rat] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( inj_on_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_897_surj__imp__inj__inv,axiom,
! [F: nat > rule] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( inj_on_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_898_surj__imp__inj__inv,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_899_surj__imp__inj__inv,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_900_surj__imp__inj__inv,axiom,
! [F: rat > rule] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( inj_on_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ top_top_set_rule ) ) ).
% surj_imp_inj_inv
thf(fact_901_surj__imp__inj__inv,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( inj_on_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_902_surj__imp__inj__inv,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_903_inj__imp__surj__inv,axiom,
! [F: rule > rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ( image_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_904_inj__imp__surj__inv,axiom,
! [F: rule > nat] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ top_top_set_nat )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_905_inj__imp__surj__inv,axiom,
! [F: rule > rat] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ( image_rat_rule @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F ) @ top_top_set_rat )
= top_top_set_rule ) ) ).
% inj_imp_surj_inv
thf(fact_906_inj__imp__surj__inv,axiom,
! [F: nat > rule] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ top_top_set_rule )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_907_inj__imp__surj__inv,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_908_inj__imp__surj__inv,axiom,
! [F: nat > rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ top_top_set_rat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_909_inj__imp__surj__inv,axiom,
! [F: rat > rule] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ( image_rule_rat @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F ) @ top_top_set_rule )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_910_inj__imp__surj__inv,axiom,
! [F: rat > nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ top_top_set_nat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_911_inj__imp__surj__inv,axiom,
! [F: rat > rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_912_image__inv__f__f,axiom,
! [F: rule > nat,A: set_rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ( image_nat_rule @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F ) @ ( image_rule_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_913_image__inv__f__f,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ( image_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ ( image_nat_rule @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_914_image__inv__f__f,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ ( image_nat_rat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_915_image__inv__f__f,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_916_image__inv__f__f,axiom,
! [F: rat > nat,A: set_rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) @ ( image_rat_nat @ F @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_917_inj__transfer,axiom,
! [F: rule > nat,P: rule > $o,X: rule] :
( ( inj_on_rule_nat @ F @ top_top_set_rule )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_rule_nat @ F @ top_top_set_rule ) )
=> ( P @ ( hilber2555471727301889379le_nat @ top_top_set_rule @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_918_inj__transfer,axiom,
! [F: rule > rat,P: rule > $o,X: rule] :
( ( inj_on_rule_rat @ F @ top_top_set_rule )
=> ( ! [Y3: rat] :
( ( member_rat @ Y3 @ ( image_rule_rat @ F @ top_top_set_rule ) )
=> ( P @ ( hilber1920341667215393643le_rat @ top_top_set_rule @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_919_inj__transfer,axiom,
! [F: rule > rule,P: rule > $o,X: rule] :
( ( inj_on_rule_rule @ F @ top_top_set_rule )
=> ( ! [Y3: rule] :
( ( member_rule @ Y3 @ ( image_rule_rule @ F @ top_top_set_rule ) )
=> ( P @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_920_inj__transfer,axiom,
! [F: nat > nat,P: nat > $o,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( P @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_921_inj__transfer,axiom,
! [F: nat > rat,P: nat > $o,X: nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ! [Y3: rat] :
( ( member_rat @ Y3 @ ( image_nat_rat @ F @ top_top_set_nat ) )
=> ( P @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_922_inj__transfer,axiom,
! [F: nat > rule,P: nat > $o,X: nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ! [Y3: rule] :
( ( member_rule @ Y3 @ ( image_nat_rule @ F @ top_top_set_nat ) )
=> ( P @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_923_inj__transfer,axiom,
! [F: rat > nat,P: rat > $o,X: rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( image_rat_nat @ F @ top_top_set_rat ) )
=> ( P @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_924_inj__transfer,axiom,
! [F: rat > rat,P: rat > $o,X: rat] :
( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ! [Y3: rat] :
( ( member_rat @ Y3 @ ( image_rat_rat @ F @ top_top_set_rat ) )
=> ( P @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_925_inj__transfer,axiom,
! [F: rat > rule,P: rat > $o,X: rat] :
( ( inj_on_rat_rule @ F @ top_top_set_rat )
=> ( ! [Y3: rule] :
( ( member_rule @ Y3 @ ( image_rat_rule @ F @ top_top_set_rat ) )
=> ( P @ ( hilber5214430877627997803t_rule @ top_top_set_rat @ F @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_926_inj__on__inv__into,axiom,
! [B2: set_rule,F: nat > rule,A: set_nat] :
( ( ord_less_eq_set_rule @ B2 @ ( image_nat_rule @ F @ A ) )
=> ( inj_on_rule_nat @ ( hilber8541579349336805475t_rule @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_927_inj__on__inv__into,axiom,
! [B2: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_nat_rat @ F @ A ) )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_928_inj__on__inv__into,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_929_inv__into__comp,axiom,
! [F: rule > nat,G: nat > rule,A: set_nat,X: nat] :
( ( inj_on_rule_nat @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_nat @ X @ ( image_rule_nat @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( hilber3633877196798814958at_nat @ A @ ( comp_rule_nat_nat @ F @ G ) @ X )
= ( comp_rule_nat_nat @ ( hilber8541579349336805475t_rule @ A @ G ) @ ( hilber2555471727301889379le_nat @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_930_inv__into__comp,axiom,
! [F: rat > nat,G: nat > rat,A: set_nat,X: nat] :
( ( inj_on_rat_nat @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_nat @ X @ ( image_rat_nat @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( hilber3633877196798814958at_nat @ A @ ( comp_rat_nat_nat @ F @ G ) @ X )
= ( comp_rat_nat_nat @ ( hilber2998747136712319222at_rat @ A @ G ) @ ( hilber3317322552863949046at_nat @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_931_inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( hilber3633877196798814958at_nat @ A @ ( comp_nat_nat_nat @ F @ G ) @ X )
= ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ A @ G ) @ ( hilber3633877196798814958at_nat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_932_inv__into__comp,axiom,
! [F: rule > rat,G: nat > rule,A: set_nat,X: rat] :
( ( inj_on_rule_rat @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_rat @ X @ ( image_rule_rat @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( hilber2998747136712319222at_rat @ A @ ( comp_rule_rat_nat @ F @ G ) @ X )
= ( comp_rule_nat_rat @ ( hilber8541579349336805475t_rule @ A @ G ) @ ( hilber1920341667215393643le_rat @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_933_inv__into__comp,axiom,
! [F: rat > rat,G: nat > rat,A: set_nat,X: rat] :
( ( inj_on_rat_rat @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_rat @ X @ ( image_rat_rat @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( hilber2998747136712319222at_rat @ A @ ( comp_rat_rat_nat @ F @ G ) @ X )
= ( comp_rat_nat_rat @ ( hilber2998747136712319222at_rat @ A @ G ) @ ( hilber2682192492777453310at_rat @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_934_inv__into__comp,axiom,
! [F: nat > rat,G: nat > nat,A: set_nat,X: rat] :
( ( inj_on_nat_rat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( hilber2998747136712319222at_rat @ A @ ( comp_nat_rat_nat @ F @ G ) @ X )
= ( comp_nat_nat_rat @ ( hilber3633877196798814958at_nat @ A @ G ) @ ( hilber2998747136712319222at_rat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_935_inv__into__comp,axiom,
! [F: rule > rule,G: nat > rule,A: set_nat,X: rule] :
( ( inj_on_rule_rule @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_rule @ X @ ( image_rule_rule @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( hilber8541579349336805475t_rule @ A @ ( comp_rule_rule_nat @ F @ G ) @ X )
= ( comp_rule_nat_rule @ ( hilber8541579349336805475t_rule @ A @ G ) @ ( hilber2978553400015838680e_rule @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_936_inv__into__comp,axiom,
! [F: rat > rule,G: nat > rat,A: set_nat,X: rule] :
( ( inj_on_rat_rule @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_rule @ X @ ( image_rat_rule @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( hilber8541579349336805475t_rule @ A @ ( comp_rat_rule_nat @ F @ G ) @ X )
= ( comp_rat_nat_rule @ ( hilber2998747136712319222at_rat @ A @ G ) @ ( hilber5214430877627997803t_rule @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_937_inv__into__comp,axiom,
! [F: nat > rule,G: nat > nat,A: set_nat,X: rule] :
( ( inj_on_nat_rule @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( hilber8541579349336805475t_rule @ A @ ( comp_nat_rule_nat @ F @ G ) @ X )
= ( comp_nat_nat_rule @ ( hilber3633877196798814958at_nat @ A @ G ) @ ( hilber8541579349336805475t_rule @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_938_bijection_Oinj__inv,axiom,
! [F: rule > rule] :
( ( hilber6733072011887318294n_rule @ F )
=> ( inj_on_rule_rule @ ( hilber2978553400015838680e_rule @ top_top_set_rule @ F ) @ top_top_set_rule ) ) ).
% bijection.inj_inv
thf(fact_939_bijection_Oinj__inv,axiom,
! [F: nat > nat] :
( ( hilber5277034221543178913on_nat @ F )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ top_top_set_nat ) ) ).
% bijection.inj_inv
thf(fact_940_bijection_Oinj__inv,axiom,
! [F: rat > rat] :
( ( hilber4641904161456683177on_rat @ F )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) @ top_top_set_rat ) ) ).
% bijection.inj_inv
thf(fact_941_all__subset__image__inj,axiom,
! [F: nat > rule,S5: set_nat,P: set_rule > $o] :
( ( ! [T4: set_rule] :
( ( ord_less_eq_set_rule @ T4 @ ( image_nat_rule @ F @ S5 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_rule @ F @ T4 ) )
=> ( P @ ( image_nat_rule @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_942_all__subset__image__inj,axiom,
! [F: nat > rat,S5: set_nat,P: set_rat > $o] :
( ( ! [T4: set_rat] :
( ( ord_less_eq_set_rat @ T4 @ ( image_nat_rat @ F @ S5 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_rat @ F @ T4 ) )
=> ( P @ ( image_nat_rat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_943_all__subset__image__inj,axiom,
! [F: nat > nat,S5: set_nat,P: set_nat > $o] :
( ( ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_nat_nat @ F @ S5 ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_nat @ F @ T4 ) )
=> ( P @ ( image_nat_nat @ F @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_944_ex__subset__image__inj,axiom,
! [F: nat > rule,S5: set_nat,P: set_rule > $o] :
( ( ? [T4: set_rule] :
( ( ord_less_eq_set_rule @ T4 @ ( image_nat_rule @ F @ S5 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_rule @ F @ T4 )
& ( P @ ( image_nat_rule @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_945_ex__subset__image__inj,axiom,
! [F: nat > rat,S5: set_nat,P: set_rat > $o] :
( ( ? [T4: set_rat] :
( ( ord_less_eq_set_rat @ T4 @ ( image_nat_rat @ F @ S5 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_rat @ F @ T4 )
& ( P @ ( image_nat_rat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_946_ex__subset__image__inj,axiom,
! [F: nat > nat,S5: set_nat,P: set_nat > $o] :
( ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ ( image_nat_nat @ F @ S5 ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S5 )
& ( inj_on_nat_nat @ F @ T4 )
& ( P @ ( image_nat_nat @ F @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_947_the__inv__into__comp,axiom,
! [F: rule > nat,G: nat > rule,A: set_nat,X: nat] :
( ( inj_on_rule_nat @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_nat @ X @ ( image_rule_nat @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( the_inv_into_nat_nat @ A @ ( comp_rule_nat_nat @ F @ G ) @ X )
= ( comp_rule_nat_nat @ ( the_in5544616208814386890t_rule @ A @ G ) @ ( the_in8781880623634246602le_nat @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_948_the__inv__into__comp,axiom,
! [F: rat > nat,G: nat > rat,A: set_nat,X: nat] :
( ( inj_on_rat_nat @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_nat @ X @ ( image_rat_nat @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( the_inv_into_nat_nat @ A @ ( comp_rat_nat_nat @ F @ G ) @ X )
= ( comp_rat_nat_nat @ ( the_inv_into_nat_rat @ A @ G ) @ ( the_inv_into_rat_nat @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_949_the__inv__into__comp,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( the_inv_into_nat_nat @ A @ ( comp_nat_nat_nat @ F @ G ) @ X )
= ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ A @ G ) @ ( the_inv_into_nat_nat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_950_the__inv__into__comp,axiom,
! [F: rule > rat,G: nat > rule,A: set_nat,X: rat] :
( ( inj_on_rule_rat @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_rat @ X @ ( image_rule_rat @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( the_inv_into_nat_rat @ A @ ( comp_rule_rat_nat @ F @ G ) @ X )
= ( comp_rule_nat_rat @ ( the_in5544616208814386890t_rule @ A @ G ) @ ( the_in8146750563547750866le_rat @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_951_the__inv__into__comp,axiom,
! [F: rat > rat,G: nat > rat,A: set_nat,X: rat] :
( ( inj_on_rat_rat @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_rat @ X @ ( image_rat_rat @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( the_inv_into_nat_rat @ A @ ( comp_rat_rat_nat @ F @ G ) @ X )
= ( comp_rat_nat_rat @ ( the_inv_into_nat_rat @ A @ G ) @ ( the_inv_into_rat_rat @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_952_the__inv__into__comp,axiom,
! [F: nat > rat,G: nat > nat,A: set_nat,X: rat] :
( ( inj_on_nat_rat @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( the_inv_into_nat_rat @ A @ ( comp_nat_rat_nat @ F @ G ) @ X )
= ( comp_nat_nat_rat @ ( the_inv_into_nat_nat @ A @ G ) @ ( the_inv_into_nat_rat @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_953_the__inv__into__comp,axiom,
! [F: rule > rule,G: nat > rule,A: set_nat,X: rule] :
( ( inj_on_rule_rule @ F @ ( image_nat_rule @ G @ A ) )
=> ( ( inj_on_nat_rule @ G @ A )
=> ( ( member_rule @ X @ ( image_rule_rule @ F @ ( image_nat_rule @ G @ A ) ) )
=> ( ( the_in5544616208814386890t_rule @ A @ ( comp_rule_rule_nat @ F @ G ) @ X )
= ( comp_rule_nat_rule @ ( the_in5544616208814386890t_rule @ A @ G ) @ ( the_in80044576880915775e_rule @ ( image_nat_rule @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_954_the__inv__into__comp,axiom,
! [F: rat > rule,G: nat > rat,A: set_nat,X: rule] :
( ( inj_on_rat_rule @ F @ ( image_nat_rat @ G @ A ) )
=> ( ( inj_on_nat_rat @ G @ A )
=> ( ( member_rule @ X @ ( image_rat_rule @ F @ ( image_nat_rat @ G @ A ) ) )
=> ( ( the_in5544616208814386890t_rule @ A @ ( comp_rat_rule_nat @ F @ G ) @ X )
= ( comp_rat_nat_rule @ ( the_inv_into_nat_rat @ A @ G ) @ ( the_in2217467737105579218t_rule @ ( image_nat_rat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_955_the__inv__into__comp,axiom,
! [F: nat > rule,G: nat > nat,A: set_nat,X: rule] :
( ( inj_on_nat_rule @ F @ ( image_nat_nat @ G @ A ) )
=> ( ( inj_on_nat_nat @ G @ A )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ ( image_nat_nat @ G @ A ) ) )
=> ( ( the_in5544616208814386890t_rule @ A @ ( comp_nat_rule_nat @ F @ G ) @ X )
= ( comp_nat_nat_rule @ ( the_inv_into_nat_nat @ A @ G ) @ ( the_in5544616208814386890t_rule @ ( image_nat_nat @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_956_the__inv__into__onto,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( image_rule_nat @ ( the_in5544616208814386890t_rule @ A @ F ) @ ( image_nat_rule @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_957_the__inv__into__onto,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A @ F ) @ ( image_nat_rat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_958_the__inv__into__onto,axiom,
! [F: rule > nat,A: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( image_nat_rule @ ( the_in8781880623634246602le_nat @ A @ F ) @ ( image_rule_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_959_the__inv__into__onto,axiom,
! [F: rat > nat,A: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A @ F ) @ ( image_rat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_960_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_961_the__inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_962_the__inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= Y2 )
=> ( ( member_nat @ X @ A )
=> ( ( the_inv_into_nat_nat @ A @ F @ Y2 )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_963_comp__fun__idem__on__axioms_Ointro,axiom,
! [S5: set_nat,F: nat > nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite3061184102382659472at_nat @ S5 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_964_comp__fun__idem__on__axioms_Ointro,axiom,
! [S5: set_rat,F: rat > nat > nat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite2744629458447793560at_nat @ S5 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_965_comp__fun__idem__on__axioms_Ointro,axiom,
! [S5: set_rule,F: rule > nat > nat] :
( ! [X2: rule] :
( ( member_rule @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite4326737663714109445le_nat @ S5 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_966_comp__fun__idem__on__axioms__def,axiom,
( finite3061184102382659472at_nat
= ( ^ [S4: set_nat,F3: nat > nat > nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_967_comp__fun__idem__on__axioms__def,axiom,
( finite2744629458447793560at_nat
= ( ^ [S4: set_rat,F3: rat > nat > nat] :
! [X3: rat] :
( ( member_rat @ X3 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_968_comp__fun__idem__on__axioms__def,axiom,
( finite4326737663714109445le_nat
= ( ^ [S4: set_rule,F3: rule > nat > nat] :
! [X3: rule] :
( ( member_rule @ X3 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F3 @ X3 ) @ ( F3 @ X3 ) )
= ( F3 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_969_the__inv__f__f,axiom,
! [F: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_970_inj__on__the__inv__into,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( inj_on_rule_nat @ ( the_in5544616208814386890t_rule @ A @ F ) @ ( image_nat_rule @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_971_inj__on__the__inv__into,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A @ F ) @ ( image_nat_rat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_972_inj__on__the__inv__into,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_973_f__the__inv__into__f,axiom,
! [F: nat > nat,A: set_nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ Y2 @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_974_f__the__inv__into__f,axiom,
! [F: nat > rat,A: set_nat,Y2: rat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( member_rat @ Y2 @ ( image_nat_rat @ F @ A ) )
=> ( ( F @ ( the_inv_into_nat_rat @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_975_f__the__inv__into__f,axiom,
! [F: nat > rule,A: set_nat,Y2: rule] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( member_rule @ Y2 @ ( image_nat_rule @ F @ A ) )
=> ( ( F @ ( the_in5544616208814386890t_rule @ A @ F @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_976_the__inv__into__into,axiom,
! [F: nat > nat,A: set_nat,X: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_977_the__inv__into__into,axiom,
! [F: rat > nat,A: set_rat,X: nat,B2: set_rat] :
( ( inj_on_rat_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_rat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_inv_into_rat_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_978_the__inv__into__into,axiom,
! [F: rule > nat,A: set_rule,X: nat,B2: set_rule] :
( ( inj_on_rule_nat @ F @ A )
=> ( ( member_nat @ X @ ( image_rule_nat @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in8781880623634246602le_nat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_979_the__inv__into__into,axiom,
! [F: nat > rat,A: set_nat,X: rat,B2: set_nat] :
( ( inj_on_nat_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_nat_rat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_inv_into_nat_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_980_the__inv__into__into,axiom,
! [F: rat > rat,A: set_rat,X: rat,B2: set_rat] :
( ( inj_on_rat_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_rat_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_inv_into_rat_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_981_the__inv__into__into,axiom,
! [F: rule > rat,A: set_rule,X: rat,B2: set_rule] :
( ( inj_on_rule_rat @ F @ A )
=> ( ( member_rat @ X @ ( image_rule_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in8146750563547750866le_rat @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_982_the__inv__into__into,axiom,
! [F: nat > rule,A: set_nat,X: rule,B2: set_nat] :
( ( inj_on_nat_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_nat_rule @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( member_nat @ ( the_in5544616208814386890t_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_983_the__inv__into__into,axiom,
! [F: rat > rule,A: set_rat,X: rule,B2: set_rat] :
( ( inj_on_rat_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_rat_rule @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ A @ B2 )
=> ( member_rat @ ( the_in2217467737105579218t_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_984_the__inv__into__into,axiom,
! [F: rule > rule,A: set_rule,X: rule,B2: set_rule] :
( ( inj_on_rule_rule @ F @ A )
=> ( ( member_rule @ X @ ( image_rule_rule @ F @ A ) )
=> ( ( ord_less_eq_set_rule @ A @ B2 )
=> ( member_rule @ ( the_in80044576880915775e_rule @ A @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_985_the__inv__f__o__f__id,axiom,
! [F: nat > nat,Z: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ top_top_set_nat @ F ) @ F @ Z )
= ( id_nat @ Z ) ) ) ).
% the_inv_f_o_f_id
thf(fact_986_inj__image__Compl__subset,axiom,
! [F: nat > rule,A: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( ord_less_eq_set_rule @ ( image_nat_rule @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus4869265918275750596t_rule @ ( image_nat_rule @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_987_inj__image__Compl__subset,axiom,
! [F: nat > rat,A: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_988_inj__image__Compl__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_989_streams__UNIV,axiom,
( ( streams_rule @ top_top_set_rule )
= top_to3705917391389534779m_rule ) ).
% streams_UNIV
thf(fact_990_streams__UNIV,axiom,
( ( streams_nat @ top_top_set_nat )
= top_to7548458143485696966am_nat ) ).
% streams_UNIV
thf(fact_991_streams__UNIV,axiom,
( ( streams_rat @ top_top_set_rat )
= top_to1168471806008019406am_rat ) ).
% streams_UNIV
thf(fact_992_Compl__iff,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_993_Compl__iff,axiom,
! [C2: rat,A: set_rat] :
( ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) )
= ( ~ ( member_rat @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_994_Compl__iff,axiom,
! [C2: rule,A: set_rule] :
( ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) )
= ( ~ ( member_rule @ C2 @ A ) ) ) ).
% Compl_iff
thf(fact_995_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_996_ComplI,axiom,
! [C2: rat,A: set_rat] :
( ~ ( member_rat @ C2 @ A )
=> ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) ) ) ).
% ComplI
thf(fact_997_ComplI,axiom,
! [C2: rule,A: set_rule] :
( ~ ( member_rule @ C2 @ A )
=> ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) ) ) ).
% ComplI
thf(fact_998_Rats__minus__iff,axiom,
! [A2: rat] :
( ( member_rat @ ( uminus_uminus_rat @ A2 ) @ field_6020823756834552118ts_rat )
= ( member_rat @ A2 @ field_6020823756834552118ts_rat ) ) ).
% Rats_minus_iff
thf(fact_999_surj__uminus,axiom,
( ( image_rat_rat @ uminus_uminus_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_uminus
thf(fact_1000_verit__negate__coefficient_I2_J,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1001_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C2 @ A ) ) ).
% ComplD
thf(fact_1002_ComplD,axiom,
! [C2: rat,A: set_rat] :
( ( member_rat @ C2 @ ( uminus2201863774496077783et_rat @ A ) )
=> ~ ( member_rat @ C2 @ A ) ) ).
% ComplD
thf(fact_1003_ComplD,axiom,
! [C2: rule,A: set_rule] :
( ( member_rule @ C2 @ ( uminus4869265918275750596t_rule @ A ) )
=> ~ ( member_rule @ C2 @ A ) ) ).
% ComplD
thf(fact_1004_fold__graph__closed__lemma,axiom,
! [G: nat > nat > nat,Z: nat,A: set_nat,X: nat,B2: set_nat,F: nat > nat > nat] :
( ( finite1441398328259824232at_nat @ G @ Z @ A @ X )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z @ A @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1005_fold__graph__closed__lemma,axiom,
! [G: nat > rat > rat,Z: rat,A: set_nat,X: rat,B2: set_rat,F: nat > rat > rat] :
( ( finite806268268173328496at_rat @ G @ Z @ A @ X )
=> ( ! [A3: nat,B3: rat] :
( ( member_nat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: rat] :
( ( member_nat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite806268268173328496at_rat @ F @ Z @ A @ X )
& ( member_rat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1006_fold__graph__closed__lemma,axiom,
! [G: nat > rule > rule,Z: rule,A: set_nat,X: rule,B2: set_rule,F: nat > rule > rule] :
( ( finite4825588098184082909t_rule @ G @ Z @ A @ X )
=> ( ! [A3: nat,B3: rule] :
( ( member_nat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: rule] :
( ( member_nat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite4825588098184082909t_rule @ F @ Z @ A @ X )
& ( member_rule @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1007_fold__graph__closed__lemma,axiom,
! [G: rat > nat > nat,Z: nat,A: set_rat,X: nat,B2: set_nat,F: rat > nat > nat] :
( ( finite1124843684324958320at_nat @ G @ Z @ A @ X )
=> ( ! [A3: rat,B3: nat] :
( ( member_rat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: nat] :
( ( member_rat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1124843684324958320at_nat @ F @ Z @ A @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1008_fold__graph__closed__lemma,axiom,
! [G: rat > rat > rat,Z: rat,A: set_rat,X: rat,B2: set_rat,F: rat > rat > rat] :
( ( finite489713624238462584at_rat @ G @ Z @ A @ X )
=> ( ! [A3: rat,B3: rat] :
( ( member_rat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: rat] :
( ( member_rat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite489713624238462584at_rat @ F @ Z @ A @ X )
& ( member_rat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1009_fold__graph__closed__lemma,axiom,
! [G: rat > rule > rule,Z: rule,A: set_rat,X: rule,B2: set_rule,F: rat > rule > rule] :
( ( finite1498439626475275237t_rule @ G @ Z @ A @ X )
=> ( ! [A3: rat,B3: rule] :
( ( member_rat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: rule] :
( ( member_rat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite1498439626475275237t_rule @ F @ Z @ A @ X )
& ( member_rule @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1010_fold__graph__closed__lemma,axiom,
! [G: rule > nat > nat,Z: nat,A: set_rule,X: nat,B2: set_nat,F: rule > nat > nat] :
( ( finite8062852513003942621le_nat @ G @ Z @ A @ X )
=> ( ! [A3: rule,B3: nat] :
( ( member_rule @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: nat] :
( ( member_rule @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite8062852513003942621le_nat @ F @ Z @ A @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1011_fold__graph__closed__lemma,axiom,
! [G: rule > rat > rat,Z: rat,A: set_rule,X: rat,B2: set_rat,F: rule > rat > rat] :
( ( finite7427722452917446885le_rat @ G @ Z @ A @ X )
=> ( ! [A3: rule,B3: rat] :
( ( member_rule @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: rat] :
( ( member_rule @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite7427722452917446885le_rat @ F @ Z @ A @ X )
& ( member_rat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1012_fold__graph__closed__lemma,axiom,
! [G: rule > rule > rule,Z: rule,A: set_rule,X: rule,B2: set_rule,F: rule > rule > rule] :
( ( finite3037048784445983058e_rule @ G @ Z @ A @ X )
=> ( ! [A3: rule,B3: rule] :
( ( member_rule @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: rule] :
( ( member_rule @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite3037048784445983058e_rule @ F @ Z @ A @ X )
& ( member_rule @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1013_fold__graph__closed__eq,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z @ A )
= ( finite1441398328259824232at_nat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1014_fold__graph__closed__eq,axiom,
! [A: set_nat,B2: set_rat,F: nat > rat > rat,G: nat > rat > rat,Z: rat] :
( ! [A3: nat,B3: rat] :
( ( member_nat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: rat] :
( ( member_nat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite806268268173328496at_rat @ F @ Z @ A )
= ( finite806268268173328496at_rat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1015_fold__graph__closed__eq,axiom,
! [A: set_nat,B2: set_rule,F: nat > rule > rule,G: nat > rule > rule,Z: rule] :
( ! [A3: nat,B3: rule] :
( ( member_nat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: nat,B3: rule] :
( ( member_nat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite4825588098184082909t_rule @ F @ Z @ A )
= ( finite4825588098184082909t_rule @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1016_fold__graph__closed__eq,axiom,
! [A: set_rat,B2: set_nat,F: rat > nat > nat,G: rat > nat > nat,Z: nat] :
( ! [A3: rat,B3: nat] :
( ( member_rat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: nat] :
( ( member_rat @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite1124843684324958320at_nat @ F @ Z @ A )
= ( finite1124843684324958320at_nat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1017_fold__graph__closed__eq,axiom,
! [A: set_rat,B2: set_rat,F: rat > rat > rat,G: rat > rat > rat,Z: rat] :
( ! [A3: rat,B3: rat] :
( ( member_rat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: rat] :
( ( member_rat @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite489713624238462584at_rat @ F @ Z @ A )
= ( finite489713624238462584at_rat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1018_fold__graph__closed__eq,axiom,
! [A: set_rat,B2: set_rule,F: rat > rule > rule,G: rat > rule > rule,Z: rule] :
( ! [A3: rat,B3: rule] :
( ( member_rat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rat,B3: rule] :
( ( member_rat @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite1498439626475275237t_rule @ F @ Z @ A )
= ( finite1498439626475275237t_rule @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1019_fold__graph__closed__eq,axiom,
! [A: set_rule,B2: set_nat,F: rule > nat > nat,G: rule > nat > nat,Z: nat] :
( ! [A3: rule,B3: nat] :
( ( member_rule @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: nat] :
( ( member_rule @ A3 @ A )
=> ( ( member_nat @ B3 @ B2 )
=> ( member_nat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite8062852513003942621le_nat @ F @ Z @ A )
= ( finite8062852513003942621le_nat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1020_fold__graph__closed__eq,axiom,
! [A: set_rule,B2: set_rat,F: rule > rat > rat,G: rule > rat > rat,Z: rat] :
( ! [A3: rule,B3: rat] :
( ( member_rule @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: rat] :
( ( member_rule @ A3 @ A )
=> ( ( member_rat @ B3 @ B2 )
=> ( member_rat @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rat @ Z @ B2 )
=> ( ( finite7427722452917446885le_rat @ F @ Z @ A )
= ( finite7427722452917446885le_rat @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1021_fold__graph__closed__eq,axiom,
! [A: set_rule,B2: set_rule,F: rule > rule > rule,G: rule > rule > rule,Z: rule] :
( ! [A3: rule,B3: rule] :
( ( member_rule @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( ( F @ A3 @ B3 )
= ( G @ A3 @ B3 ) ) ) )
=> ( ! [A3: rule,B3: rule] :
( ( member_rule @ A3 @ A )
=> ( ( member_rule @ B3 @ B2 )
=> ( member_rule @ ( G @ A3 @ B3 ) @ B2 ) ) )
=> ( ( member_rule @ Z @ B2 )
=> ( ( finite3037048784445983058e_rule @ F @ Z @ A )
= ( finite3037048784445983058e_rule @ G @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1022_smap__streams,axiom,
! [S2: stream_nat,A: set_nat,F: nat > nat,B2: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_nat @ ( smap_nat_nat @ F @ S2 ) @ ( streams_nat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1023_smap__streams,axiom,
! [S2: stream_nat,A: set_nat,F: nat > rat,B2: set_rat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rat @ ( smap_nat_rat @ F @ S2 ) @ ( streams_rat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1024_smap__streams,axiom,
! [S2: stream_nat,A: set_nat,F: nat > rule,B2: set_rule] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rule @ ( smap_nat_rule @ F @ S2 ) @ ( streams_rule @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1025_smap__streams,axiom,
! [S2: stream_rat,A: set_rat,F: rat > nat,B2: set_nat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ A ) )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_nat @ ( smap_rat_nat @ F @ S2 ) @ ( streams_nat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1026_smap__streams,axiom,
! [S2: stream_rat,A: set_rat,F: rat > rat,B2: set_rat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ A ) )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rat @ ( smap_rat_rat @ F @ S2 ) @ ( streams_rat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1027_smap__streams,axiom,
! [S2: stream_rat,A: set_rat,F: rat > rule,B2: set_rule] :
( ( member_stream_rat @ S2 @ ( streams_rat @ A ) )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rule @ ( smap_rat_rule @ F @ S2 ) @ ( streams_rule @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1028_smap__streams,axiom,
! [S2: stream_rule,A: set_rule,F: rule > nat,B2: set_nat] :
( ( member_stream_rule @ S2 @ ( streams_rule @ A ) )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_nat @ ( smap_rule_nat @ F @ S2 ) @ ( streams_nat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1029_smap__streams,axiom,
! [S2: stream_rule,A: set_rule,F: rule > rat,B2: set_rat] :
( ( member_stream_rule @ S2 @ ( streams_rule @ A ) )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rat @ ( smap_rule_rat @ F @ S2 ) @ ( streams_rat @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1030_smap__streams,axiom,
! [S2: stream_rule,A: set_rule,F: rule > rule,B2: set_rule] :
( ( member_stream_rule @ S2 @ ( streams_rule @ A ) )
=> ( ! [X2: rule] :
( ( member_rule @ X2 @ A )
=> ( member_rule @ ( F @ X2 ) @ B2 ) )
=> ( member_stream_rule @ ( smap_rule_rule @ F @ S2 ) @ ( streams_rule @ B2 ) ) ) ) ).
% smap_streams
thf(fact_1031_streams__iff__snth,axiom,
! [S2: stream_rat,X5: set_rat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ X5 ) )
= ( ! [N3: nat] : ( member_rat @ ( snth_rat @ S2 @ N3 ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_1032_streams__iff__snth,axiom,
! [S2: stream_rule,X5: set_rule] :
( ( member_stream_rule @ S2 @ ( streams_rule @ X5 ) )
= ( ! [N3: nat] : ( member_rule @ ( snth_rule @ S2 @ N3 ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_1033_streams__iff__snth,axiom,
! [S2: stream_nat,X5: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ X5 ) )
= ( ! [N3: nat] : ( member_nat @ ( snth_nat @ S2 @ N3 ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_1034_snth__in,axiom,
! [S2: stream_rat,X5: set_rat,N2: nat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ X5 ) )
=> ( member_rat @ ( snth_rat @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_1035_snth__in,axiom,
! [S2: stream_rule,X5: set_rule,N2: nat] :
( ( member_stream_rule @ S2 @ ( streams_rule @ X5 ) )
=> ( member_rule @ ( snth_rule @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_1036_snth__in,axiom,
! [S2: stream_nat,X5: set_nat,N2: nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ X5 ) )
=> ( member_nat @ ( snth_nat @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_1037_sset__streams,axiom,
! [S2: stream_rule,A: set_rule] :
( ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A )
=> ( member_stream_rule @ S2 @ ( streams_rule @ A ) ) ) ).
% sset_streams
thf(fact_1038_sset__streams,axiom,
! [S2: stream_rat,A: set_rat] :
( ( ord_less_eq_set_rat @ ( sset_rat @ S2 ) @ A )
=> ( member_stream_rat @ S2 @ ( streams_rat @ A ) ) ) ).
% sset_streams
thf(fact_1039_sset__streams,axiom,
! [S2: stream_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sset_nat @ S2 ) @ A )
=> ( member_stream_nat @ S2 @ ( streams_nat @ A ) ) ) ).
% sset_streams
thf(fact_1040_streams__sset,axiom,
! [S2: stream_rule,A: set_rule] :
( ( member_stream_rule @ S2 @ ( streams_rule @ A ) )
=> ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_1041_streams__sset,axiom,
! [S2: stream_rat,A: set_rat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ A ) )
=> ( ord_less_eq_set_rat @ ( sset_rat @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_1042_streams__sset,axiom,
! [S2: stream_nat,A: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
=> ( ord_less_eq_set_nat @ ( sset_nat @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_1043_streams__iff__sset,axiom,
! [S2: stream_rule,A: set_rule] :
( ( member_stream_rule @ S2 @ ( streams_rule @ A ) )
= ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_1044_streams__iff__sset,axiom,
! [S2: stream_rat,A: set_rat] :
( ( member_stream_rat @ S2 @ ( streams_rat @ A ) )
= ( ord_less_eq_set_rat @ ( sset_rat @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_1045_streams__iff__sset,axiom,
! [S2: stream_nat,A: set_nat] :
( ( member_stream_nat @ S2 @ ( streams_nat @ A ) )
= ( ord_less_eq_set_nat @ ( sset_nat @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_1046_surj__Compl__image__subset,axiom,
! [F: rule > rule,A: set_rule] :
( ( ( image_rule_rule @ F @ top_top_set_rule )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_rule_rule @ F @ A ) ) @ ( image_rule_rule @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1047_surj__Compl__image__subset,axiom,
! [F: rule > nat,A: set_rule] :
( ( ( image_rule_nat @ F @ top_top_set_rule )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rule_nat @ F @ A ) ) @ ( image_rule_nat @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1048_surj__Compl__image__subset,axiom,
! [F: rule > rat,A: set_rule] :
( ( ( image_rule_rat @ F @ top_top_set_rule )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rule_rat @ F @ A ) ) @ ( image_rule_rat @ F @ ( uminus4869265918275750596t_rule @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1049_surj__Compl__image__subset,axiom,
! [F: nat > rule,A: set_nat] :
( ( ( image_nat_rule @ F @ top_top_set_nat )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_nat_rule @ F @ A ) ) @ ( image_nat_rule @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1050_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1051_surj__Compl__image__subset,axiom,
! [F: nat > rat,A: set_nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F @ A ) ) @ ( image_nat_rat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1052_surj__Compl__image__subset,axiom,
! [F: rat > rule,A: set_rat] :
( ( ( image_rat_rule @ F @ top_top_set_rat )
= top_top_set_rule )
=> ( ord_less_eq_set_rule @ ( uminus4869265918275750596t_rule @ ( image_rat_rule @ F @ A ) ) @ ( image_rat_rule @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1053_surj__Compl__image__subset,axiom,
! [F: rat > nat,A: set_rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F @ A ) ) @ ( image_rat_nat @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1054_surj__Compl__image__subset,axiom,
! [F: rat > rat,A: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F @ A ) ) @ ( image_rat_rat @ F @ ( uminus2201863774496077783et_rat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1055_neg__less__iff__less,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_1056_neg__le__iff__le,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_1057_le__imp__neg__le,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1058_minus__le__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1059_le__minus__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1060_minus__less__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1061_less__minus__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1062_stream_Omap__o__corec,axiom,
! [F: nat > nat,G: nat > nat,Ga: nat > $o,Gb: nat > stream_nat,Gc: nat > nat] :
( ( comp_s2553592672699509493at_nat @ ( smap_nat_nat @ F ) @ ( corec_stream_nat_nat @ G @ Ga @ Gb @ Gc ) )
= ( corec_stream_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ Ga @ ( comp_s2553592672699509493at_nat @ ( smap_nat_nat @ F ) @ Gb ) @ Gc ) ) ).
% stream.map_o_corec
thf(fact_1063_inj__imp__bij__betw__inv,axiom,
! [F: nat > rule,M: set_nat] :
( ( inj_on_nat_rule @ F @ top_top_set_nat )
=> ( bij_betw_rule_nat @ ( hilber8541579349336805475t_rule @ top_top_set_nat @ F ) @ ( image_nat_rule @ F @ M ) @ M ) ) ).
% inj_imp_bij_betw_inv
thf(fact_1064_inj__imp__bij__betw__inv,axiom,
! [F: nat > rat,M: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( bij_betw_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) @ ( image_nat_rat @ F @ M ) @ M ) ) ).
% inj_imp_bij_betw_inv
thf(fact_1065_inj__imp__bij__betw__inv,axiom,
! [F: nat > nat,M: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ ( image_nat_nat @ F @ M ) @ M ) ) ).
% inj_imp_bij_betw_inv
thf(fact_1066_strict__mono__inv__on__range,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ ord_less_nat @ ord_less_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1067_strict__mono__inv__on__range,axiom,
! [F: nat > rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_nat @ ord_less_rat @ F )
=> ( monotone_on_rat_nat @ ( image_nat_rat @ F @ top_top_set_nat ) @ ord_less_rat @ ord_less_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1068_strict__mono__inv__on__range,axiom,
! [F: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( monotone_on_nat_rat @ ( image_rat_nat @ F @ top_top_set_rat ) @ ord_less_nat @ ord_less_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1069_strict__mono__inv__on__range,axiom,
! [F: rat > rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_rat @ ord_less_rat @ F )
=> ( monotone_on_rat_rat @ ( image_rat_rat @ F @ top_top_set_rat ) @ ord_less_rat @ ord_less_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1070_bij__betw__id,axiom,
! [A: set_nat] : ( bij_betw_nat_nat @ id_nat @ A @ A ) ).
% bij_betw_id
thf(fact_1071_ord_Omono__onD,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1072_ord_Omono__onD,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat,R: rule,S2: rule] :
( ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_rule @ R @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1073_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1074_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > rat,R: nat,S2: nat] :
( ( monotone_on_nat_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1075_ord_Omono__onD,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > rat,R: rat,S2: rat] :
( ( monotone_on_rat_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1076_ord_Omono__onD,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > rat,R: rule,S2: rule] :
( ( monotone_on_rule_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
=> ( ( member_rule @ R @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1077_ord_Omono__onI,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1078_ord_Omono__onI,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat] :
( ! [R3: rule,S6: rule] :
( ( member_rule @ R3 @ A )
=> ( ( member_rule @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1079_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_1080_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > rat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_rat @ A @ Less_eq @ ord_less_eq_rat @ F ) ) ).
% ord.mono_onI
thf(fact_1081_ord_Omono__onI,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > rat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_rat @ A @ Less_eq @ ord_less_eq_rat @ F ) ) ).
% ord.mono_onI
thf(fact_1082_ord_Omono__onI,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > rat] :
( ! [R3: rule,S6: rule] :
( ( member_rule @ R3 @ A )
=> ( ( member_rule @ S6 @ A )
=> ( ( Less_eq @ R3 @ S6 )
=> ( ord_less_eq_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rule_rat @ A @ Less_eq @ ord_less_eq_rat @ F ) ) ).
% ord.mono_onI
thf(fact_1083_ord_Omono__on__def,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R4: rat,S: rat] :
( ( ( member_rat @ R4 @ A )
& ( member_rat @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1084_ord_Omono__on__def,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > nat] :
( ( monotone_on_rule_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R4: rule,S: rule] :
( ( ( member_rule @ R4 @ A )
& ( member_rule @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1085_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R4: nat,S: nat] :
( ( ( member_nat @ R4 @ A )
& ( member_nat @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1086_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > rat] :
( ( monotone_on_nat_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
= ( ! [R4: nat,S: nat] :
( ( ( member_nat @ R4 @ A )
& ( member_nat @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1087_ord_Omono__on__def,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > rat] :
( ( monotone_on_rat_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
= ( ! [R4: rat,S: rat] :
( ( ( member_rat @ R4 @ A )
& ( member_rat @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1088_ord_Omono__on__def,axiom,
! [A: set_rule,Less_eq: rule > rule > $o,F: rule > rat] :
( ( monotone_on_rule_rat @ A @ Less_eq @ ord_less_eq_rat @ F )
= ( ! [R4: rule,S: rule] :
( ( ( member_rule @ R4 @ A )
& ( member_rule @ S @ A )
& ( Less_eq @ R4 @ S ) )
=> ( ord_less_eq_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1089_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1090_mono__onD,axiom,
! [A: set_nat,F: nat > rat,R: nat,S2: nat] :
( ( monotone_on_nat_rat @ A @ ord_less_eq_nat @ ord_less_eq_rat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1091_mono__onD,axiom,
! [A: set_rat,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1092_mono__onD,axiom,
! [A: set_rat,F: rat > rat,R: rat,S2: rat] :
( ( monotone_on_rat_rat @ A @ ord_less_eq_rat @ ord_less_eq_rat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R @ S2 )
=> ( ord_less_eq_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1093_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( ord_less_eq_nat @ R3 @ S6 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_1094_mono__onI,axiom,
! [A: set_nat,F: nat > rat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( ord_less_eq_nat @ R3 @ S6 )
=> ( ord_less_eq_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_rat @ A @ ord_less_eq_nat @ ord_less_eq_rat @ F ) ) ).
% mono_onI
thf(fact_1095_mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( ord_less_eq_rat @ R3 @ S6 )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_1096_mono__onI,axiom,
! [A: set_rat,F: rat > rat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( ord_less_eq_rat @ R3 @ S6 )
=> ( ord_less_eq_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_rat @ A @ ord_less_eq_rat @ ord_less_eq_rat @ F ) ) ).
% mono_onI
thf(fact_1097_monotone__on__subset,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B2: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( monotone_on_nat_nat @ B2 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_1098_mono__inv,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) ) ) ) ).
% mono_inv
thf(fact_1099_mono__inv,axiom,
! [F: nat > rat] :
( ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ F )
=> ( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F ) ) ) ) ).
% mono_inv
thf(fact_1100_mono__inv,axiom,
! [F: rat > nat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ( monotone_on_nat_rat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F ) ) ) ) ).
% mono_inv
thf(fact_1101_mono__inv,axiom,
! [F: rat > rat] :
( ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ F )
=> ( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ( monotone_on_rat_rat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F ) ) ) ) ).
% mono_inv
thf(fact_1102_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% bij_betw_imp_inj_on
thf(fact_1103_bij__betw__the__inv__into,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( bij_betw_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ B2 @ A ) ) ).
% bij_betw_the_inv_into
thf(fact_1104_f__the__inv__into__f__bij__betw,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,X: nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( member_nat @ X @ B2 ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A @ F @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_1105_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_1106_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y2 )
=> ( Ordb @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ).
% monotoneD
thf(fact_1107_involuntory__imp__bij,axiom,
! [F: rule > rule] :
( ! [X2: rule] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ).
% involuntory_imp_bij
thf(fact_1108_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X2: nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_1109_involuntory__imp__bij,axiom,
! [F: rat > rat] :
( ! [X2: rat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ).
% involuntory_imp_bij
thf(fact_1110_bij__pointE,axiom,
! [F: rule > rule,Y2: rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
=> ~ ! [X2: rule] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rule] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1111_bij__pointE,axiom,
! [F: rule > nat,Y2: nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
=> ~ ! [X2: rule] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rule] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1112_bij__pointE,axiom,
! [F: rule > rat,Y2: rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
=> ~ ! [X2: rule] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rule] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1113_bij__pointE,axiom,
! [F: nat > rule,Y2: rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
=> ~ ! [X2: nat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1114_bij__pointE,axiom,
! [F: nat > nat,Y2: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X2: nat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1115_bij__pointE,axiom,
! [F: nat > rat,Y2: rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
=> ~ ! [X2: nat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: nat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1116_bij__pointE,axiom,
! [F: rat > rule,Y2: rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
=> ~ ! [X2: rat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1117_bij__pointE,axiom,
! [F: rat > nat,Y2: nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
=> ~ ! [X2: rat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1118_bij__pointE,axiom,
! [F: rat > rat,Y2: rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
=> ~ ! [X2: rat] :
( ( Y2
= ( F @ X2 ) )
=> ~ ! [X9: rat] :
( ( Y2
= ( F @ X9 ) )
=> ( X9 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_1119_bij__iff,axiom,
! [F: rule > rule] :
( ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y: rule] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rule] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1120_bij__iff,axiom,
! [F: rule > nat] :
( ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y: rule] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rule] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1121_bij__iff,axiom,
! [F: rule > rat] :
( ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y: rule] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rule] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1122_bij__iff,axiom,
! [F: nat > rule] :
( ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y: nat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1123_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y: nat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1124_bij__iff,axiom,
! [F: nat > rat] :
( ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y: nat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1125_bij__iff,axiom,
! [F: rat > rule] :
( ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule )
= ( ! [X3: rule] :
? [Y: rat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1126_bij__iff,axiom,
! [F: rat > nat] :
( ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y: rat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1127_bij__iff,axiom,
! [F: rat > rat] :
( ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat )
= ( ! [X3: rat] :
? [Y: rat] :
( ( ( F @ Y )
= X3 )
& ! [Z4: rat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_1128_bij__betw__imp__surj__on,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( ( image_nat_rule @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_1129_bij__betw__imp__surj__on,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( ( image_nat_rat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_1130_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( image_nat_nat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_1131_ord_Ostrict__mono__onD,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1132_ord_Ostrict__mono__onD,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat,R: rule,S2: rule] :
( ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_rule @ R @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1133_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1134_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > rat,R: nat,S2: nat] :
( ( monotone_on_nat_rat @ A @ Less @ ord_less_rat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1135_ord_Ostrict__mono__onD,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > rat,R: rat,S2: rat] :
( ( monotone_on_rat_rat @ A @ Less @ ord_less_rat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1136_ord_Ostrict__mono__onD,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > rat,R: rule,S2: rule] :
( ( monotone_on_rule_rat @ A @ Less @ ord_less_rat @ F )
=> ( ( member_rule @ R @ A )
=> ( ( member_rule @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_1137_ord_Ostrict__mono__onI,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1138_ord_Ostrict__mono__onI,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat] :
( ! [R3: rule,S6: rule] :
( ( member_rule @ R3 @ A )
=> ( ( member_rule @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1139_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1140_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > rat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_rat @ A @ Less @ ord_less_rat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1141_ord_Ostrict__mono__onI,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > rat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_rat @ A @ Less @ ord_less_rat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1142_ord_Ostrict__mono__onI,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > rat] :
( ! [R3: rule,S6: rule] :
( ( member_rule @ R3 @ A )
=> ( ( member_rule @ S6 @ A )
=> ( ( Less @ R3 @ S6 )
=> ( ord_less_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rule_rat @ A @ Less @ ord_less_rat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_1143_ord_Ostrict__mono__on__def,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R4: rat,S: rat] :
( ( ( member_rat @ R4 @ A )
& ( member_rat @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1144_ord_Ostrict__mono__on__def,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > nat] :
( ( monotone_on_rule_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R4: rule,S: rule] :
( ( ( member_rule @ R4 @ A )
& ( member_rule @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1145_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R4: nat,S: nat] :
( ( ( member_nat @ R4 @ A )
& ( member_nat @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_nat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1146_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > rat] :
( ( monotone_on_nat_rat @ A @ Less @ ord_less_rat @ F )
= ( ! [R4: nat,S: nat] :
( ( ( member_nat @ R4 @ A )
& ( member_nat @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1147_ord_Ostrict__mono__on__def,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > rat] :
( ( monotone_on_rat_rat @ A @ Less @ ord_less_rat @ F )
= ( ! [R4: rat,S: rat] :
( ( ( member_rat @ R4 @ A )
& ( member_rat @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1148_ord_Ostrict__mono__on__def,axiom,
! [A: set_rule,Less: rule > rule > $o,F: rule > rat] :
( ( monotone_on_rule_rat @ A @ Less @ ord_less_rat @ F )
= ( ! [R4: rule,S: rule] :
( ( ( member_rule @ R4 @ A )
& ( member_rule @ S @ A )
& ( Less @ R4 @ S ) )
=> ( ord_less_rat @ ( F @ R4 ) @ ( F @ S ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_1149_strict__mono__onD,axiom,
! [A: set_nat,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1150_strict__mono__onD,axiom,
! [A: set_nat,F: nat > rat,R: nat,S2: nat] :
( ( monotone_on_nat_rat @ A @ ord_less_nat @ ord_less_rat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_nat @ R @ S2 )
=> ( ord_less_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1151_strict__mono__onD,axiom,
! [A: set_rat,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1152_strict__mono__onD,axiom,
! [A: set_rat,F: rat > rat,R: rat,S2: rat] :
( ( monotone_on_rat_rat @ A @ ord_less_rat @ ord_less_rat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R @ S2 )
=> ( ord_less_rat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_1153_strict__mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( ord_less_nat @ R3 @ S6 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_1154_strict__mono__onI,axiom,
! [A: set_nat,F: nat > rat] :
( ! [R3: nat,S6: nat] :
( ( member_nat @ R3 @ A )
=> ( ( member_nat @ S6 @ A )
=> ( ( ord_less_nat @ R3 @ S6 )
=> ( ord_less_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_nat_rat @ A @ ord_less_nat @ ord_less_rat @ F ) ) ).
% strict_mono_onI
thf(fact_1155_strict__mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( ord_less_rat @ R3 @ S6 )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_1156_strict__mono__onI,axiom,
! [A: set_rat,F: rat > rat] :
( ! [R3: rat,S6: rat] :
( ( member_rat @ R3 @ A )
=> ( ( member_rat @ S6 @ A )
=> ( ( ord_less_rat @ R3 @ S6 )
=> ( ord_less_rat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) )
=> ( monotone_on_rat_rat @ A @ ord_less_rat @ ord_less_rat @ F ) ) ).
% strict_mono_onI
thf(fact_1157_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y2: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( Y2 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1158_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > rat,X: nat,Y2: nat] :
( ( monotone_on_nat_rat @ A @ ord_less_nat @ ord_less_rat @ F )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( Y2 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1159_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > nat,X: rat,Y2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( Y2 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1160_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > rat,X: rat,Y2: rat] :
( ( monotone_on_rat_rat @ A @ ord_less_rat @ ord_less_rat @ F )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( Y2 = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_1161_bij__betw__iff__bijections,axiom,
( bij_betw_rat_nat
= ( ^ [F3: rat > nat,A6: set_rat,B6: set_nat] :
? [G2: nat > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1162_bij__betw__iff__bijections,axiom,
( bij_betw_rule_nat
= ( ^ [F3: rule > nat,A6: set_rule,B6: set_nat] :
? [G2: nat > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A6 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1163_bij__betw__iff__bijections,axiom,
( bij_betw_nat_rat
= ( ^ [F3: nat > rat,A6: set_nat,B6: set_rat] :
? [G2: rat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B6 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1164_bij__betw__iff__bijections,axiom,
( bij_betw_rat_rat
= ( ^ [F3: rat > rat,A6: set_rat,B6: set_rat] :
? [G2: rat > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B6 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1165_bij__betw__iff__bijections,axiom,
( bij_betw_rule_rat
= ( ^ [F3: rule > rat,A6: set_rule,B6: set_rat] :
? [G2: rat > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A6 )
=> ( ( member_rat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B6 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1166_bij__betw__iff__bijections,axiom,
( bij_betw_nat_rule
= ( ^ [F3: nat > rule,A6: set_nat,B6: set_rule] :
? [G2: rule > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B6 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1167_bij__betw__iff__bijections,axiom,
( bij_betw_rat_rule
= ( ^ [F3: rat > rule,A6: set_rat,B6: set_rule] :
? [G2: rule > rat] :
( ! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B6 )
=> ( ( member_rat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1168_bij__betw__iff__bijections,axiom,
( bij_betw_rule_rule
= ( ^ [F3: rule > rule,A6: set_rule,B6: set_rule] :
? [G2: rule > rule] :
( ! [X3: rule] :
( ( member_rule @ X3 @ A6 )
=> ( ( member_rule @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: rule] :
( ( member_rule @ X3 @ B6 )
=> ( ( member_rule @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1169_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A6: set_nat,B6: set_nat] :
? [G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_nat @ ( F3 @ X3 ) @ B6 )
& ( ( G2 @ ( F3 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_nat @ ( G2 @ X3 ) @ A6 )
& ( ( F3 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_1170_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A6: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F3: nat > nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ! [Y: nat] :
( ( member_nat @ Y @ A6 )
=> ( ( Orda2 @ X3 @ Y )
=> ( Ordb2 @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_1171_bij__betw__apply,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat,A2: nat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1172_bij__betw__apply,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule,A2: nat] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1173_bij__betw__apply,axiom,
! [F: rat > nat,A: set_rat,B2: set_nat,A2: rat] :
( ( bij_betw_rat_nat @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1174_bij__betw__apply,axiom,
! [F: rat > rat,A: set_rat,B2: set_rat,A2: rat] :
( ( bij_betw_rat_rat @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1175_bij__betw__apply,axiom,
! [F: rat > rule,A: set_rat,B2: set_rule,A2: rat] :
( ( bij_betw_rat_rule @ F @ A @ B2 )
=> ( ( member_rat @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1176_bij__betw__apply,axiom,
! [F: rule > nat,A: set_rule,B2: set_nat,A2: rule] :
( ( bij_betw_rule_nat @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1177_bij__betw__apply,axiom,
! [F: rule > rat,A: set_rule,B2: set_rat,A2: rule] :
( ( bij_betw_rule_rat @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_rat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1178_bij__betw__apply,axiom,
! [F: rule > rule,A: set_rule,B2: set_rule,A2: rule] :
( ( bij_betw_rule_rule @ F @ A @ B2 )
=> ( ( member_rule @ A2 @ A )
=> ( member_rule @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1179_bij__betw__apply,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_1180_bij__betw__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat,A7: set_nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A @ A7 )
= ( bij_betw_nat_nat @ G @ A @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_1181_bij__betw__ball,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,Phi: nat > $o] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( Phi @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( Phi @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_1182_monotone__onI,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_1183_monotone__onD,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y2: nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( Orda @ X @ Y2 )
=> ( Ordb @ ( F @ X ) @ ( F @ Y2 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_1184_bij__betw__inv,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ? [G4: nat > nat] : ( bij_betw_nat_nat @ G4 @ B2 @ A ) ) ).
% bij_betw_inv
thf(fact_1185_bij__betwE,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_1186_bij__betw__id__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ id_nat @ A @ B2 )
= ( A = B2 ) ) ).
% bij_betw_id_iff
thf(fact_1187_bij__betw__comp__iff,axiom,
! [F: nat > nat,A: set_nat,A7: set_nat,F4: nat > nat,A8: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ A7 )
=> ( ( bij_betw_nat_nat @ F4 @ A7 @ A8 )
= ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ F4 @ F ) @ A @ A8 ) ) ) ).
% bij_betw_comp_iff
thf(fact_1188_bij__betw__trans,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,G: nat > nat,C: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( bij_betw_nat_nat @ G @ B2 @ C )
=> ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A @ C ) ) ) ).
% bij_betw_trans
thf(fact_1189_bij__betw__inv__into,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( bij_betw_nat_nat @ ( hilber3633877196798814958at_nat @ A @ F ) @ B2 @ A ) ) ).
% bij_betw_inv_into
thf(fact_1190_inv__into__inv__into__eq,axiom,
! [F: nat > nat,A: set_nat,A7: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ A7 )
=> ( ( member_nat @ A2 @ A )
=> ( ( hilber3633877196798814958at_nat @ A7 @ ( hilber3633877196798814958at_nat @ A @ F ) @ A2 )
= ( F @ A2 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_1191_bij__betw__inv__into__left,axiom,
! [F: nat > nat,A: set_nat,A7: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ A7 )
=> ( ( member_nat @ A2 @ A )
=> ( ( hilber3633877196798814958at_nat @ A @ F @ ( F @ A2 ) )
= A2 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_1192_bij__betw__inv__into__right,axiom,
! [F: nat > nat,A: set_nat,A7: set_nat,A5: nat] :
( ( bij_betw_nat_nat @ F @ A @ A7 )
=> ( ( member_nat @ A5 @ A7 )
=> ( ( F @ ( hilber3633877196798814958at_nat @ A @ F @ A5 ) )
= A5 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_1193_bij__betw__Pow,axiom,
! [F: nat > rule,A: set_nat,B2: set_rule] :
( ( bij_betw_nat_rule @ F @ A @ B2 )
=> ( bij_be6872371095225530105t_rule @ ( image_nat_rule @ F ) @ ( pow_nat @ A ) @ ( pow_rule @ B2 ) ) ) ).
% bij_betw_Pow
thf(fact_1194_bij__betw__Pow,axiom,
! [F: nat > rat,A: set_nat,B2: set_rat] :
( ( bij_betw_nat_rat @ F @ A @ B2 )
=> ( bij_be9153158031321299212et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A ) @ ( pow_rat @ B2 ) ) ) ).
% bij_betw_Pow
thf(fact_1195_bij__betw__Pow,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( bij_be3438014552859920132et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) @ ( pow_nat @ B2 ) ) ) ).
% bij_betw_Pow
thf(fact_1196_bijI_H,axiom,
! [F: rule > rule] :
( ! [X2: rule,Y3: rule] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rule] :
? [X4: rule] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rule_rule @ F @ top_top_set_rule @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1197_bijI_H,axiom,
! [F: rule > nat] :
( ! [X2: rule,Y3: rule] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: nat] :
? [X4: rule] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rule_nat @ F @ top_top_set_rule @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1198_bijI_H,axiom,
! [F: rule > rat] :
( ! [X2: rule,Y3: rule] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rat] :
? [X4: rule] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rule_rat @ F @ top_top_set_rule @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1199_bijI_H,axiom,
! [F: nat > rule] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rule] :
? [X4: nat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_nat_rule @ F @ top_top_set_nat @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1200_bijI_H,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: nat] :
? [X4: nat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1201_bijI_H,axiom,
! [F: nat > rat] :
( ! [X2: nat,Y3: nat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rat] :
? [X4: nat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_nat_rat @ F @ top_top_set_nat @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1202_bijI_H,axiom,
! [F: rat > rule] :
( ! [X2: rat,Y3: rat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rule] :
? [X4: rat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rat_rule @ F @ top_top_set_rat @ top_top_set_rule ) ) ) ).
% bijI'
thf(fact_1203_bijI_H,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y3: rat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: nat] :
? [X4: rat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rat_nat @ F @ top_top_set_rat @ top_top_set_nat ) ) ) ).
% bijI'
thf(fact_1204_bijI_H,axiom,
! [F: rat > rat] :
( ! [X2: rat,Y3: rat] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) )
=> ( ! [Y3: rat] :
? [X4: rat] :
( Y3
= ( F @ X4 ) )
=> ( bij_betw_rat_rat @ F @ top_top_set_rat @ top_top_set_rat ) ) ) ).
% bijI'
thf(fact_1205_mono__on__greaterD,axiom,
! [A: set_rat,G: rat > rat,X: rat,Y2: rat] :
( ( monotone_on_rat_rat @ A @ ord_less_eq_rat @ ord_less_eq_rat @ G )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ ( G @ Y2 ) @ ( G @ X ) )
=> ( ord_less_rat @ Y2 @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_1206_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1207_bij__betw__Suc,axiom,
! [M: set_nat,N: set_nat] :
( ( bij_betw_nat_nat @ suc @ M @ N )
= ( ( image_nat_nat @ suc @ M )
= N ) ) ).
% bij_betw_Suc
thf(fact_1208_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1209_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1210_Suc__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1211_Suc__mono,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1212_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1213_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1214_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_1215_not__less__less__Suc__eq,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1216_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1217_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1218_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1219_Suc__less__SucD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1220_less__antisym,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
=> ( M3 = N2 ) ) ) ).
% less_antisym
thf(fact_1221_Suc__less__eq2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M3 )
= ( ? [M5: nat] :
( ( M3
= ( suc @ M5 ) )
& ( ord_less_nat @ N2 @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1222_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1223_not__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M3 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1224_less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1225_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1226_less__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1227_less__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M3 @ N2 )
=> ( M3 = N2 ) ) ) ).
% less_SucE
thf(fact_1228_Suc__lessI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ( suc @ M3 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1229_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1230_Suc__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_lessD
thf(fact_1231_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1232_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1233_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_1234_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1235_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R5: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X2: nat] : ( R5 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z2: nat] :
( ( R5 @ X2 @ Y3 )
=> ( ( R5 @ Y3 @ Z2 )
=> ( R5 @ X2 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R5 @ N4 @ ( suc @ N4 ) )
=> ( R5 @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1236_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ M3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1237_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1238_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1239_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1240_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1241_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M7: nat] :
( M6
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_1242_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1243_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1244_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1245_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1246_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1247_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1248_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1249_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1250_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1251_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1252_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1253_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1254_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1255_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1256_unbounded__k__infinite,axiom,
! [K: nat,S5: set_nat] :
( ! [M7: nat] :
( ( ord_less_nat @ K @ M7 )
=> ? [N5: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ( member_nat @ N5 @ S5 ) ) )
=> ~ ( finite_finite_nat @ S5 ) ) ).
% unbounded_k_infinite
thf(fact_1257_infinite__nat__iff__unbounded,axiom,
! [S5: set_nat] :
( ( ~ ( finite_finite_nat @ S5 ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ( member_nat @ N3 @ S5 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1258_infinite__nat__iff__unbounded__le,axiom,
! [S5: set_nat] :
( ( ~ ( finite_finite_nat @ S5 ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( member_nat @ N3 @ S5 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1259_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1260_infinite__enumerate,axiom,
! [S5: set_nat] :
( ~ ( finite_finite_nat @ S5 )
=> ? [R3: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R3 )
& ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S5 ) ) ) ).
% infinite_enumerate
thf(fact_1261_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1262_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1263_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M7: nat] :
( ( P @ M7 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1264_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1265_strict__mono__enumerate,axiom,
! [S5: set_nat] :
( ~ ( finite_finite_nat @ S5 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S5 ) ) ) ).
% strict_mono_enumerate
thf(fact_1266_le__enumerate,axiom,
! [S5: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S5 )
=> ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S5 @ N2 ) ) ) ).
% le_enumerate
thf(fact_1267_range__enumerate,axiom,
! [S5: set_nat] :
( ~ ( finite_finite_nat @ S5 )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S5 ) @ top_top_set_nat )
= S5 ) ) ).
% range_enumerate
thf(fact_1268_bij__enumerate,axiom,
! [S5: set_nat] :
( ~ ( finite_finite_nat @ S5 )
=> ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S5 ) @ top_top_set_nat @ S5 ) ) ).
% bij_enumerate
thf(fact_1269_card__le__Suc__Max,axiom,
! [S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S5 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S5 ) ) ) ) ).
% card_le_Suc_Max
% Conjectures (1)
thf(conj_0,conjecture,
( ( sset_rule @ ( fair_f4564919574533178778m_rule @ rule_of_nat ) )
= top_top_set_rule ) ).
%------------------------------------------------------------------------------