%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP072^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:19 AM UTC 2026
% Result : Theorem 4.05s 2.24s
% Output : Refutation 4.05s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 11
% Syntax : Number of formulae : 66 ( 19 unt; 0 typ; 3 def)
% Number of atoms : 255 ( 109 equ; 0 cnn)
% Maximal formula atoms : 6 ( 3 avg)
% Number of connectives : 363 ( 69 ~; 47 |; 11 &; 221 @)
% ( 6 <=>; 8 =>; 0 <=; 1 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 14 con; 0-2 aty)
% Number of variables : 87 ( 18 ^; 62 !; 7 ?; 87 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_HF_Mirabelle_hf: $tType ).
thf(type_def_6,type,
nat: $tType ).
thf(type_def_8,type,
set_nat: $tType ).
thf(type_def_9,type,
set_se933006839lle_hf: $tType ).
thf(type_def_10,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_0,type,
set_se933006839lle_hf: $tType ).
thf(func_def_1,type,
set_HF_Mirabelle_hf: $tType ).
thf(func_def_2,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_3,type,
set_nat: $tType ).
thf(func_def_4,type,
nat: $tType ).
thf(func_def_5,type,
finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
thf(func_def_6,type,
finite_card_nat: set_nat > nat ).
thf(func_def_7,type,
finite90088345lle_hf: set_se933006839lle_hf > nat ).
thf(func_def_8,type,
finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
thf(func_def_9,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_10,type,
finite1450550360lle_hf: set_se933006839lle_hf > $o ).
thf(func_def_11,type,
inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
thf(func_def_12,type,
inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
thf(func_def_13,type,
inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
thf(func_def_14,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_15,type,
inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
thf(func_def_16,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_17,type,
minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_18,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_19,type,
minus_500612048lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(func_def_20,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(func_def_21,type,
zero_zero_nat: nat ).
thf(func_def_22,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_23,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(func_def_24,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_25,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_26,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_27,type,
lattic710307446hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_28,type,
lattic1974000059at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_29,type,
semiring_1_Nats_nat: set_nat ).
thf(func_def_30,type,
bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
thf(func_def_31,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_32,type,
bot_bo554042810e_hf_o: set_HF_Mirabelle_hf > $o ).
thf(func_def_34,type,
bot_bot_nat: nat ).
thf(func_def_35,type,
bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
thf(func_def_36,type,
bot_bot_set_nat: set_nat ).
thf(func_def_37,type,
bot_bo2093393035lle_hf: set_se933006839lle_hf ).
thf(func_def_38,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_39,type,
ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_40,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_41,type,
ord_le2016357975lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > $o ).
thf(func_def_42,type,
semiri2115134414it_nat: nat > nat > nat ).
thf(func_def_43,type,
semiri2013084963it_nat: nat > nat > nat ).
thf(func_def_44,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(func_def_45,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_46,type,
collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
thf(func_def_47,type,
image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_48,type,
image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
thf(func_def_49,type,
image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
thf(func_def_50,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_51,type,
image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
thf(func_def_52,type,
image_1514960916lle_hf: ( set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ) > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(func_def_53,type,
insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_54,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_55,type,
insert1636143089lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
thf(func_def_56,type,
is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
thf(func_def_57,type,
is_sin1448700567lle_hf: set_HF_Mirabelle_hf > $o ).
thf(func_def_58,type,
the_el1104322134lle_hf: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_59,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_60,type,
member_nat: nat > set_nat > $o ).
thf(func_def_61,type,
member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
thf(func_def_62,type,
z: hF_Mirabelle_hf ).
thf(func_def_77,type,
sK0: hF_Mirabelle_hf ).
thf(func_def_78,type,
sK1: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_79,type,
sK2: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(f1050,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1030,f1035,f1040,f1043,f1049]) ).
thf(f1049,plain,
( ~ spl3_1
| spl3_2 ),
inference(avatar_contradiction_clause,[status(thm)],[f1048]) ).
thf(f1048,plain,
( $false
| ~ spl3_1
| spl3_2 ),
inference(subsumption_resolution,[status(thm)],[f1047,f1028]) ).
thf(f1028,plain,
( ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
!= z )
| spl3_2 ),
inference(avatar_component_clause,[status(thm)],[f1027]) ).
thf(f1027,definition,
( spl3_2
<=> ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z ) ),
introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition]) ).
thf(f1047,plain,
( ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z )
| ~ spl3_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1045]) ).
thf(f1045,plain,
( ( $true != $true )
| ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z )
| ~ spl3_1 ),
inference(superposition,[status(thm)],[f1025,f1044]) ).
thf(f1044,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
= ( member1367349282lle_hf @ ( sK2 @ X0 ) @ ( hF_Mirabelle_hfset @ X0 ) ) )
| ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= X0 ) ),
inference(superposition,[status(thm)],[f1021,f1005]) ).
thf(f1005,plain,
! [X0: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
= X0 )
| ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= X0 ) ),
inference(definition_unfolding,[status(thm)],[f992,f985]) ).
thf(f985,plain,
( zero_z189798548lle_hf
= ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,axiom,
( zero_z189798548lle_hf
= ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_9_Zero__hf__def) ).
thf(f992,plain,
! [X0: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X0 )
| ( ( hF_Mirabelle_hinsert @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
= X0 ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f980,plain,
! [X0: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X0 )
| ( ( ( hF_Mirabelle_hinsert @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
= X0 )
& ( ( hF_Mirabelle_hmem @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2])],[f973,f979]) ).
thf(f979,plain,
! [X0: hF_Mirabelle_hf] :
( ? [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X2 @ X1 )
= X0 )
& ( $true
!= ( hF_Mirabelle_hmem @ X2 @ X1 ) ) )
=> ( ( ( hF_Mirabelle_hinsert @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
= X0 )
& ( ( hF_Mirabelle_hmem @ ( sK2 @ X0 ) @ ( sK1 @ X0 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f973,plain,
! [X0: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X0 )
| ? [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X2 @ X1 )
= X0 )
& ( $true
!= ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f620]) ).
thf(f620,plain,
! [X0: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf != X0 )
=> ~ ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X2 @ X1 )
= X0 )
=> ( $true
= ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f619]) ).
thf(f619,plain,
! [X0: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf != X0 )
=> ~ ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X2 @ X1 )
= X0 )
=> ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X6: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf != X6 )
=> ~ ! [X8: hF_Mirabelle_hf,X7: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X7 @ X8 )
= X6 )
=> ( hF_Mirabelle_hmem @ X7 @ X8 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_3_hf__cases) ).
thf(f1021,plain,
! [X2: hF_Mirabelle_hf,X0: hF_Mirabelle_hf] :
( $true
= ( member1367349282lle_hf @ X2 @ ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ X2 @ X0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1013]) ).
thf(f1013,plain,
! [X2: hF_Mirabelle_hf,X0: hF_Mirabelle_hf] :
( $true
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X2
@ ( hF_Mirabelle_hinsert @ X2 @ X0 ) ) ),
inference(equality_resolution,[status(thm)],[f1008]) ).
thf(f1008,plain,
! [X2: hF_Mirabelle_hf,X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $true
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X2
@ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) )
| ( X1 != X2 ) ),
inference(definition_unfolding,[status(thm)],[f996,f997]) ).
thf(f997,plain,
( hF_Mirabelle_hmem
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f420]) ).
thf(f420,plain,
( hF_Mirabelle_hmem
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f419]) ).
thf(f419,plain,
( hF_Mirabelle_hmem
= ( ^ [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( hF_Mirabelle_hmem
= ( ^ [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ ( hF_Mirabelle_hfset @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_5_hmem__def) ).
thf(f996,plain,
! [X2: hF_Mirabelle_hf,X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) )
| ( X1 != X2 ) ),
inference(cnf_transformation,[status(thm)],[f982]) ).
thf(f982,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( $true
= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) )
| ( ( X1 != X2 )
& ( $true
!= ( hF_Mirabelle_hmem @ X2 @ X0 ) ) ) )
& ( ( X1 = X2 )
| ( $true
= ( hF_Mirabelle_hmem @ X2 @ X0 ) )
| ( $true
!= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) ) ) ),
inference(flattening,[status(thm)],[f981]) ).
thf(f981,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( $true
= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) )
| ( ( X1 != X2 )
& ( $true
!= ( hF_Mirabelle_hmem @ X2 @ X0 ) ) ) )
& ( ( X1 = X2 )
| ( $true
= ( hF_Mirabelle_hmem @ X2 @ X0 ) )
| ( $true
!= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f598]) ).
thf(f598,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) ) )
<=> ( ( X1 = X2 )
| ( $true
= ( hF_Mirabelle_hmem @ X2 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f597]) ).
thf(f597,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_hinsert @ X1 @ X0 ) )
= ( hF_Mirabelle_hmem @ X2 @ X0 ) )
| ( X1 = X2 ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X10: hF_Mirabelle_hf,X9: hF_Mirabelle_hf,X5: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X5 @ ( hF_Mirabelle_hinsert @ X9 @ X10 ) )
= ( hF_Mirabelle_hmem @ X5 @ X10 ) )
| ( X5 = X9 ) ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_4_hmem__hinsert) ).
thf(f1025,plain,
( ! [X1: hF_Mirabelle_hf] :
( $true
!= ( member1367349282lle_hf @ X1 @ ( hF_Mirabelle_hfset @ z ) ) )
| ~ spl3_1 ),
inference(avatar_component_clause,[status(thm)],[f1024]) ).
thf(f1024,definition,
( spl3_1
<=> ! [X1: hF_Mirabelle_hf] :
( $true
!= ( member1367349282lle_hf @ X1 @ ( hF_Mirabelle_hfset @ z ) ) ) ),
introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition]) ).
thf(f1043,plain,
( ~ spl3_1
| ~ spl3_3 ),
inference(avatar_contradiction_clause,[status(thm)],[f1042]) ).
thf(f1042,plain,
( $false
| ~ spl3_1
| ~ spl3_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1041]) ).
thf(f1041,plain,
( ( $true != $true )
| ~ spl3_1
| ~ spl3_3 ),
inference(superposition,[status(thm)],[f1025,f1034]) ).
thf(f1034,plain,
( ( $true
= ( member1367349282lle_hf @ sK0 @ ( hF_Mirabelle_hfset @ z ) ) )
| ~ spl3_3 ),
inference(avatar_component_clause,[status(thm)],[f1032]) ).
thf(f1032,definition,
( spl3_3
<=> ( $true
= ( member1367349282lle_hf @ sK0 @ ( hF_Mirabelle_hfset @ z ) ) ) ),
introduced(definition,[new_symbols(naming,[spl3_3])],[avatar_definition]) ).
thf(f1040,plain,
( spl3_1
| ~ spl3_2 ),
inference(avatar_split_clause,[status(thm)],[f1039,f1027,f1024]) ).
thf(f1039,plain,
( ! [X0: hF_Mirabelle_hf] :
( $true
!= ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ z ) ) )
| ~ spl3_2 ),
inference(backward_demodulation,[status(thm)],[f1022,f1029]) ).
thf(f1029,plain,
( ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z )
| ~ spl3_2 ),
inference(avatar_component_clause,[status(thm)],[f1027]) ).
thf(f1022,plain,
! [X0: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f1000]) ).
thf(f1000,plain,
! [X0: hF_Mirabelle_hf] :
( $true
!= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X0
@ ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ),
inference(definition_unfolding,[status(thm)],[f986,f997,f985]) ).
thf(f986,plain,
! [X0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf )
!= $true ),
inference(cnf_transformation,[status(thm)],[f970]) ).
thf(f970,plain,
! [X0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf )
!= $true ),
inference(flattening,[status(thm)],[f552]) ).
thf(f552,plain,
! [X0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf )
!= $true ),
inference(fool_elimination,[status(thm)],[f551]) ).
thf(f551,plain,
! [X0: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X5: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X5 @ zero_z189798548lle_hf ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_2_hmem__hempty) ).
thf(f1035,plain,
( spl3_3
| ~ spl3_2 ),
inference(avatar_split_clause,[status(thm)],[f1016,f1027,f1032]) ).
thf(f1016,plain,
( ( $true
= ( member1367349282lle_hf @ sK0 @ ( hF_Mirabelle_hfset @ z ) ) )
| ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
!= z ) ),
inference(beta_eta_normalization,[status(thm)],[f998]) ).
thf(f998,plain,
( ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
!= z )
| ( ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ sK0
@ z )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f984,f985,f997]) ).
thf(f984,plain,
( ( zero_z189798548lle_hf != z )
| ( $true
= ( hF_Mirabelle_hmem @ sK0 @ z ) ) ),
inference(cnf_transformation,[status(thm)],[f977]) ).
thf(f977,plain,
( ( ( zero_z189798548lle_hf != z )
| ( $true
= ( hF_Mirabelle_hmem @ sK0 @ z ) ) )
& ( ( zero_z189798548lle_hf = z )
| ! [X1: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X1 @ z ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f975,f976]) ).
thf(f976,plain,
( ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ z ) )
=> ( $true
= ( hF_Mirabelle_hmem @ sK0 @ z ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f975,plain,
( ( ( zero_z189798548lle_hf != z )
| ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ z ) ) )
& ( ( zero_z189798548lle_hf = z )
| ! [X1: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X1 @ z ) ) ) ),
inference(rectify,[status(thm)],[f974]) ).
thf(f974,plain,
( ( ( zero_z189798548lle_hf != z )
| ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ z ) ) )
& ( ( zero_z189798548lle_hf = z )
| ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ z ) ) ) ),
inference(nnf_transformation,[status(thm)],[f972]) ).
thf(f972,plain,
( ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ z ) )
<~> ( zero_z189798548lle_hf = z ) ),
inference(ennf_transformation,[status(thm)],[f969]) ).
thf(f969,plain,
~ ( ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ z ) )
<=> ( zero_z189798548lle_hf = z ) ),
inference(flattening,[status(thm)],[f870]) ).
thf(f870,plain,
~ ( ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ z ) )
<=> ( zero_z189798548lle_hf = z ) ),
inference(fool_elimination,[status(thm)],[f869]) ).
thf(f869,plain,
~ ! [X0: hF_Mirabelle_hf] :
( ( ~ ( hF_Mirabelle_hmem @ X0 @ z ) )
= ( zero_z189798548lle_hf = z ) ),
inference(rectify,[status(thm)],[f353]) ).
thf(f353,negated_conjecture,
~ ! [X4: hF_Mirabelle_hf] :
( ( ~ ( hF_Mirabelle_hmem @ X4 @ z ) )
= ( zero_z189798548lle_hf = z ) ),
inference(negated_conjecture,[status(cth)],[f352]) ).
thf(f352,conjecture,
! [X4: hF_Mirabelle_hf] :
( ( ~ ( hF_Mirabelle_hmem @ X4 @ z ) )
= ( zero_z189798548lle_hf = z ) ),
file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',conj_0) ).
thf(f1030,plain,
( spl3_1
| spl3_2 ),
inference(avatar_split_clause,[status(thm)],[f1018,f1027,f1024]) ).
thf(f1018,plain,
! [X1: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z )
| ( $true
!= ( member1367349282lle_hf @ X1 @ ( hF_Mirabelle_hfset @ z ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f999]) ).
thf(f999,plain,
! [X1: hF_Mirabelle_hf] :
( ( $true
!= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X1
@ z ) )
| ( ( hF_Mirabelle_HF @ bot_bo53200981lle_hf )
= z ) ),
inference(definition_unfolding,[status(thm)],[f983,f985,f997]) ).
thf(f983,plain,
! [X1: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = z )
| ( $true
!= ( hF_Mirabelle_hmem @ X1 @ z ) ) ),
inference(cnf_transformation,[status(thm)],[f977]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP072^1 : TPTP v9.2.1. Released v7.5.0.
% 0.10/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Feb 24 08:33:32 EST 2026
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.25/0.46 ---- Original DTF file ---
% 0.25/0.46 thf(spec,logic,$$dhol).
% 0.25/0.46 %------------------------------------------------------------------------------
% 0.25/0.46 % File : ITP072^1 : TPTP v9.2.1. Released v7.5.0.
% 0.25/0.46 % Domain : Interactive Theorem Proving
% 0.25/0.46 % Problem : Sledgehammer HF problem prob_113__5324706_1
% 0.25/0.46 % Version : Especial.
% 0.25/0.46 % English :
% 0.25/0.46
% 0.25/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.25/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.25/0.46 % Source : [Des21]
% 0.25/0.46 % Names : HF/prob_113__5324706_1 [Des21]
% 0.25/0.46
% 0.25/0.46 % Status : Theorem
% 0.25/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.25/0.46 % Syntax : Number of formulae : 415 ( 154 unt; 63 typ; 0 def)
% 0.25/0.46 % Number of atoms : 1084 ( 383 equ; 0 cnn)
% 0.25/0.46 % Maximal formula atoms : 12 ( 3 avg)
% 0.25/0.46 % Number of connectives : 2828 ( 164 ~; 16 |; 87 &;2035 @)
% 0.25/0.46 % ( 0 <=>; 526 =>; 0 <=; 0 <~>)
% 0.25/0.46 % Maximal formula depth : 18 ( 6 avg)
% 0.25/0.46 % Number of types : 6 ( 5 usr)
% 0.25/0.46 % Number of type conns : 221 ( 221 >; 0 *; 0 +; 0 <<)
% 0.25/0.46 % Number of symbols : 59 ( 58 usr; 9 con; 0-2 aty)
% 0.25/0.46 % Number of variables : 898 ( 41 ^; 811 !; 46 ?; 898 :)
% 0.25/0.46 % SPC : TH0_THM_EQU_NAR
% 0.25/0.46
% 0.25/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:36:17.372
% 0.25/0.46 %------------------------------------------------------------------------------
% 0.25/0.46 % Could-be-implicit typings (5)
% 0.25/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.25/0.46 set_se933006839lle_hf: $tType ).
% 0.25/0.46
% 0.25/0.46 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 set_HF_Mirabelle_hf: $tType ).
% 0.25/0.46
% 0.25/0.46 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 hF_Mirabelle_hf: $tType ).
% 0.25/0.46
% 0.25/0.46 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.25/0.46 set_nat: $tType ).
% 0.25/0.46
% 0.25/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.25/0.46 nat: $tType ).
% 0.25/0.46
% 0.25/0.46 % Explicit typings (58)
% 0.25/0.46 thf(sy_c_Finite__Set_Ocard_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.25/0.46 finite_card_nat: set_nat > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 finite90088345lle_hf: set_se933006839lle_hf > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Finite__Set_Ofinite_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.25/0.46 finite_finite_nat: set_nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 finite1450550360lle_hf: set_se933006839lle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.25/0.46 inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.25/0.46 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.25/0.46 minus_minus_nat: nat > nat > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.25/0.46 minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.25/0.46 minus_500612048lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.25/0.46 zero_zero_nat: nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.25/0.46 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.25/0.46 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.25/0.46 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.25/0.46 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.25/0.46 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.25/0.46 lattic710307446hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.25/0.46 lattic1974000059at_nat: ( nat > nat ) > set_nat > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Nat__Onat,type,
% 0.25/0.46 semiring_1_Nats_nat: set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.25/0.46 bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.25/0.46 bot_bot_nat_o: nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_M_Eo_J,type,
% 0.25/0.46 bot_bo554042810e_hf_o: set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.25/0.46 bot_bot_o: $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.25/0.46 bot_bot_nat: nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.25/0.46 bot_bot_set_nat: set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.25/0.46 bot_bo2093393035lle_hf: set_se933006839lle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.25/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.25/0.46 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.25/0.46 ord_le2016357975lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit_001t__Nat__Onat,type,
% 0.25/0.46 semiri2115134414it_nat: nat > nat > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat,type,
% 0.25/0.46 semiri2013084963it_nat: nat > nat > nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.25/0.46 collect_nat: ( nat > $o ) > set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.25/0.46 image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.25/0.46 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 image_1514960916lle_hf: ( set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ) > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.25/0.46 insert_nat: nat > set_nat > set_nat ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Oinsert_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 insert1636143089lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Ois__singleton_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 is_sin1448700567lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_Set_Othe__elem_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 the_el1104322134lle_hf: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.25/0.46 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_member_001t__Nat__Onat,type,
% 0.25/0.46 member_nat: nat > set_nat > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_c_member_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.25/0.46 member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
% 0.25/0.46
% 0.25/0.46 thf(sy_v_z,type,
% 0.25/0.46 z: hF_Mirabelle_hf ).
% 0.25/0.46
% 0.25/0.46 % Relevant facts (351)
% 0.25/0.46 thf(fact_0_hf__ext,axiom,
% 0.25/0.46 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.25/0.46 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.25/0.46 ! [X: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( hF_Mirabelle_hmem @ X @ A )
% 0.25/0.46 = ( hF_Mirabelle_hmem @ X @ B ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % hf_ext
% 0.25/0.46 thf(fact_1_hemptyE,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % hemptyE
% 0.25/0.46 thf(fact_2_hmem__hempty,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % hmem_hempty
% 0.25/0.46 thf(fact_3_hf__cases,axiom,
% 0.25/0.46 ! [Y2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( Y2 != zero_z189798548lle_hf )
% 0.25/0.46 => ~ ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( Y2
% 0.25/0.46 = ( hF_Mirabelle_hinsert @ A3 @ B2 ) )
% 0.25/0.46 => ( hF_Mirabelle_hmem @ A3 @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % hf_cases
% 0.25/0.46 thf(fact_4_hmem__hinsert,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B3 @ C ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % hmem_hinsert
% 0.25/0.46 thf(fact_5_hmem__def,axiom,
% 0.25/0.46 ( hF_Mirabelle_hmem
% 0.25/0.46 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % hmem_def
% 0.25/0.46 thf(fact_6_zero__reorient,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( zero_z189798548lle_hf = X2 )
% 0.25/0.46 = ( X2 = zero_z189798548lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % zero_reorient
% 0.25/0.46 thf(fact_7_zero__reorient,axiom,
% 0.25/0.46 ! [X2: nat] :
% 0.25/0.46 ( ( zero_zero_nat = X2 )
% 0.25/0.46 = ( X2 = zero_zero_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % zero_reorient
% 0.25/0.46 thf(fact_8_Abs__hf__0,axiom,
% 0.25/0.46 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.25/0.46 = zero_z189798548lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % Abs_hf_0
% 0.25/0.46 thf(fact_9_Zero__hf__def,axiom,
% 0.25/0.46 ( zero_z189798548lle_hf
% 0.25/0.46 = ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % Zero_hf_def
% 0.25/0.46 thf(fact_10_push__bit__of__0,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( semiri2013084963it_nat @ N @ zero_zero_nat )
% 0.25/0.46 = zero_zero_nat ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_of_0
% 0.25/0.46 thf(fact_11_push__bit__eq__0__iff,axiom,
% 0.25/0.46 ! [N: nat,A2: nat] :
% 0.25/0.46 ( ( ( semiri2013084963it_nat @ N @ A2 )
% 0.25/0.46 = zero_zero_nat )
% 0.25/0.46 = ( A2 = zero_zero_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_eq_0_iff
% 0.25/0.46 thf(fact_12_drop__bit__of__0,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( semiri2115134414it_nat @ N @ zero_zero_nat )
% 0.25/0.46 = zero_zero_nat ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_of_0
% 0.25/0.46 thf(fact_13_Nats__0,axiom,
% 0.25/0.46 member_nat @ zero_zero_nat @ semiring_1_Nats_nat ).
% 0.25/0.46
% 0.25/0.46 % Nats_0
% 0.25/0.46 thf(fact_14_HF__hfset,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.25/0.46 = A2 ) ).
% 0.25/0.46
% 0.25/0.46 % HF_hfset
% 0.25/0.46 thf(fact_15_empty__iff,axiom,
% 0.25/0.46 ! [C: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1490636632lle_hf @ C @ bot_bo2093393035lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % empty_iff
% 0.25/0.46 thf(fact_16_empty__iff,axiom,
% 0.25/0.46 ! [C: nat] :
% 0.25/0.46 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.25/0.46
% 0.25/0.46 % empty_iff
% 0.25/0.46 thf(fact_17_empty__iff,axiom,
% 0.25/0.46 ! [C: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % empty_iff
% 0.25/0.46 thf(fact_18_all__not__in__conv,axiom,
% 0.25/0.46 ! [A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( ! [X: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1490636632lle_hf @ X @ A4 ) )
% 0.25/0.46 = ( A4 = bot_bo2093393035lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % all_not_in_conv
% 0.25/0.46 thf(fact_19_all__not__in__conv,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ( ! [X: nat] :
% 0.25/0.46 ~ ( member_nat @ X @ A4 ) )
% 0.25/0.46 = ( A4 = bot_bot_set_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % all_not_in_conv
% 0.25/0.46 thf(fact_20_all__not__in__conv,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ! [X: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1367349282lle_hf @ X @ A4 ) )
% 0.25/0.46 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % all_not_in_conv
% 0.25/0.46 thf(fact_21_Collect__empty__eq,axiom,
% 0.25/0.46 ! [P: nat > $o] :
% 0.25/0.46 ( ( ( collect_nat @ P )
% 0.25/0.46 = bot_bot_set_nat )
% 0.25/0.46 = ( ! [X: nat] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_empty_eq
% 0.25/0.46 thf(fact_22_Collect__empty__eq,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( ( collec1758573718lle_hf @ P )
% 0.25/0.46 = bot_bo2093393035lle_hf )
% 0.25/0.46 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_empty_eq
% 0.25/0.46 thf(fact_23_Collect__empty__eq,axiom,
% 0.25/0.46 ! [P: hF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( ( collec2046588256lle_hf @ P )
% 0.25/0.46 = bot_bo53200981lle_hf )
% 0.25/0.46 = ( ! [X: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_empty_eq
% 0.25/0.46 thf(fact_24_empty__Collect__eq,axiom,
% 0.25/0.46 ! [P: nat > $o] :
% 0.25/0.46 ( ( bot_bot_set_nat
% 0.25/0.46 = ( collect_nat @ P ) )
% 0.25/0.46 = ( ! [X: nat] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % empty_Collect_eq
% 0.25/0.46 thf(fact_25_empty__Collect__eq,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( bot_bo2093393035lle_hf
% 0.25/0.46 = ( collec1758573718lle_hf @ P ) )
% 0.25/0.46 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % empty_Collect_eq
% 0.25/0.46 thf(fact_26_empty__Collect__eq,axiom,
% 0.25/0.46 ! [P: hF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( bot_bo53200981lle_hf
% 0.25/0.46 = ( collec2046588256lle_hf @ P ) )
% 0.25/0.46 = ( ! [X: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( P @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % empty_Collect_eq
% 0.25/0.46 thf(fact_27_bot__apply,axiom,
% 0.25/0.46 ( bot_bo1263054448e_hf_o
% 0.25/0.46 = ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_apply
% 0.25/0.46 thf(fact_28_bot__apply,axiom,
% 0.25/0.46 ( bot_bo554042810e_hf_o
% 0.25/0.46 = ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_apply
% 0.25/0.46 thf(fact_29_hinsert__def,axiom,
% 0.25/0.46 ( hF_Mirabelle_hinsert
% 0.25/0.46 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % hinsert_def
% 0.25/0.46 thf(fact_30_hfset__HF,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.46 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
% 0.25/0.46 = A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % hfset_HF
% 0.25/0.46 thf(fact_31_zero__natural_Orsp,axiom,
% 0.25/0.46 zero_zero_nat = zero_zero_nat ).
% 0.25/0.46
% 0.25/0.46 % zero_natural.rsp
% 0.25/0.46 thf(fact_32_emptyE,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1490636632lle_hf @ A2 @ bot_bo2093393035lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % emptyE
% 0.25/0.46 thf(fact_33_emptyE,axiom,
% 0.25/0.46 ! [A2: nat] :
% 0.25/0.46 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 0.25/0.46
% 0.25/0.46 % emptyE
% 0.25/0.46 thf(fact_34_emptyE,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1367349282lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % emptyE
% 0.25/0.46 thf(fact_35_equals0D,axiom,
% 0.25/0.46 ! [A4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4 = bot_bo2093393035lle_hf )
% 0.25/0.46 => ~ ( member1490636632lle_hf @ A2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0D
% 0.25/0.46 thf(fact_36_equals0D,axiom,
% 0.25/0.46 ! [A4: set_nat,A2: nat] :
% 0.25/0.46 ( ( A4 = bot_bot_set_nat )
% 0.25/0.46 => ~ ( member_nat @ A2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0D
% 0.25/0.46 thf(fact_37_equals0D,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4 = bot_bo53200981lle_hf )
% 0.25/0.46 => ~ ( member1367349282lle_hf @ A2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0D
% 0.25/0.46 thf(fact_38_insert__absorb2,axiom,
% 0.25/0.46 ! [X2: nat,A4: set_nat] :
% 0.25/0.46 ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A4 ) )
% 0.25/0.46 = ( insert_nat @ X2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb2
% 0.25/0.46 thf(fact_39_insert__absorb2,axiom,
% 0.25/0.46 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ X2 @ A4 ) )
% 0.25/0.46 = ( insert1636143089lle_hf @ X2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb2
% 0.25/0.46 thf(fact_40_insert__absorb2,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ X2 @ A4 ) )
% 0.25/0.46 = ( insert9649339lle_hf @ X2 @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb2
% 0.25/0.46 thf(fact_41_insert__iff,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,A4: set_nat] :
% 0.25/0.46 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 | ( member_nat @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_iff
% 0.25/0.46 thf(fact_42_insert__iff,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 | ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_iff
% 0.25/0.46 thf(fact_43_insert__iff,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 | ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_iff
% 0.25/0.46 thf(fact_44_insertCI,axiom,
% 0.25/0.46 ! [A2: nat,B4: set_nat,B3: nat] :
% 0.25/0.46 ( ( ~ ( member_nat @ A2 @ B4 )
% 0.25/0.46 => ( A2 = B3 ) )
% 0.25/0.46 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertCI
% 0.25/0.46 thf(fact_45_insertCI,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ~ ( member1490636632lle_hf @ A2 @ B4 )
% 0.25/0.46 => ( A2 = B3 ) )
% 0.25/0.46 => ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertCI
% 0.25/0.46 thf(fact_46_insertCI,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
% 0.25/0.46 => ( A2 = B3 ) )
% 0.25/0.46 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertCI
% 0.25/0.46 thf(fact_47_singletonI,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonI
% 0.25/0.46 thf(fact_48_singletonI,axiom,
% 0.25/0.46 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonI
% 0.25/0.46 thf(fact_49_singletonI,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonI
% 0.25/0.46 thf(fact_50_mk__disjoint__insert,axiom,
% 0.25/0.46 ! [A2: nat,A4: set_nat] :
% 0.25/0.46 ( ( member_nat @ A2 @ A4 )
% 0.25/0.46 => ? [B5: set_nat] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert_nat @ A2 @ B5 ) )
% 0.25/0.46 & ~ ( member_nat @ A2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mk_disjoint_insert
% 0.25/0.46 thf(fact_51_mk__disjoint__insert,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ A4 )
% 0.25/0.46 => ? [B5: set_se933006839lle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert1636143089lle_hf @ A2 @ B5 ) )
% 0.25/0.46 & ~ ( member1490636632lle_hf @ A2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mk_disjoint_insert
% 0.25/0.46 thf(fact_52_mk__disjoint__insert,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.25/0.46 => ? [B5: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert9649339lle_hf @ A2 @ B5 ) )
% 0.25/0.46 & ~ ( member1367349282lle_hf @ A2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mk_disjoint_insert
% 0.25/0.46 thf(fact_53_insert__commute,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat,A4: set_nat] :
% 0.25/0.46 ( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A4 ) )
% 0.25/0.46 = ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_commute
% 0.25/0.46 thf(fact_54_insert__commute,axiom,
% 0.25/0.46 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ Y2 @ A4 ) )
% 0.25/0.46 = ( insert1636143089lle_hf @ Y2 @ ( insert1636143089lle_hf @ X2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_commute
% 0.25/0.46 thf(fact_55_insert__commute,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ Y2 @ A4 ) )
% 0.25/0.46 = ( insert9649339lle_hf @ Y2 @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_commute
% 0.25/0.46 thf(fact_56_insert__eq__iff,axiom,
% 0.25/0.46 ! [A2: nat,A4: set_nat,B3: nat,B4: set_nat] :
% 0.25/0.46 ( ~ ( member_nat @ A2 @ A4 )
% 0.25/0.46 => ( ~ ( member_nat @ B3 @ B4 )
% 0.25/0.46 => ( ( ( insert_nat @ A2 @ A4 )
% 0.25/0.46 = ( insert_nat @ B3 @ B4 ) )
% 0.25/0.46 = ( ( ( A2 = B3 )
% 0.25/0.46 => ( A4 = B4 ) )
% 0.25/0.46 & ( ( A2 != B3 )
% 0.25/0.46 => ? [C2: set_nat] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert_nat @ B3 @ C2 ) )
% 0.25/0.46 & ~ ( member_nat @ B3 @ C2 )
% 0.25/0.46 & ( B4
% 0.25/0.46 = ( insert_nat @ A2 @ C2 ) )
% 0.25/0.46 & ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_eq_iff
% 0.25/0.46 thf(fact_57_insert__eq__iff,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] :
% 0.25/0.46 ( ~ ( member1490636632lle_hf @ A2 @ A4 )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ B3 @ B4 )
% 0.25/0.46 => ( ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.25/0.46 = ( insert1636143089lle_hf @ B3 @ B4 ) )
% 0.25/0.46 = ( ( ( A2 = B3 )
% 0.25/0.46 => ( A4 = B4 ) )
% 0.25/0.46 & ( ( A2 != B3 )
% 0.25/0.46 => ? [C2: set_se933006839lle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert1636143089lle_hf @ B3 @ C2 ) )
% 0.25/0.46 & ~ ( member1490636632lle_hf @ B3 @ C2 )
% 0.25/0.46 & ( B4
% 0.25/0.46 = ( insert1636143089lle_hf @ A2 @ C2 ) )
% 0.25/0.46 & ~ ( member1490636632lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_eq_iff
% 0.25/0.46 thf(fact_58_insert__eq__iff,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ~ ( member1367349282lle_hf @ A2 @ A4 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ B3 @ B4 )
% 0.25/0.46 => ( ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.25/0.46 = ( insert9649339lle_hf @ B3 @ B4 ) )
% 0.25/0.46 = ( ( ( A2 = B3 )
% 0.25/0.46 => ( A4 = B4 ) )
% 0.25/0.46 & ( ( A2 != B3 )
% 0.25/0.46 => ? [C2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert9649339lle_hf @ B3 @ C2 ) )
% 0.25/0.46 & ~ ( member1367349282lle_hf @ B3 @ C2 )
% 0.25/0.46 & ( B4
% 0.25/0.46 = ( insert9649339lle_hf @ A2 @ C2 ) )
% 0.25/0.46 & ~ ( member1367349282lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_eq_iff
% 0.25/0.46 thf(fact_59_insert__absorb,axiom,
% 0.25/0.46 ! [A2: nat,A4: set_nat] :
% 0.25/0.46 ( ( member_nat @ A2 @ A4 )
% 0.25/0.46 => ( ( insert_nat @ A2 @ A4 )
% 0.25/0.46 = A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb
% 0.25/0.46 thf(fact_60_insert__absorb,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ A4 )
% 0.25/0.46 => ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.25/0.46 = A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb
% 0.25/0.46 thf(fact_61_insert__absorb,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.25/0.46 => ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.25/0.46 = A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_absorb
% 0.25/0.46 thf(fact_62_insert__ident,axiom,
% 0.25/0.46 ! [X2: nat,A4: set_nat,B4: set_nat] :
% 0.25/0.46 ( ~ ( member_nat @ X2 @ A4 )
% 0.25/0.46 => ( ~ ( member_nat @ X2 @ B4 )
% 0.25/0.46 => ( ( ( insert_nat @ X2 @ A4 )
% 0.25/0.46 = ( insert_nat @ X2 @ B4 ) )
% 0.25/0.46 = ( A4 = B4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_ident
% 0.25/0.46 thf(fact_63_insert__ident,axiom,
% 0.25/0.46 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.25/0.46 ( ~ ( member1490636632lle_hf @ X2 @ A4 )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ X2 @ B4 )
% 0.25/0.46 => ( ( ( insert1636143089lle_hf @ X2 @ A4 )
% 0.25/0.46 = ( insert1636143089lle_hf @ X2 @ B4 ) )
% 0.25/0.46 = ( A4 = B4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_ident
% 0.25/0.46 thf(fact_64_insert__ident,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.25/0.46 => ( ( ( insert9649339lle_hf @ X2 @ A4 )
% 0.25/0.46 = ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.25/0.46 = ( A4 = B4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_ident
% 0.25/0.46 thf(fact_65_Set_Oset__insert,axiom,
% 0.25/0.46 ! [X2: nat,A4: set_nat] :
% 0.25/0.46 ( ( member_nat @ X2 @ A4 )
% 0.25/0.46 => ~ ! [B5: set_nat] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert_nat @ X2 @ B5 ) )
% 0.25/0.46 => ( member_nat @ X2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Set.set_insert
% 0.25/0.46 thf(fact_66_Set_Oset__insert,axiom,
% 0.25/0.46 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.25/0.46 => ~ ! [B5: set_se933006839lle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert1636143089lle_hf @ X2 @ B5 ) )
% 0.25/0.46 => ( member1490636632lle_hf @ X2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Set.set_insert
% 0.25/0.46 thf(fact_67_Set_Oset__insert,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.46 => ~ ! [B5: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4
% 0.25/0.46 = ( insert9649339lle_hf @ X2 @ B5 ) )
% 0.25/0.46 => ( member1367349282lle_hf @ X2 @ B5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Set.set_insert
% 0.25/0.46 thf(fact_68_insertI2,axiom,
% 0.25/0.46 ! [A2: nat,B4: set_nat,B3: nat] :
% 0.25/0.46 ( ( member_nat @ A2 @ B4 )
% 0.25/0.46 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI2
% 0.25/0.46 thf(fact_69_insertI2,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ B4 )
% 0.25/0.46 => ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI2
% 0.25/0.46 thf(fact_70_insertI2,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.25/0.46 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI2
% 0.25/0.46 thf(fact_71_insertI1,axiom,
% 0.25/0.46 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI1
% 0.25/0.46 thf(fact_72_insertI1,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ B4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI1
% 0.25/0.46 thf(fact_73_insertI1,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertI1
% 0.25/0.46 thf(fact_74_insertE,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,A4: set_nat] :
% 0.25/0.46 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
% 0.25/0.46 => ( ( A2 != B3 )
% 0.25/0.46 => ( member_nat @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertE
% 0.25/0.46 thf(fact_75_insertE,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
% 0.25/0.46 => ( ( A2 != B3 )
% 0.25/0.46 => ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertE
% 0.25/0.46 thf(fact_76_insertE,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
% 0.25/0.46 => ( ( A2 != B3 )
% 0.25/0.46 => ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insertE
% 0.25/0.46 thf(fact_77_finite__cases,axiom,
% 0.25/0.46 ! [F: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F )
% 0.25/0.46 => ( ( F != bot_bo2093393035lle_hf )
% 0.25/0.46 => ~ ! [A5: set_se933006839lle_hf,X3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( F
% 0.25/0.46 = ( insert1636143089lle_hf @ X3 @ A5 ) )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ X3 @ A5 )
% 0.25/0.46 => ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_cases
% 0.25/0.46 thf(fact_78_finite__cases,axiom,
% 0.25/0.46 ! [F: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F )
% 0.25/0.46 => ( ( F != bot_bot_set_nat )
% 0.25/0.46 => ~ ! [A5: set_nat,X3: nat] :
% 0.25/0.46 ( ( F
% 0.25/0.46 = ( insert_nat @ X3 @ A5 ) )
% 0.25/0.46 => ( ~ ( member_nat @ X3 @ A5 )
% 0.25/0.46 => ~ ( finite_finite_nat @ A5 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_cases
% 0.25/0.46 thf(fact_79_finite__cases,axiom,
% 0.25/0.46 ! [F: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F )
% 0.25/0.46 => ( ( F != bot_bo53200981lle_hf )
% 0.25/0.46 => ~ ! [A5: set_HF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( F
% 0.25/0.46 = ( insert9649339lle_hf @ X3 @ A5 ) )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ X3 @ A5 )
% 0.25/0.46 => ~ ( finite586181922lle_hf @ A5 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_cases
% 0.25/0.46 thf(fact_80_bot__set__def,axiom,
% 0.25/0.46 ( bot_bot_set_nat
% 0.25/0.46 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_set_def
% 0.25/0.46 thf(fact_81_bot__set__def,axiom,
% 0.25/0.46 ( bot_bo2093393035lle_hf
% 0.25/0.46 = ( collec1758573718lle_hf @ bot_bo554042810e_hf_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_set_def
% 0.25/0.46 thf(fact_82_bot__set__def,axiom,
% 0.25/0.46 ( bot_bo53200981lle_hf
% 0.25/0.46 = ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_set_def
% 0.25/0.46 thf(fact_83_singleton__inject,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf )
% 0.25/0.46 = ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
% 0.25/0.46 => ( A2 = B3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_inject
% 0.25/0.46 thf(fact_84_singleton__inject,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat] :
% 0.25/0.46 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 0.25/0.46 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.25/0.46 => ( A2 = B3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_inject
% 0.25/0.46 thf(fact_85_singleton__inject,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.25/0.46 = ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 => ( A2 = B3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_inject
% 0.25/0.46 thf(fact_86_insert__not__empty,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.25/0.46 != bot_bo2093393035lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % insert_not_empty
% 0.25/0.46 thf(fact_87_insert__not__empty,axiom,
% 0.25/0.46 ! [A2: nat,A4: set_nat] :
% 0.25/0.46 ( ( insert_nat @ A2 @ A4 )
% 0.25/0.46 != bot_bot_set_nat ) ).
% 0.25/0.46
% 0.25/0.46 % insert_not_empty
% 0.25/0.46 thf(fact_88_insert__not__empty,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.25/0.46 != bot_bo53200981lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % insert_not_empty
% 0.25/0.46 thf(fact_89_doubleton__eq__iff,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf,D: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert1636143089lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
% 0.25/0.46 = ( insert1636143089lle_hf @ C @ ( insert1636143089lle_hf @ D @ bot_bo2093393035lle_hf ) ) )
% 0.25/0.46 = ( ( ( A2 = C )
% 0.25/0.46 & ( B3 = D ) )
% 0.25/0.46 | ( ( A2 = D )
% 0.25/0.46 & ( B3 = C ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % doubleton_eq_iff
% 0.25/0.46 thf(fact_90_doubleton__eq__iff,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.25/0.46 ( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.25/0.46 = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.25/0.46 = ( ( ( A2 = C )
% 0.25/0.46 & ( B3 = D ) )
% 0.25/0.46 | ( ( A2 = D )
% 0.25/0.46 & ( B3 = C ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % doubleton_eq_iff
% 0.25/0.46 thf(fact_91_doubleton__eq__iff,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert9649339lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 = ( insert9649339lle_hf @ C @ ( insert9649339lle_hf @ D @ bot_bo53200981lle_hf ) ) )
% 0.25/0.46 = ( ( ( A2 = C )
% 0.25/0.46 & ( B3 = D ) )
% 0.25/0.46 | ( ( A2 = D )
% 0.25/0.46 & ( B3 = C ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % doubleton_eq_iff
% 0.25/0.46 thf(fact_92_mem__Collect__eq,axiom,
% 0.25/0.46 ! [A2: nat,P: nat > $o] :
% 0.25/0.46 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.25/0.46 = ( P @ A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % mem_Collect_eq
% 0.25/0.46 thf(fact_93_mem__Collect__eq,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
% 0.25/0.46 = ( P @ A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % mem_Collect_eq
% 0.25/0.46 thf(fact_94_mem__Collect__eq,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( member1490636632lle_hf @ A2 @ ( collec1758573718lle_hf @ P ) )
% 0.25/0.46 = ( P @ A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % mem_Collect_eq
% 0.25/0.46 thf(fact_95_Collect__mem__eq,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ( collect_nat
% 0.25/0.46 @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
% 0.25/0.46 = A4 ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_mem_eq
% 0.25/0.46 thf(fact_96_Collect__mem__eq,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( collec2046588256lle_hf
% 0.25/0.46 @ ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
% 0.25/0.46 = A4 ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_mem_eq
% 0.25/0.46 thf(fact_97_Collect__mem__eq,axiom,
% 0.25/0.46 ! [A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( collec1758573718lle_hf
% 0.25/0.46 @ ^ [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
% 0.25/0.46 = A4 ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_mem_eq
% 0.25/0.46 thf(fact_98_Collect__cong,axiom,
% 0.25/0.46 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ! [X3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( P @ X3 )
% 0.25/0.46 = ( Q @ X3 ) )
% 0.25/0.46 => ( ( collec2046588256lle_hf @ P )
% 0.25/0.46 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_cong
% 0.25/0.46 thf(fact_99_Collect__cong,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( P @ X3 )
% 0.25/0.46 = ( Q @ X3 ) )
% 0.25/0.46 => ( ( collec1758573718lle_hf @ P )
% 0.25/0.46 = ( collec1758573718lle_hf @ Q ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_cong
% 0.25/0.46 thf(fact_100_singleton__iff,axiom,
% 0.25/0.46 ! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
% 0.25/0.46 = ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_iff
% 0.25/0.46 thf(fact_101_singleton__iff,axiom,
% 0.25/0.46 ! [B3: nat,A2: nat] :
% 0.25/0.46 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.25/0.46 = ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_iff
% 0.25/0.46 thf(fact_102_singleton__iff,axiom,
% 0.25/0.46 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 = ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_iff
% 0.25/0.46 thf(fact_103_singletonD,axiom,
% 0.25/0.46 ! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
% 0.25/0.46 => ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonD
% 0.25/0.46 thf(fact_104_singletonD,axiom,
% 0.25/0.46 ! [B3: nat,A2: nat] :
% 0.25/0.46 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.25/0.46 => ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonD
% 0.25/0.46 thf(fact_105_singletonD,axiom,
% 0.25/0.46 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 => ( B3 = A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % singletonD
% 0.25/0.46 thf(fact_106_bot__nat__def,axiom,
% 0.25/0.46 bot_bot_nat = zero_zero_nat ).
% 0.25/0.46
% 0.25/0.46 % bot_nat_def
% 0.25/0.46 thf(fact_107_finite__hfset,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_hfset
% 0.25/0.46 thf(fact_108_bot__fun__def,axiom,
% 0.25/0.46 ( bot_bo1263054448e_hf_o
% 0.25/0.46 = ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_fun_def
% 0.25/0.46 thf(fact_109_bot__fun__def,axiom,
% 0.25/0.46 ( bot_bo554042810e_hf_o
% 0.25/0.46 = ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_fun_def
% 0.25/0.46 thf(fact_110_ex__in__conv,axiom,
% 0.25/0.46 ! [A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( ? [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
% 0.25/0.46 = ( A4 != bot_bo2093393035lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % ex_in_conv
% 0.25/0.46 thf(fact_111_ex__in__conv,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
% 0.25/0.46 = ( A4 != bot_bot_set_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % ex_in_conv
% 0.25/0.46 thf(fact_112_ex__in__conv,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ? [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
% 0.25/0.46 = ( A4 != bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % ex_in_conv
% 0.25/0.46 thf(fact_113_equals0I,axiom,
% 0.25/0.46 ! [A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ! [Y3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1490636632lle_hf @ Y3 @ A4 )
% 0.25/0.46 => ( A4 = bot_bo2093393035lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0I
% 0.25/0.46 thf(fact_114_equals0I,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ! [Y3: nat] :
% 0.25/0.46 ~ ( member_nat @ Y3 @ A4 )
% 0.25/0.46 => ( A4 = bot_bot_set_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0I
% 0.25/0.46 thf(fact_115_equals0I,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ! [Y3: hF_Mirabelle_hf] :
% 0.25/0.46 ~ ( member1367349282lle_hf @ Y3 @ A4 )
% 0.25/0.46 => ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % equals0I
% 0.25/0.46 thf(fact_116_finite__insert,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ ( insert1636143089lle_hf @ A2 @ A4 ) )
% 0.25/0.46 = ( finite1450550360lle_hf @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_insert
% 0.25/0.46 thf(fact_117_finite__insert,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.25/0.46 = ( finite586181922lle_hf @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_insert
% 0.25/0.46 thf(fact_118_finite__insert,axiom,
% 0.25/0.46 ! [A2: nat,A4: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) )
% 0.25/0.46 = ( finite_finite_nat @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_insert
% 0.25/0.46 thf(fact_119_finite_Ocases,axiom,
% 0.25/0.46 ! [A2: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ A2 )
% 0.25/0.46 => ( ( A2 != bot_bo2093393035lle_hf )
% 0.25/0.46 => ~ ! [A5: set_se933006839lle_hf] :
% 0.25/0.46 ( ? [A3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( A2
% 0.25/0.46 = ( insert1636143089lle_hf @ A3 @ A5 ) )
% 0.25/0.46 => ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.cases
% 0.25/0.46 thf(fact_120_finite_Ocases,axiom,
% 0.25/0.46 ! [A2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A2 )
% 0.25/0.46 => ( ( A2 != bot_bot_set_nat )
% 0.25/0.46 => ~ ! [A5: set_nat] :
% 0.25/0.46 ( ? [A3: nat] :
% 0.25/0.46 ( A2
% 0.25/0.46 = ( insert_nat @ A3 @ A5 ) )
% 0.25/0.46 => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.cases
% 0.25/0.46 thf(fact_121_finite_Ocases,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ A2 )
% 0.25/0.46 => ( ( A2 != bot_bo53200981lle_hf )
% 0.25/0.46 => ~ ! [A5: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ? [A3: hF_Mirabelle_hf] :
% 0.25/0.46 ( A2
% 0.25/0.46 = ( insert9649339lle_hf @ A3 @ A5 ) )
% 0.25/0.46 => ~ ( finite586181922lle_hf @ A5 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.cases
% 0.25/0.46 thf(fact_122_finite_Osimps,axiom,
% 0.25/0.46 ( finite1450550360lle_hf
% 0.25/0.46 = ( ^ [A: set_se933006839lle_hf] :
% 0.25/0.46 ( ( A = bot_bo2093393035lle_hf )
% 0.25/0.46 | ? [A6: set_se933006839lle_hf,B: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A
% 0.25/0.46 = ( insert1636143089lle_hf @ B @ A6 ) )
% 0.25/0.46 & ( finite1450550360lle_hf @ A6 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.simps
% 0.25/0.46 thf(fact_123_finite_Osimps,axiom,
% 0.25/0.46 ( finite_finite_nat
% 0.25/0.46 = ( ^ [A: set_nat] :
% 0.25/0.46 ( ( A = bot_bot_set_nat )
% 0.25/0.46 | ? [A6: set_nat,B: nat] :
% 0.25/0.46 ( ( A
% 0.25/0.46 = ( insert_nat @ B @ A6 ) )
% 0.25/0.46 & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.simps
% 0.25/0.46 thf(fact_124_finite_Osimps,axiom,
% 0.25/0.46 ( finite586181922lle_hf
% 0.25/0.46 = ( ^ [A: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A = bot_bo53200981lle_hf )
% 0.25/0.46 | ? [A6: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( A
% 0.25/0.46 = ( insert9649339lle_hf @ B @ A6 ) )
% 0.25/0.46 & ( finite586181922lle_hf @ A6 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.simps
% 0.25/0.46 thf(fact_125_finite__induct,axiom,
% 0.25/0.46 ! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F )
% 0.25/0.46 => ( ( P @ bot_bo2093393035lle_hf )
% 0.25/0.46 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F2 )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_induct
% 0.25/0.46 thf(fact_126_finite__induct,axiom,
% 0.25/0.46 ! [F: set_nat,P: set_nat > $o] :
% 0.25/0.46 ( ( finite_finite_nat @ F )
% 0.25/0.46 => ( ( P @ bot_bot_set_nat )
% 0.25/0.46 => ( ! [X3: nat,F2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F2 )
% 0.25/0.46 => ( ~ ( member_nat @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_induct
% 0.25/0.46 thf(fact_127_finite__induct,axiom,
% 0.25/0.46 ! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F )
% 0.25/0.46 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F2 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_induct
% 0.25/0.46 thf(fact_128_finite_Oinducts,axiom,
% 0.25/0.46 ! [X2: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ X2 )
% 0.25/0.46 => ( ( P @ bot_bo2093393035lle_hf )
% 0.25/0.46 => ( ! [A5: set_se933006839lle_hf,A3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ A5 )
% 0.25/0.46 => ( ( P @ A5 )
% 0.25/0.46 => ( P @ ( insert1636143089lle_hf @ A3 @ A5 ) ) ) )
% 0.25/0.46 => ( P @ X2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.inducts
% 0.25/0.46 thf(fact_129_finite_Oinducts,axiom,
% 0.25/0.46 ! [X2: set_nat,P: set_nat > $o] :
% 0.25/0.46 ( ( finite_finite_nat @ X2 )
% 0.25/0.46 => ( ( P @ bot_bot_set_nat )
% 0.25/0.46 => ( ! [A5: set_nat,A3: nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A5 )
% 0.25/0.46 => ( ( P @ A5 )
% 0.25/0.46 => ( P @ ( insert_nat @ A3 @ A5 ) ) ) )
% 0.25/0.46 => ( P @ X2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.inducts
% 0.25/0.46 thf(fact_130_finite_Oinducts,axiom,
% 0.25/0.46 ! [X2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( finite586181922lle_hf @ X2 )
% 0.25/0.46 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [A5: set_HF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ A5 )
% 0.25/0.46 => ( ( P @ A5 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ A3 @ A5 ) ) ) )
% 0.25/0.46 => ( P @ X2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.inducts
% 0.25/0.46 thf(fact_131_finite__ne__induct,axiom,
% 0.25/0.46 ! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F )
% 0.25/0.46 => ( ( F != bot_bo2093393035lle_hf )
% 0.25/0.46 => ( ! [X3: set_HF_Mirabelle_hf] : ( P @ ( insert1636143089lle_hf @ X3 @ bot_bo2093393035lle_hf ) )
% 0.25/0.46 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F2 )
% 0.25/0.46 => ( ( F2 != bot_bo2093393035lle_hf )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_ne_induct
% 0.25/0.46 thf(fact_132_finite__ne__induct,axiom,
% 0.25/0.46 ! [F: set_nat,P: set_nat > $o] :
% 0.25/0.46 ( ( finite_finite_nat @ F )
% 0.25/0.46 => ( ( F != bot_bot_set_nat )
% 0.25/0.46 => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.25/0.46 => ( ! [X3: nat,F2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F2 )
% 0.25/0.46 => ( ( F2 != bot_bot_set_nat )
% 0.25/0.46 => ( ~ ( member_nat @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_ne_induct
% 0.25/0.46 thf(fact_133_finite__ne__induct,axiom,
% 0.25/0.46 ! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F )
% 0.25/0.46 => ( ( F != bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F2 )
% 0.25/0.46 => ( ( F2 != bot_bo53200981lle_hf )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_ne_induct
% 0.25/0.46 thf(fact_134_infinite__finite__induct,axiom,
% 0.25/0.46 ! [P: set_se933006839lle_hf > $o,A4: set_se933006839lle_hf] :
% 0.25/0.46 ( ! [A5: set_se933006839lle_hf] :
% 0.25/0.46 ( ~ ( finite1450550360lle_hf @ A5 )
% 0.25/0.46 => ( P @ A5 ) )
% 0.25/0.46 => ( ( P @ bot_bo2093393035lle_hf )
% 0.25/0.46 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.25/0.46 ( ( finite1450550360lle_hf @ F2 )
% 0.25/0.46 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ A4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_finite_induct
% 0.25/0.46 thf(fact_135_infinite__finite__induct,axiom,
% 0.25/0.46 ! [P: set_nat > $o,A4: set_nat] :
% 0.25/0.46 ( ! [A5: set_nat] :
% 0.25/0.46 ( ~ ( finite_finite_nat @ A5 )
% 0.25/0.46 => ( P @ A5 ) )
% 0.25/0.46 => ( ( P @ bot_bot_set_nat )
% 0.25/0.46 => ( ! [X3: nat,F2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F2 )
% 0.25/0.46 => ( ~ ( member_nat @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ A4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_finite_induct
% 0.25/0.46 thf(fact_136_infinite__finite__induct,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ! [A5: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ~ ( finite586181922lle_hf @ A5 )
% 0.25/0.46 => ( P @ A5 ) )
% 0.25/0.46 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F2 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
% 0.25/0.46 => ( P @ A4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_finite_induct
% 0.25/0.46 thf(fact_137_the__elem__eq,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( the_el1104322134lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 = X2 ) ).
% 0.25/0.46
% 0.25/0.46 % the_elem_eq
% 0.25/0.46 thf(fact_138_finite_OemptyI,axiom,
% 0.25/0.46 finite_finite_nat @ bot_bot_set_nat ).
% 0.25/0.46
% 0.25/0.46 % finite.emptyI
% 0.25/0.46 thf(fact_139_finite_OemptyI,axiom,
% 0.25/0.46 finite586181922lle_hf @ bot_bo53200981lle_hf ).
% 0.25/0.46
% 0.25/0.46 % finite.emptyI
% 0.25/0.46 thf(fact_140_infinite__imp__nonempty,axiom,
% 0.25/0.46 ! [S: set_nat] :
% 0.25/0.46 ( ~ ( finite_finite_nat @ S )
% 0.25/0.46 => ( S != bot_bot_set_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_imp_nonempty
% 0.25/0.46 thf(fact_141_infinite__imp__nonempty,axiom,
% 0.25/0.46 ! [S: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ~ ( finite586181922lle_hf @ S )
% 0.25/0.46 => ( S != bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_imp_nonempty
% 0.25/0.46 thf(fact_142_finite_OinsertI,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.46 => ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.insertI
% 0.25/0.46 thf(fact_143_finite_OinsertI,axiom,
% 0.25/0.46 ! [A4: set_nat,A2: nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A4 )
% 0.25/0.46 => ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite.insertI
% 0.25/0.46 thf(fact_144_bot__empty__eq,axiom,
% 0.25/0.46 ( bot_bo1263054448e_hf_o
% 0.25/0.46 = ( ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_empty_eq
% 0.25/0.46 thf(fact_145_Collect__empty__eq__bot,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( ( collec1758573718lle_hf @ P )
% 0.25/0.46 = bot_bo2093393035lle_hf )
% 0.25/0.46 = ( P = bot_bo554042810e_hf_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_empty_eq_bot
% 0.25/0.46 thf(fact_146_Collect__empty__eq__bot,axiom,
% 0.25/0.46 ! [P: hF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( ( collec2046588256lle_hf @ P )
% 0.25/0.46 = bot_bo53200981lle_hf )
% 0.25/0.46 = ( P = bot_bo1263054448e_hf_o ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_empty_eq_bot
% 0.25/0.46 thf(fact_147_is__singleton__the__elem,axiom,
% 0.25/0.46 ( is_sin1448700567lle_hf
% 0.25/0.46 = ( ^ [A6: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( A6
% 0.25/0.46 = ( insert9649339lle_hf @ ( the_el1104322134lle_hf @ A6 ) @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % is_singleton_the_elem
% 0.25/0.46 thf(fact_148_is__singletonI,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf] : ( is_sin1448700567lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % is_singletonI
% 0.25/0.46 thf(fact_149_Set_Ois__empty__def,axiom,
% 0.25/0.46 ( is_emp566801209lle_hf
% 0.25/0.46 = ( ^ [A6: set_HF_Mirabelle_hf] : ( A6 = bot_bo53200981lle_hf ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Set.is_empty_def
% 0.25/0.46 thf(fact_150_is__singleton__def,axiom,
% 0.25/0.46 ( is_sin1448700567lle_hf
% 0.25/0.46 = ( ^ [A6: set_HF_Mirabelle_hf] :
% 0.25/0.46 ? [X: hF_Mirabelle_hf] :
% 0.25/0.46 ( A6
% 0.25/0.46 = ( insert9649339lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % is_singleton_def
% 0.25/0.46 thf(fact_151_is__singletonE,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( is_sin1448700567lle_hf @ A4 )
% 0.25/0.46 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.25/0.46 ( A4
% 0.25/0.46 != ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % is_singletonE
% 0.25/0.46 thf(fact_152_is__singletonI_H,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( A4 != bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.25/0.46 => ( ( member1367349282lle_hf @ Y3 @ A4 )
% 0.25/0.46 => ( X3 = Y3 ) ) )
% 0.25/0.46 => ( is_sin1448700567lle_hf @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % is_singletonI'
% 0.25/0.46 thf(fact_153_card__0__eq,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A4 )
% 0.25/0.46 => ( ( ( finite_card_nat @ A4 )
% 0.25/0.46 = zero_zero_nat )
% 0.25/0.46 = ( A4 = bot_bot_set_nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % card_0_eq
% 0.25/0.46 thf(fact_154_card__0__eq,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.46 => ( ( ( finite1213132899lle_hf @ A4 )
% 0.25/0.46 = zero_zero_nat )
% 0.25/0.46 = ( A4 = bot_bo53200981lle_hf ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % card_0_eq
% 0.25/0.46 thf(fact_155_finite__subset__induct,axiom,
% 0.25/0.46 ! [F: set_nat,A4: set_nat,P: set_nat > $o] :
% 0.25/0.46 ( ( finite_finite_nat @ F )
% 0.25/0.46 => ( ( ord_less_eq_set_nat @ F @ A4 )
% 0.25/0.46 => ( ( P @ bot_bot_set_nat )
% 0.25/0.46 => ( ! [A3: nat,F2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F2 )
% 0.25/0.46 => ( ( member_nat @ A3 @ A4 )
% 0.25/0.46 => ( ~ ( member_nat @ A3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset_induct
% 0.25/0.46 thf(fact_156_finite__subset__induct,axiom,
% 0.25/0.46 ! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F )
% 0.25/0.46 => ( ( ord_le432112161lle_hf @ F @ A4 )
% 0.25/0.46 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F2 )
% 0.25/0.46 => ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ A3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset_induct
% 0.25/0.46 thf(fact_157_finite__subset__induct_H,axiom,
% 0.25/0.46 ! [F: set_nat,A4: set_nat,P: set_nat > $o] :
% 0.25/0.46 ( ( finite_finite_nat @ F )
% 0.25/0.46 => ( ( ord_less_eq_set_nat @ F @ A4 )
% 0.25/0.46 => ( ( P @ bot_bot_set_nat )
% 0.25/0.46 => ( ! [A3: nat,F2: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ F2 )
% 0.25/0.46 => ( ( member_nat @ A3 @ A4 )
% 0.25/0.46 => ( ( ord_less_eq_set_nat @ F2 @ A4 )
% 0.25/0.46 => ( ~ ( member_nat @ A3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset_induct'
% 0.25/0.46 thf(fact_158_finite__subset__induct_H,axiom,
% 0.25/0.46 ! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F )
% 0.25/0.46 => ( ( ord_le432112161lle_hf @ F @ A4 )
% 0.25/0.46 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ F2 )
% 0.25/0.46 => ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.25/0.46 => ( ( ord_le432112161lle_hf @ F2 @ A4 )
% 0.25/0.46 => ( ~ ( member1367349282lle_hf @ A3 @ F2 )
% 0.25/0.46 => ( ( P @ F2 )
% 0.25/0.46 => ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) ) )
% 0.25/0.46 => ( P @ F ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset_induct'
% 0.25/0.46 thf(fact_159_inj__on__HF,axiom,
% 0.25/0.46 inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% 0.25/0.46
% 0.25/0.46 % inj_on_HF
% 0.25/0.46 thf(fact_160_order__refl,axiom,
% 0.25/0.46 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.25/0.46
% 0.25/0.46 % order_refl
% 0.25/0.46 thf(fact_161_subsetI,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ! [X3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.25/0.46 => ( member1367349282lle_hf @ X3 @ B4 ) )
% 0.25/0.46 => ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % subsetI
% 0.25/0.46 thf(fact_162_le__zero__eq,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.25/0.46 = ( N = zero_zero_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % le_zero_eq
% 0.25/0.46 thf(fact_163_subset__empty,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf )
% 0.25/0.46 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % subset_empty
% 0.25/0.46 thf(fact_164_empty__subsetI,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A4 ) ).
% 0.25/0.46
% 0.25/0.46 % empty_subsetI
% 0.25/0.46 thf(fact_165_insert__subset,axiom,
% 0.25/0.46 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
% 0.25/0.46 = ( ( member1367349282lle_hf @ X2 @ B4 )
% 0.25/0.46 & ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_subset
% 0.25/0.46 thf(fact_166_singleton__insert__inj__eq_H,axiom,
% 0.25/0.46 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.25/0.46 = ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 & ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_insert_inj_eq'
% 0.25/0.46 thf(fact_167_singleton__insert__inj__eq,axiom,
% 0.25/0.46 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf )
% 0.25/0.46 = ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.25/0.46 = ( ( A2 = B3 )
% 0.25/0.46 & ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % singleton_insert_inj_eq
% 0.25/0.46 thf(fact_168_card_Oempty,axiom,
% 0.25/0.46 ( ( finite1213132899lle_hf @ bot_bo53200981lle_hf )
% 0.25/0.46 = zero_zero_nat ) ).
% 0.25/0.46
% 0.25/0.46 % card.empty
% 0.25/0.46 thf(fact_169_card_Oinfinite,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ~ ( finite586181922lle_hf @ A4 )
% 0.25/0.46 => ( ( finite1213132899lle_hf @ A4 )
% 0.25/0.46 = zero_zero_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % card.infinite
% 0.25/0.46 thf(fact_170_card_Oinfinite,axiom,
% 0.25/0.46 ! [A4: set_nat] :
% 0.25/0.46 ( ~ ( finite_finite_nat @ A4 )
% 0.25/0.46 => ( ( finite_card_nat @ A4 )
% 0.25/0.46 = zero_zero_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % card.infinite
% 0.25/0.46 thf(fact_171_dual__order_Oantisym,axiom,
% 0.25/0.46 ! [B3: nat,A2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( A2 = B3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % dual_order.antisym
% 0.25/0.46 thf(fact_172_dual__order_Oeq__iff,axiom,
% 0.25/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.25/0.46 = ( ^ [A: nat,B: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.25/0.46 & ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % dual_order.eq_iff
% 0.25/0.46 thf(fact_173_dual__order_Otrans,axiom,
% 0.25/0.46 ! [B3: nat,A2: nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.25/0.46 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % dual_order.trans
% 0.25/0.46 thf(fact_174_linorder__wlog,axiom,
% 0.25/0.46 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.25/0.46 ( ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.25/0.46 => ( P @ A3 @ B2 ) )
% 0.25/0.46 => ( ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( P @ B2 @ A3 )
% 0.25/0.46 => ( P @ A3 @ B2 ) )
% 0.25/0.46 => ( P @ A2 @ B3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % linorder_wlog
% 0.25/0.46 thf(fact_175_dual__order_Orefl,axiom,
% 0.25/0.46 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.25/0.46
% 0.25/0.46 % dual_order.refl
% 0.25/0.46 thf(fact_176_order__trans,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat,Z2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.25/0.46 => ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order_trans
% 0.25/0.46 thf(fact_177_order__class_Oorder_Oantisym,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.25/0.46 => ( A2 = B3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order_class.order.antisym
% 0.25/0.46 thf(fact_178_ord__le__eq__trans,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( ( B3 = C )
% 0.25/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % ord_le_eq_trans
% 0.25/0.46 thf(fact_179_ord__eq__le__trans,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,C: nat] :
% 0.25/0.46 ( ( A2 = B3 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.25/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % ord_eq_le_trans
% 0.25/0.46 thf(fact_180_order__class_Oorder_Oeq__iff,axiom,
% 0.25/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.25/0.46 = ( ^ [A: nat,B: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.25/0.46 & ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order_class.order.eq_iff
% 0.25/0.46 thf(fact_181_antisym__conv,axiom,
% 0.25/0.46 ! [Y2: nat,X2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 = ( X2 = Y2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % antisym_conv
% 0.25/0.46 thf(fact_182_le__cases3,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat,Z2: nat] :
% 0.25/0.46 ( ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
% 0.25/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
% 0.25/0.46 => ( ( ( ord_less_eq_nat @ X2 @ Z2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
% 0.25/0.46 => ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
% 0.25/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
% 0.25/0.46 => ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
% 0.25/0.46 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % le_cases3
% 0.25/0.46 thf(fact_183_order_Otrans,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.25/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order.trans
% 0.25/0.46 thf(fact_184_le__cases,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat] :
% 0.25/0.46 ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % le_cases
% 0.25/0.46 thf(fact_185_eq__refl,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat] :
% 0.25/0.46 ( ( X2 = Y2 )
% 0.25/0.46 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_refl
% 0.25/0.46 thf(fact_186_linear,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % linear
% 0.25/0.46 thf(fact_187_antisym,axiom,
% 0.25/0.46 ! [X2: nat,Y2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.25/0.46 => ( X2 = Y2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % antisym
% 0.25/0.46 thf(fact_188_eq__iff,axiom,
% 0.25/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.25/0.46 = ( ^ [X: nat,Y4: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.25/0.46 & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_iff
% 0.25/0.46 thf(fact_189_ord__le__eq__subst,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( ( ( F3 @ B3 )
% 0.25/0.46 = C )
% 0.25/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % ord_le_eq_subst
% 0.25/0.46 thf(fact_190_ord__eq__le__subst,axiom,
% 0.25/0.46 ! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
% 0.25/0.46 ( ( A2
% 0.25/0.46 = ( F3 @ B3 ) )
% 0.25/0.46 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.25/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.25/0.46 => ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % ord_eq_le_subst
% 0.25/0.46 thf(fact_191_order__subst2,axiom,
% 0.25/0.46 ! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
% 0.25/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order_subst2
% 0.25/0.46 thf(fact_192_order__subst1,axiom,
% 0.25/0.46 ! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
% 0.25/0.46 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.25/0.46 => ( ! [X3: nat,Y3: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.25/0.46 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.25/0.46 => ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % order_subst1
% 0.25/0.46 thf(fact_193_Collect__mono__iff,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) )
% 0.25/0.46 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( P @ X )
% 0.25/0.46 => ( Q @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_mono_iff
% 0.25/0.46 thf(fact_194_Collect__mono,axiom,
% 0.25/0.46 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.25/0.46 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( P @ X3 )
% 0.25/0.46 => ( Q @ X3 ) )
% 0.25/0.46 => ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Collect_mono
% 0.25/0.46 thf(fact_195_subset__iff,axiom,
% 0.25/0.46 ( ord_le432112161lle_hf
% 0.25/0.46 = ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
% 0.25/0.46 ! [T: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ T @ A6 )
% 0.25/0.46 => ( member1367349282lle_hf @ T @ B6 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % subset_iff
% 0.25/0.46 thf(fact_196_subset__eq,axiom,
% 0.25/0.46 ( ord_le432112161lle_hf
% 0.25/0.46 = ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
% 0.25/0.46 ! [X: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( member1367349282lle_hf @ X @ A6 )
% 0.25/0.46 => ( member1367349282lle_hf @ X @ B6 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % subset_eq
% 0.25/0.46 thf(fact_197_subsetD,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.46 => ( ( member1367349282lle_hf @ C @ A4 )
% 0.25/0.46 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % subsetD
% 0.25/0.46 thf(fact_198_in__mono,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.46 => ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.46 => ( member1367349282lle_hf @ X2 @ B4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % in_mono
% 0.25/0.46 thf(fact_199_card__subset__eq,axiom,
% 0.25/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.46 => ( ( ( finite1213132899lle_hf @ A4 )
% 0.25/0.46 = ( finite1213132899lle_hf @ B4 ) )
% 0.25/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % card_subset_eq
% 0.25/0.46 thf(fact_200_card__subset__eq,axiom,
% 0.25/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ B4 )
% 0.25/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.25/0.46 => ( ( ( finite_card_nat @ A4 )
% 0.25/0.46 = ( finite_card_nat @ B4 ) )
% 0.25/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % card_subset_eq
% 0.25/0.46 thf(fact_201_infinite__arbitrarily__large,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,N: nat] :
% 0.25/0.46 ( ~ ( finite586181922lle_hf @ A4 )
% 0.25/0.46 => ? [B5: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( finite586181922lle_hf @ B5 )
% 0.25/0.46 & ( ( finite1213132899lle_hf @ B5 )
% 0.25/0.46 = N )
% 0.25/0.46 & ( ord_le432112161lle_hf @ B5 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_arbitrarily_large
% 0.25/0.46 thf(fact_202_infinite__arbitrarily__large,axiom,
% 0.25/0.46 ! [A4: set_nat,N: nat] :
% 0.25/0.46 ( ~ ( finite_finite_nat @ A4 )
% 0.25/0.46 => ? [B5: set_nat] :
% 0.25/0.46 ( ( finite_finite_nat @ B5 )
% 0.25/0.46 & ( ( finite_card_nat @ B5 )
% 0.25/0.46 = N )
% 0.25/0.46 & ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_arbitrarily_large
% 0.25/0.46 thf(fact_203_zero__le,axiom,
% 0.25/0.46 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.25/0.46
% 0.25/0.46 % zero_le
% 0.25/0.46 thf(fact_204_le__numeral__extra_I3_J,axiom,
% 0.25/0.46 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.25/0.46
% 0.25/0.46 % le_numeral_extra(3)
% 0.25/0.46 thf(fact_205_bot_Oextremum,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A2 ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum
% 0.25/0.46 thf(fact_206_bot_Oextremum,axiom,
% 0.25/0.46 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum
% 0.25/0.46 thf(fact_207_bot_Oextremum__unique,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.25/0.46 = ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum_unique
% 0.25/0.46 thf(fact_208_bot_Oextremum__unique,axiom,
% 0.25/0.46 ! [A2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.25/0.46 = ( A2 = bot_bot_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum_unique
% 0.25/0.46 thf(fact_209_bot_Oextremum__uniqueI,axiom,
% 0.25/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.25/0.46 => ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum_uniqueI
% 0.25/0.46 thf(fact_210_bot_Oextremum__uniqueI,axiom,
% 0.25/0.46 ! [A2: nat] :
% 0.25/0.46 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.25/0.46 => ( A2 = bot_bot_nat ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot.extremum_uniqueI
% 0.25/0.46 thf(fact_211_finite__has__maximal2,axiom,
% 0.25/0.46 ! [A4: set_nat,A2: nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A4 )
% 0.25/0.46 => ( ( member_nat @ A2 @ A4 )
% 0.25/0.46 => ? [X3: nat] :
% 0.25/0.46 ( ( member_nat @ X3 @ A4 )
% 0.25/0.46 & ( ord_less_eq_nat @ A2 @ X3 )
% 0.25/0.46 & ! [Xa: nat] :
% 0.25/0.46 ( ( member_nat @ Xa @ A4 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ X3 @ Xa )
% 0.25/0.46 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_has_maximal2
% 0.25/0.46 thf(fact_212_finite__has__minimal2,axiom,
% 0.25/0.46 ! [A4: set_nat,A2: nat] :
% 0.25/0.46 ( ( finite_finite_nat @ A4 )
% 0.25/0.46 => ( ( member_nat @ A2 @ A4 )
% 0.25/0.46 => ? [X3: nat] :
% 0.25/0.46 ( ( member_nat @ X3 @ A4 )
% 0.25/0.46 & ( ord_less_eq_nat @ X3 @ A2 )
% 0.25/0.46 & ! [Xa: nat] :
% 0.25/0.46 ( ( member_nat @ Xa @ A4 )
% 0.25/0.46 => ( ( ord_less_eq_nat @ Xa @ X3 )
% 0.25/0.46 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_has_minimal2
% 0.25/0.46 thf(fact_213_finite__subset,axiom,
% 0.25/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.25/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset
% 0.25/0.46 thf(fact_214_finite__subset,axiom,
% 0.25/0.46 ! [A4: set_nat,B4: set_nat] :
% 0.25/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.25/0.46 => ( ( finite_finite_nat @ B4 )
% 0.25/0.46 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % finite_subset
% 0.25/0.46 thf(fact_215_infinite__super,axiom,
% 0.25/0.46 ! [S: set_HF_Mirabelle_hf,T2: set_HF_Mirabelle_hf] :
% 0.25/0.46 ( ( ord_le432112161lle_hf @ S @ T2 )
% 0.25/0.46 => ( ~ ( finite586181922lle_hf @ S )
% 0.25/0.46 => ~ ( finite586181922lle_hf @ T2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_super
% 0.25/0.46 thf(fact_216_infinite__super,axiom,
% 0.25/0.46 ! [S: set_nat,T2: set_nat] :
% 0.25/0.46 ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.25/0.46 => ( ~ ( finite_finite_nat @ S )
% 0.25/0.46 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % infinite_super
% 0.25/0.46 thf(fact_217_rev__finite__subset,axiom,
% 0.25/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % rev_finite_subset
% 0.25/0.47 thf(fact_218_rev__finite__subset,axiom,
% 0.25/0.47 ! [B4: set_nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.25/0.47 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % rev_finite_subset
% 0.25/0.47 thf(fact_219_insert__mono,axiom,
% 0.25/0.47 ! [C3: set_HF_Mirabelle_hf,D2: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ C3 @ D2 )
% 0.25/0.47 => ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ A2 @ C3 ) @ ( insert9649339lle_hf @ A2 @ D2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_mono
% 0.25/0.47 thf(fact_220_subset__insert,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.25/0.47 = ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_insert
% 0.25/0.47 thf(fact_221_subset__insertI,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] : ( ord_le432112161lle_hf @ B4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_insertI
% 0.25/0.47 thf(fact_222_subset__insertI2,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_insertI2
% 0.25/0.47 thf(fact_223_finite__has__minimal,axiom,
% 0.25/0.47 ! [A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( A4 != bot_bot_set_nat )
% 0.25/0.47 => ? [X3: nat] :
% 0.25/0.47 ( ( member_nat @ X3 @ A4 )
% 0.25/0.47 & ! [Xa: nat] :
% 0.25/0.47 ( ( member_nat @ Xa @ A4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ Xa @ X3 )
% 0.25/0.47 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_has_minimal
% 0.25/0.47 thf(fact_224_finite__has__maximal,axiom,
% 0.25/0.47 ! [A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( A4 != bot_bot_set_nat )
% 0.25/0.47 => ? [X3: nat] :
% 0.25/0.47 ( ( member_nat @ X3 @ A4 )
% 0.25/0.47 & ! [Xa: nat] :
% 0.25/0.47 ( ( member_nat @ Xa @ A4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ X3 @ Xa )
% 0.25/0.47 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_has_maximal
% 0.25/0.47 thf(fact_225_subset__singleton__iff,axiom,
% 0.25/0.47 ! [X4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ X4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.25/0.47 = ( ( X4 = bot_bo53200981lle_hf )
% 0.25/0.47 | ( X4
% 0.25/0.47 = ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_singleton_iff
% 0.25/0.47 thf(fact_226_subset__singletonD,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.25/0.47 => ( ( A4 = bot_bo53200981lle_hf )
% 0.25/0.47 | ( A4
% 0.25/0.47 = ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_singletonD
% 0.25/0.47 thf(fact_227_card__eq__0__iff,axiom,
% 0.25/0.47 ! [A4: set_nat] :
% 0.25/0.47 ( ( ( finite_card_nat @ A4 )
% 0.25/0.47 = zero_zero_nat )
% 0.25/0.47 = ( ( A4 = bot_bot_set_nat )
% 0.25/0.47 | ~ ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_eq_0_iff
% 0.25/0.47 thf(fact_228_card__eq__0__iff,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( finite1213132899lle_hf @ A4 )
% 0.25/0.47 = zero_zero_nat )
% 0.25/0.47 = ( ( A4 = bot_bo53200981lle_hf )
% 0.25/0.47 | ~ ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_eq_0_iff
% 0.25/0.47 thf(fact_229_inj__on__empty,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] : ( inj_on811196232lle_hf @ F3 @ bot_bo2093393035lle_hf ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_empty
% 0.25/0.47 thf(fact_230_finite__ranking__induct,axiom,
% 0.25/0.47 ! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
% 0.25/0.47 ( ( finite_finite_nat @ S )
% 0.25/0.47 => ( ( P @ bot_bot_set_nat )
% 0.25/0.47 => ( ! [X3: nat,S2: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ S2 )
% 0.25/0.47 => ( ! [Y5: nat] :
% 0.25/0.47 ( ( member_nat @ Y5 @ S2 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
% 0.25/0.47 => ( ( P @ S2 )
% 0.25/0.47 => ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
% 0.25/0.47 => ( P @ S ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_ranking_induct
% 0.25/0.47 thf(fact_231_finite__ranking__induct,axiom,
% 0.25/0.47 ! [S: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o,F3: hF_Mirabelle_hf > nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ S )
% 0.25/0.47 => ( ( P @ bot_bo53200981lle_hf )
% 0.25/0.47 => ( ! [X3: hF_Mirabelle_hf,S2: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ S2 )
% 0.25/0.47 => ( ! [Y5: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ Y5 @ S2 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
% 0.25/0.47 => ( ( P @ S2 )
% 0.25/0.47 => ( P @ ( insert9649339lle_hf @ X3 @ S2 ) ) ) ) )
% 0.25/0.47 => ( P @ S ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_ranking_induct
% 0.25/0.47 thf(fact_232_inj__on__subset,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ B4 @ A4 )
% 0.25/0.47 => ( inj_on811196232lle_hf @ F3 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_subset
% 0.25/0.47 thf(fact_233_subset__inj__on,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ B4 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_inj_on
% 0.25/0.47 thf(fact_234_insert__subsetI,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ X4 @ A4 )
% 0.25/0.47 => ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ X4 ) @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_subsetI
% 0.25/0.47 thf(fact_235_subset__emptyI,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ! [X3: hF_Mirabelle_hf] :
% 0.25/0.47 ~ ( member1367349282lle_hf @ X3 @ A4 )
% 0.25/0.47 => ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_emptyI
% 0.25/0.47 thf(fact_236_bot__nat__0_Oextremum,axiom,
% 0.25/0.47 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 0.25/0.47
% 0.25/0.47 % bot_nat_0.extremum
% 0.25/0.47 thf(fact_237_le0,axiom,
% 0.25/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.25/0.47
% 0.25/0.47 % le0
% 0.25/0.47 thf(fact_238_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.25/0.47 ! [A2: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.25/0.47 => ( A2 = zero_zero_nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % bot_nat_0.extremum_uniqueI
% 0.25/0.47 thf(fact_239_bot__nat__0_Oextremum__unique,axiom,
% 0.25/0.47 ! [A2: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.25/0.47 = ( A2 = zero_zero_nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % bot_nat_0.extremum_unique
% 0.25/0.47 thf(fact_240_le__0__eq,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.25/0.47 = ( N = zero_zero_nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_0_eq
% 0.25/0.47 thf(fact_241_less__eq__nat_Osimps_I1_J,axiom,
% 0.25/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.25/0.47
% 0.25/0.47 % less_eq_nat.simps(1)
% 0.25/0.47 thf(fact_242_finite__if__finite__subsets__card__bdd,axiom,
% 0.25/0.47 ! [F: set_HF_Mirabelle_hf,C3: nat] :
% 0.25/0.47 ( ! [G: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ G @ F )
% 0.25/0.47 => ( ( finite586181922lle_hf @ G )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G ) @ C3 ) ) )
% 0.25/0.47 => ( ( finite586181922lle_hf @ F )
% 0.25/0.47 & ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F ) @ C3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_if_finite_subsets_card_bdd
% 0.25/0.47 thf(fact_243_finite__if__finite__subsets__card__bdd,axiom,
% 0.25/0.47 ! [F: set_nat,C3: nat] :
% 0.25/0.47 ( ! [G: set_nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ G @ F )
% 0.25/0.47 => ( ( finite_finite_nat @ G )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
% 0.25/0.47 => ( ( finite_finite_nat @ F )
% 0.25/0.47 & ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_if_finite_subsets_card_bdd
% 0.25/0.47 thf(fact_244_card__seteq,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ B4 ) @ ( finite1213132899lle_hf @ A4 ) )
% 0.25/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_seteq
% 0.25/0.47 thf(fact_245_card__seteq,axiom,
% 0.25/0.47 ! [B4: set_nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
% 0.25/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_seteq
% 0.25/0.47 thf(fact_246_card__mono,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_mono
% 0.25/0.47 thf(fact_247_card__mono,axiom,
% 0.25/0.47 ! [B4: set_nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_mono
% 0.25/0.47 thf(fact_248_card__insert__le,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_insert_le
% 0.25/0.47 thf(fact_249_card__insert__le,axiom,
% 0.25/0.47 ! [A4: set_nat,X2: nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_insert_le
% 0.25/0.47 thf(fact_250_inj__on__inverseI,axiom,
% 0.25/0.47 ! [A4: set_se933006839lle_hf,G2: hF_Mirabelle_hf > set_HF_Mirabelle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ X3 @ A4 )
% 0.25/0.47 => ( ( G2 @ ( F3 @ X3 ) )
% 0.25/0.47 = X3 ) )
% 0.25/0.47 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_inverseI
% 0.25/0.47 thf(fact_251_inj__on__contraD,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( X2 != Y2 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.25/0.47 => ( ( F3 @ X2 )
% 0.25/0.47 != ( F3 @ Y2 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_contraD
% 0.25/0.47 thf(fact_252_inj__on__eq__iff,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.25/0.47 => ( ( ( F3 @ X2 )
% 0.25/0.47 = ( F3 @ Y2 ) )
% 0.25/0.47 = ( X2 = Y2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_eq_iff
% 0.25/0.47 thf(fact_253_inj__on__cong,axiom,
% 0.25/0.47 ! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,G2: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ! [A3: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ A3 @ A4 )
% 0.25/0.47 => ( ( F3 @ A3 )
% 0.25/0.47 = ( G2 @ A3 ) ) )
% 0.25/0.47 => ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 = ( inj_on811196232lle_hf @ G2 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_cong
% 0.25/0.47 thf(fact_254_inj__on__def,axiom,
% 0.25/0.47 ( inj_on811196232lle_hf
% 0.25/0.47 = ( ^ [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A6: set_se933006839lle_hf] :
% 0.25/0.47 ! [X: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ X @ A6 )
% 0.25/0.47 => ! [Y4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ Y4 @ A6 )
% 0.25/0.47 => ( ( ( F4 @ X )
% 0.25/0.47 = ( F4 @ Y4 ) )
% 0.25/0.47 => ( X = Y4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_def
% 0.25/0.47 thf(fact_255_inj__onI,axiom,
% 0.25/0.47 ! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ! [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ X3 @ A4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ Y3 @ A4 )
% 0.25/0.47 => ( ( ( F3 @ X3 )
% 0.25/0.47 = ( F3 @ Y3 ) )
% 0.25/0.47 => ( X3 = Y3 ) ) ) )
% 0.25/0.47 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_onI
% 0.25/0.47 thf(fact_256_inj__onD,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( ( F3 @ X2 )
% 0.25/0.47 = ( F3 @ Y2 ) )
% 0.25/0.47 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.25/0.47 => ( X2 = Y2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_onD
% 0.25/0.47 thf(fact_257_obtain__subset__with__card__n,axiom,
% 0.25/0.47 ! [N: nat,S: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_less_eq_nat @ N @ ( finite1213132899lle_hf @ S ) )
% 0.25/0.47 => ~ ! [T3: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ T3 @ S )
% 0.25/0.47 => ( ( ( finite1213132899lle_hf @ T3 )
% 0.25/0.47 = N )
% 0.25/0.47 => ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % obtain_subset_with_card_n
% 0.25/0.47 thf(fact_258_obtain__subset__with__card__n,axiom,
% 0.25/0.47 ! [N: nat,S: set_nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 0.25/0.47 => ~ ! [T3: set_nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ T3 @ S )
% 0.25/0.47 => ( ( ( finite_card_nat @ T3 )
% 0.25/0.47 = N )
% 0.25/0.47 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % obtain_subset_with_card_n
% 0.25/0.47 thf(fact_259_card__le__if__inj__on__rel,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ! [A3: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.25/0.47 => ? [B7: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ B7 @ B4 )
% 0.25/0.47 & ( R @ A3 @ B7 ) ) )
% 0.25/0.47 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ B2 @ B4 )
% 0.25/0.47 => ( ( R @ A1 @ B2 )
% 0.25/0.47 => ( ( R @ A22 @ B2 )
% 0.25/0.47 => ( A1 = A22 ) ) ) ) ) )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_if_inj_on_rel
% 0.25/0.47 thf(fact_260_card__le__if__inj__on__rel,axiom,
% 0.25/0.47 ! [B4: set_nat,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > nat > $o] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ! [A3: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.25/0.47 => ? [B7: nat] :
% 0.25/0.47 ( ( member_nat @ B7 @ B4 )
% 0.25/0.47 & ( R @ A3 @ B7 ) ) )
% 0.25/0.47 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: nat] :
% 0.25/0.47 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.25/0.47 => ( ( member_nat @ B2 @ B4 )
% 0.25/0.47 => ( ( R @ A1 @ B2 )
% 0.25/0.47 => ( ( R @ A22 @ B2 )
% 0.25/0.47 => ( A1 = A22 ) ) ) ) ) )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_if_inj_on_rel
% 0.25/0.47 thf(fact_261_arg__min__least,axiom,
% 0.25/0.47 ! [S: set_nat,Y2: nat,F3: nat > nat] :
% 0.25/0.47 ( ( finite_finite_nat @ S )
% 0.25/0.47 => ( ( S != bot_bot_set_nat )
% 0.25/0.47 => ( ( member_nat @ Y2 @ S )
% 0.25/0.47 => ( ord_less_eq_nat @ ( F3 @ ( lattic1974000059at_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % arg_min_least
% 0.25/0.47 thf(fact_262_arg__min__least,axiom,
% 0.25/0.47 ! [S: set_HF_Mirabelle_hf,Y2: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ S )
% 0.25/0.47 => ( ( S != bot_bo53200981lle_hf )
% 0.25/0.47 => ( ( member1367349282lle_hf @ Y2 @ S )
% 0.25/0.47 => ( ord_less_eq_nat @ ( F3 @ ( lattic710307446hf_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % arg_min_least
% 0.25/0.47 thf(fact_263_bounded__Max__nat,axiom,
% 0.25/0.47 ! [P: nat > $o,X2: nat,M: nat] :
% 0.25/0.47 ( ( P @ X2 )
% 0.25/0.47 => ( ! [X3: nat] :
% 0.25/0.47 ( ( P @ X3 )
% 0.25/0.47 => ( ord_less_eq_nat @ X3 @ M ) )
% 0.25/0.47 => ~ ! [M2: nat] :
% 0.25/0.47 ( ( P @ M2 )
% 0.25/0.47 => ~ ! [X5: nat] :
% 0.25/0.47 ( ( P @ X5 )
% 0.25/0.47 => ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % bounded_Max_nat
% 0.25/0.47 thf(fact_264_Nat_Oex__has__greatest__nat,axiom,
% 0.25/0.47 ! [P: nat > $o,K: nat,B3: nat] :
% 0.25/0.47 ( ( P @ K )
% 0.25/0.47 => ( ! [Y3: nat] :
% 0.25/0.47 ( ( P @ Y3 )
% 0.25/0.47 => ( ord_less_eq_nat @ Y3 @ B3 ) )
% 0.25/0.47 => ? [X3: nat] :
% 0.25/0.47 ( ( P @ X3 )
% 0.25/0.47 & ! [Y5: nat] :
% 0.25/0.47 ( ( P @ Y5 )
% 0.25/0.47 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Nat.ex_has_greatest_nat
% 0.25/0.47 thf(fact_265_nat__le__linear,axiom,
% 0.25/0.47 ! [M3: nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ M3 @ N )
% 0.25/0.47 | ( ord_less_eq_nat @ N @ M3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % nat_le_linear
% 0.25/0.47 thf(fact_266_le__antisym,axiom,
% 0.25/0.47 ! [M3: nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ M3 @ N )
% 0.25/0.47 => ( ( ord_less_eq_nat @ N @ M3 )
% 0.25/0.47 => ( M3 = N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_antisym
% 0.25/0.47 thf(fact_267_eq__imp__le,axiom,
% 0.25/0.47 ! [M3: nat,N: nat] :
% 0.25/0.47 ( ( M3 = N )
% 0.25/0.47 => ( ord_less_eq_nat @ M3 @ N ) ) ).
% 0.25/0.47
% 0.25/0.47 % eq_imp_le
% 0.25/0.47 thf(fact_268_le__trans,axiom,
% 0.25/0.47 ! [I: nat,J: nat,K: nat] :
% 0.25/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.25/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.25/0.47 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_trans
% 0.25/0.47 thf(fact_269_le__refl,axiom,
% 0.25/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.25/0.47
% 0.25/0.47 % le_refl
% 0.25/0.47 thf(fact_270_finite__nat__set__iff__bounded__le,axiom,
% 0.25/0.47 ( finite_finite_nat
% 0.25/0.47 = ( ^ [N2: set_nat] :
% 0.25/0.47 ? [M4: nat] :
% 0.25/0.47 ! [X: nat] :
% 0.25/0.47 ( ( member_nat @ X @ N2 )
% 0.25/0.47 => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_nat_set_iff_bounded_le
% 0.25/0.47 thf(fact_271_card__Diff1__le,axiom,
% 0.25/0.47 ! [A4: set_nat,X2: nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_Diff1_le
% 0.25/0.47 thf(fact_272_card__Diff1__le,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) @ ( finite1213132899lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_Diff1_le
% 0.25/0.47 thf(fact_273_card__le__inj,axiom,
% 0.25/0.47 ! [A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite1450550360lle_hf @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.25/0.47 => ? [F5: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F5 @ A4 ) @ B4 )
% 0.25/0.47 & ( inj_on811196232lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_inj
% 0.25/0.47 thf(fact_274_card__le__inj,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.25/0.47 => ? [F5: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F5 @ A4 ) @ B4 )
% 0.25/0.47 & ( inj_on755450110lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_inj
% 0.25/0.47 thf(fact_275_card__le__inj,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.25/0.47 => ? [F5: hF_Mirabelle_hf > nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F5 @ A4 ) @ B4 )
% 0.25/0.47 & ( inj_on1874279374hf_nat @ F5 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_inj
% 0.25/0.47 thf(fact_276_card__le__inj,axiom,
% 0.25/0.47 ! [A4: set_nat,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.25/0.47 => ? [F5: nat > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F5 @ A4 ) @ B4 )
% 0.25/0.47 & ( inj_on1988990670lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_inj
% 0.25/0.47 thf(fact_277_card__le__inj,axiom,
% 0.25/0.47 ! [A4: set_nat,B4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.25/0.47 => ? [F5: nat > nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B4 )
% 0.25/0.47 & ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_inj
% 0.25/0.47 thf(fact_278_card__inj__on__le,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ B4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_inj_on_le
% 0.25/0.47 thf(fact_279_image__eqI,axiom,
% 0.25/0.47 ! [B3: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( B3
% 0.25/0.47 = ( F3 @ X2 ) )
% 0.25/0.47 => ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( member1367349282lle_hf @ B3 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % image_eqI
% 0.25/0.47 thf(fact_280_DiffI,axiom,
% 0.25/0.47 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ C @ A4 )
% 0.25/0.47 => ( ~ ( member1367349282lle_hf @ C @ B4 )
% 0.25/0.47 => ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % DiffI
% 0.25/0.47 thf(fact_281_Diff__iff,axiom,
% 0.25/0.47 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.25/0.47 = ( ( member1367349282lle_hf @ C @ A4 )
% 0.25/0.47 & ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_iff
% 0.25/0.47 thf(fact_282_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.25/0.47 ! [A2: nat] :
% 0.25/0.47 ( ( minus_minus_nat @ A2 @ A2 )
% 0.25/0.47 = zero_zero_nat ) ).
% 0.25/0.47
% 0.25/0.47 % cancel_comm_monoid_add_class.diff_cancel
% 0.25/0.47 thf(fact_283_diff__zero,axiom,
% 0.25/0.47 ! [A2: nat] :
% 0.25/0.47 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 0.25/0.47 = A2 ) ).
% 0.25/0.47
% 0.25/0.47 % diff_zero
% 0.25/0.47 thf(fact_284_zero__diff,axiom,
% 0.25/0.47 ! [A2: nat] :
% 0.25/0.47 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 0.25/0.47 = zero_zero_nat ) ).
% 0.25/0.47
% 0.25/0.47 % zero_diff
% 0.25/0.47 thf(fact_285_image__empty,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( image_1743964010lle_hf @ F3 @ bot_bo53200981lle_hf )
% 0.25/0.47 = bot_bo53200981lle_hf ) ).
% 0.25/0.47
% 0.25/0.47 % image_empty
% 0.25/0.47 thf(fact_286_empty__is__image,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( bot_bo53200981lle_hf
% 0.25/0.47 = ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.47
% 0.25/0.47 % empty_is_image
% 0.25/0.47 thf(fact_287_image__is__empty,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( image_1743964010lle_hf @ F3 @ A4 )
% 0.25/0.47 = bot_bo53200981lle_hf )
% 0.25/0.47 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.25/0.47
% 0.25/0.47 % image_is_empty
% 0.25/0.47 thf(fact_288_finite__imageI,axiom,
% 0.25/0.47 ! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ F )
% 0.25/0.47 => ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageI
% 0.25/0.47 thf(fact_289_finite__imageI,axiom,
% 0.25/0.47 ! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ F )
% 0.25/0.47 => ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageI
% 0.25/0.47 thf(fact_290_finite__imageI,axiom,
% 0.25/0.47 ! [F: set_nat,H: nat > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite_finite_nat @ F )
% 0.25/0.47 => ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageI
% 0.25/0.47 thf(fact_291_finite__imageI,axiom,
% 0.25/0.47 ! [F: set_nat,H: nat > nat] :
% 0.25/0.47 ( ( finite_finite_nat @ F )
% 0.25/0.47 => ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageI
% 0.25/0.47 thf(fact_292_image__insert,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( image_1743964010lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.25/0.47 = ( insert9649339lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % image_insert
% 0.25/0.47 thf(fact_293_insert__image,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( insert9649339lle_hf @ ( F3 @ X2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_image
% 0.25/0.47 thf(fact_294_Diff__empty,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( minus_1450406810lle_hf @ A4 @ bot_bo53200981lle_hf )
% 0.25/0.47 = A4 ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_empty
% 0.25/0.47 thf(fact_295_empty__Diff,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A4 )
% 0.25/0.47 = bot_bo53200981lle_hf ) ).
% 0.25/0.47
% 0.25/0.47 % empty_Diff
% 0.25/0.47 thf(fact_296_Diff__cancel,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( minus_1450406810lle_hf @ A4 @ A4 )
% 0.25/0.47 = bot_bo53200981lle_hf ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_cancel
% 0.25/0.47 thf(fact_297_finite__Diff,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff
% 0.25/0.47 thf(fact_298_finite__Diff,axiom,
% 0.25/0.47 ! [A4: set_nat,B4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff
% 0.25/0.47 thf(fact_299_finite__Diff2,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.25/0.47 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff2
% 0.25/0.47 thf(fact_300_finite__Diff2,axiom,
% 0.25/0.47 ! [B4: set_nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
% 0.25/0.47 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff2
% 0.25/0.47 thf(fact_301_Diff__insert0,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.25/0.47 = ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_insert0
% 0.25/0.47 thf(fact_302_insert__Diff1,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ X2 @ B4 )
% 0.25/0.47 => ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
% 0.25/0.47 = ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_Diff1
% 0.25/0.47 thf(fact_303_Diff__eq__empty__iff,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( minus_1450406810lle_hf @ A4 @ B4 )
% 0.25/0.47 = bot_bo53200981lle_hf )
% 0.25/0.47 = ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_eq_empty_iff
% 0.25/0.47 thf(fact_304_insert__Diff__single,axiom,
% 0.25/0.47 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
% 0.25/0.47 = ( insert9649339lle_hf @ A2 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_Diff_single
% 0.25/0.47 thf(fact_305_finite__Diff__insert,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) )
% 0.25/0.47 = ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff_insert
% 0.25/0.47 thf(fact_306_finite__Diff__insert,axiom,
% 0.25/0.47 ! [A4: set_nat,A2: nat,B4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A2 @ B4 ) ) )
% 0.25/0.47 = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_Diff_insert
% 0.25/0.47 thf(fact_307_inj__on__insert,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ ( insert1636143089lle_hf @ A2 @ A4 ) )
% 0.25/0.47 = ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 & ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_insert
% 0.25/0.47 thf(fact_308_inj__on__insert,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on755450110lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.25/0.47 = ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.25/0.47 & ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_insert
% 0.25/0.47 thf(fact_309_inj__on__image__set__diff,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ C3 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ ( minus_500612048lle_hf @ A4 @ B4 ) @ C3 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ B4 @ C3 )
% 0.25/0.47 => ( ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ B4 ) )
% 0.25/0.47 = ( minus_1450406810lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ ( image_899003828lle_hf @ F3 @ B4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_image_set_diff
% 0.25/0.47 thf(fact_310_image__subsetI,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ! [X3: hF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.25/0.47 => ( member1367349282lle_hf @ ( F3 @ X3 ) @ B4 ) )
% 0.25/0.47 => ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % image_subsetI
% 0.25/0.47 thf(fact_311_finite__surj,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_surj
% 0.25/0.47 thf(fact_312_finite__surj,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_nat,F3: hF_Mirabelle_hf > nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ A4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.25/0.47 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_surj
% 0.25/0.47 thf(fact_313_finite__surj,axiom,
% 0.25/0.47 ! [A4: set_nat,B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_surj
% 0.25/0.47 thf(fact_314_finite__surj,axiom,
% 0.25/0.47 ! [A4: set_nat,B4: set_nat,F3: nat > nat] :
% 0.25/0.47 ( ( finite_finite_nat @ A4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.25/0.47 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_surj
% 0.25/0.47 thf(fact_315_finite__subset__image,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ? [C4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.25/0.47 & ( finite586181922lle_hf @ C4 )
% 0.25/0.47 & ( B4
% 0.25/0.47 = ( image_1743964010lle_hf @ F3 @ C4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_subset_image
% 0.25/0.47 thf(fact_316_finite__subset__image,axiom,
% 0.25/0.47 ! [B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ? [C4: set_nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.25/0.47 & ( finite_finite_nat @ C4 )
% 0.25/0.47 & ( B4
% 0.25/0.47 = ( image_246164834lle_hf @ F3 @ C4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_subset_image
% 0.25/0.47 thf(fact_317_finite__subset__image,axiom,
% 0.25/0.47 ! [B4: set_nat,F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.25/0.47 => ? [C4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.25/0.47 & ( finite586181922lle_hf @ C4 )
% 0.25/0.47 & ( B4
% 0.25/0.47 = ( image_131453538hf_nat @ F3 @ C4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_subset_image
% 0.25/0.47 thf(fact_318_finite__subset__image,axiom,
% 0.25/0.47 ! [B4: set_nat,F3: nat > nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B4 )
% 0.25/0.47 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.25/0.47 => ? [C4: set_nat] :
% 0.25/0.47 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.25/0.47 & ( finite_finite_nat @ C4 )
% 0.25/0.47 & ( B4
% 0.25/0.47 = ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_subset_image
% 0.25/0.47 thf(fact_319_ex__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.47 ( ( ? [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 & ( P @ B6 ) ) )
% 0.25/0.47 = ( ? [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ A4 )
% 0.25/0.47 & ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % ex_finite_subset_image
% 0.25/0.47 thf(fact_320_ex__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.47 ( ( ? [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.25/0.47 & ( P @ B6 ) ) )
% 0.25/0.47 = ( ? [B6: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ A4 )
% 0.25/0.47 & ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % ex_finite_subset_image
% 0.25/0.47 thf(fact_321_ex__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.25/0.47 ( ( ? [B6: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.25/0.47 & ( P @ B6 ) ) )
% 0.25/0.47 = ( ? [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ A4 )
% 0.25/0.47 & ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % ex_finite_subset_image
% 0.25/0.47 thf(fact_322_ex__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.25/0.47 ( ( ? [B6: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.25/0.47 & ( P @ B6 ) ) )
% 0.25/0.47 = ( ? [B6: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ A4 )
% 0.25/0.47 & ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % ex_finite_subset_image
% 0.25/0.47 thf(fact_323_all__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.47 ( ( ! [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) )
% 0.25/0.47 => ( P @ B6 ) ) )
% 0.25/0.47 = ( ! [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ A4 ) )
% 0.25/0.47 => ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % all_finite_subset_image
% 0.25/0.47 thf(fact_324_all__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.25/0.47 ( ( ! [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) ) )
% 0.25/0.47 => ( P @ B6 ) ) )
% 0.25/0.47 = ( ! [B6: set_nat] :
% 0.25/0.47 ( ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ A4 ) )
% 0.25/0.47 => ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % all_finite_subset_image
% 0.25/0.47 thf(fact_325_all__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.25/0.47 ( ( ! [B6: set_nat] :
% 0.25/0.47 ( ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) ) )
% 0.25/0.47 => ( P @ B6 ) ) )
% 0.25/0.47 = ( ! [B6: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ( finite586181922lle_hf @ B6 )
% 0.25/0.47 & ( ord_le432112161lle_hf @ B6 @ A4 ) )
% 0.25/0.47 => ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % all_finite_subset_image
% 0.25/0.47 thf(fact_326_all__finite__subset__image,axiom,
% 0.25/0.47 ! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.25/0.47 ( ( ! [B6: set_nat] :
% 0.25/0.47 ( ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) ) )
% 0.25/0.47 => ( P @ B6 ) ) )
% 0.25/0.47 = ( ! [B6: set_nat] :
% 0.25/0.47 ( ( ( finite_finite_nat @ B6 )
% 0.25/0.47 & ( ord_less_eq_set_nat @ B6 @ A4 ) )
% 0.25/0.47 => ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % all_finite_subset_image
% 0.25/0.47 thf(fact_327_finite__imageD,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageD
% 0.25/0.47 thf(fact_328_finite__imageD,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageD
% 0.25/0.47 thf(fact_329_finite__imageD,axiom,
% 0.25/0.47 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.25/0.47 ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ( ( inj_on1988990670lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageD
% 0.25/0.47 thf(fact_330_finite__imageD,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.25/0.47 => ( ( inj_on1874279374hf_nat @ F3 @ A4 )
% 0.25/0.47 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageD
% 0.25/0.47 thf(fact_331_finite__imageD,axiom,
% 0.25/0.47 ! [F3: nat > nat,A4: set_nat] :
% 0.25/0.47 ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
% 0.25/0.47 => ( ( inj_on_nat_nat @ F3 @ A4 )
% 0.25/0.47 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_imageD
% 0.25/0.47 thf(fact_332_finite__image__iff,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_image_iff
% 0.25/0.47 thf(fact_333_finite__image__iff,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_image_iff
% 0.25/0.47 thf(fact_334_finite__image__iff,axiom,
% 0.25/0.47 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.25/0.47 ( ( inj_on1988990670lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_image_iff
% 0.25/0.47 thf(fact_335_finite__image__iff,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on1874279374hf_nat @ F3 @ A4 )
% 0.25/0.47 => ( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.25/0.47 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_image_iff
% 0.25/0.47 thf(fact_336_finite__image__iff,axiom,
% 0.25/0.47 ! [F3: nat > nat,A4: set_nat] :
% 0.25/0.47 ( ( inj_on_nat_nat @ F3 @ A4 )
% 0.25/0.47 => ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
% 0.25/0.47 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_image_iff
% 0.25/0.47 thf(fact_337_inj__on__image__eq__iff,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ C3 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ A4 @ C3 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ B4 @ C3 )
% 0.25/0.47 => ( ( ( image_899003828lle_hf @ F3 @ A4 )
% 0.25/0.47 = ( image_899003828lle_hf @ F3 @ B4 ) )
% 0.25/0.47 = ( A4 = B4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_image_eq_iff
% 0.25/0.47 thf(fact_338_inj__on__image__mem__iff,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ B4 )
% 0.25/0.47 => ( ( member1490636632lle_hf @ A2 @ B4 )
% 0.25/0.47 => ( ( ord_le2016357975lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( member1490636632lle_hf @ A2 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_image_mem_iff
% 0.25/0.47 thf(fact_339_inj__on__image__mem__iff,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on755450110lle_hf @ F3 @ B4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.25/0.47 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.25/0.47 => ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( member1367349282lle_hf @ A2 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_on_image_mem_iff
% 0.25/0.47 thf(fact_340_inj__img__insertE,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.25/0.47 => ( ( ( insert9649339lle_hf @ X2 @ B4 )
% 0.25/0.47 = ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ~ ! [X6: set_HF_Mirabelle_hf,A7: set_se933006839lle_hf] :
% 0.25/0.47 ( ~ ( member1490636632lle_hf @ X6 @ A7 )
% 0.25/0.47 => ( ( A4
% 0.25/0.47 = ( insert1636143089lle_hf @ X6 @ A7 ) )
% 0.25/0.47 => ( ( X2
% 0.25/0.47 = ( F3 @ X6 ) )
% 0.25/0.47 => ( B4
% 0.25/0.47 != ( image_899003828lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_img_insertE
% 0.25/0.47 thf(fact_341_inj__img__insertE,axiom,
% 0.25/0.47 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.25/0.47 => ( ( ( insert9649339lle_hf @ X2 @ B4 )
% 0.25/0.47 = ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.25/0.47 => ~ ! [X6: hF_Mirabelle_hf,A7: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ~ ( member1367349282lle_hf @ X6 @ A7 )
% 0.25/0.47 => ( ( A4
% 0.25/0.47 = ( insert9649339lle_hf @ X6 @ A7 ) )
% 0.25/0.47 => ( ( X2
% 0.25/0.47 = ( F3 @ X6 ) )
% 0.25/0.47 => ( B4
% 0.25/0.47 != ( image_1743964010lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % inj_img_insertE
% 0.25/0.47 thf(fact_342_card__image,axiom,
% 0.25/0.47 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.25/0.47 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.25/0.47 => ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.25/0.47 = ( finite90088345lle_hf @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_image
% 0.25/0.47 thf(fact_343_Diff__insert__absorb,axiom,
% 0.25/0.47 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 => ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.25/0.47 = A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_insert_absorb
% 0.25/0.47 thf(fact_344_Diff__insert2,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.25/0.47 = ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_insert2
% 0.25/0.47 thf(fact_345_insert__Diff,axiom,
% 0.25/0.47 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.25/0.47 => ( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
% 0.25/0.47 = A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % insert_Diff
% 0.25/0.47 thf(fact_346_Diff__insert,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.25/0.47 = ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Diff_insert
% 0.25/0.47 thf(fact_347_subset__Diff__insert,axiom,
% 0.25/0.47 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,C3: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ ( insert9649339lle_hf @ X2 @ C3 ) ) )
% 0.25/0.47 = ( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ C3 ) )
% 0.25/0.47 & ~ ( member1367349282lle_hf @ X2 @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_Diff_insert
% 0.25/0.47 thf(fact_348_in__image__insert__iff,axiom,
% 0.25/0.47 ! [B4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ! [C4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1490636632lle_hf @ C4 @ B4 )
% 0.25/0.47 => ~ ( member1367349282lle_hf @ X2 @ C4 ) )
% 0.25/0.47 => ( ( member1490636632lle_hf @ A4 @ ( image_1514960916lle_hf @ ( insert9649339lle_hf @ X2 ) @ B4 ) )
% 0.25/0.47 = ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.25/0.47 & ( member1490636632lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_image_insert_iff
% 0.25/0.47 thf(fact_349_DiffE,axiom,
% 0.25/0.47 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.25/0.47 => ~ ( ( member1367349282lle_hf @ C @ A4 )
% 0.25/0.47 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % DiffE
% 0.25/0.47 thf(fact_350_DiffD1,axiom,
% 0.25/0.47 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.25/0.47 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.25/0.47 => ( member1367349282lle_hf @ C @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % DiffD1
% 0.25/0.47
% 0.25/0.47 % Conjectures (1)
% 0.25/0.47 thf(conj_0,conjecture,
% 0.25/0.47 ( ( z = zero_z189798548lle_hf )
% 0.25/0.47 = ( ! [X: hF_Mirabelle_hf] :
% 0.25/0.47 ~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
% 0.25/0.47
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 ------------------------
% 3.58/2.03 ---- Embedded in THF ---
% 0.26/2.03 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.26/2.03 %%% Generated on Tue Feb 24 08:33:33 EST 2026
% 0.26/2.03 %%% using '$$dhol' embedding, version 1.3.0.
% 0.26/2.03 %%% Logic specification used:
% 0.26/2.03 %%% thf(spec, logic, $$dhol).
% 0.26/2.03
% 0.26/2.03 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2DTF20378.p
% See solution above
% 0.26/2.03 ------------------------
% 0.26/2.08 ---- Cleaned THF ---
% 0.26/2.08 thf(ty_n_t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.26/2.08 set_se933006839lle_hf: $tType ).
% 0.26/2.08
% 0.26/2.08 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 set_HF_Mirabelle_hf: $tType ).
% 0.26/2.08
% 0.26/2.08 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 hF_Mirabelle_hf: $tType ).
% 0.26/2.08
% 0.26/2.08 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.26/2.08 set_nat: $tType ).
% 0.26/2.08
% 0.26/2.08 thf(ty_n_t__Nat__Onat,type,
% 0.26/2.08 nat: $tType ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ocard_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.26/2.08 finite_card_nat: set_nat > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 finite90088345lle_hf: set_se933006839lle_hf > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ofinite_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.26/2.08 finite_finite_nat: set_nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 finite1450550360lle_hf: set_se933006839lle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.26/2.08 inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.26/2.08 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.26/2.08 minus_minus_nat: nat > nat > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.26/2.08 minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.26/2.08 minus_500612048lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.26/2.08 zero_zero_nat: nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.26/2.08 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.26/2.08 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.26/2.08 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.26/2.08 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.26/2.08 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.26/2.08 lattic710307446hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.26/2.08 lattic1974000059at_nat: ( nat > nat ) > set_nat > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Nat__Onat,type,
% 0.26/2.08 semiring_1_Nats_nat: set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.26/2.08 bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.26/2.08 bot_bot_nat_o: nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_M_Eo_J,type,
% 0.26/2.08 bot_bo554042810e_hf_o: set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.26/2.08 bot_bot_o: $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.26/2.08 bot_bot_nat: nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.26/2.08 bot_bot_set_nat: set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.26/2.08 bot_bo2093393035lle_hf: set_se933006839lle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.26/2.08 ord_less_eq_nat: nat > nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.26/2.08 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.26/2.08 ord_le2016357975lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit_001t__Nat__Onat,type,
% 0.26/2.08 semiri2115134414it_nat: nat > nat > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat,type,
% 0.26/2.08 semiri2013084963it_nat: nat > nat > nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.26/2.08 collect_nat: ( nat > $o ) > set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_OCollect_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.26/2.08 image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.26/2.08 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 image_1514960916lle_hf: ( set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ) > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.26/2.08 insert_nat: nat > set_nat > set_nat ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Oinsert_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 insert1636143089lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > set_se933006839lle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Ois__singleton_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 is_sin1448700567lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_Set_Othe__elem_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 the_el1104322134lle_hf: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/2.08 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_member_001t__Nat__Onat,type,
% 0.26/2.08 member_nat: nat > set_nat > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_c_member_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/2.08 member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
% 0.26/2.08
% 0.26/2.08 thf(sy_v_z,type,
% 0.26/2.08 z: hF_Mirabelle_hf ).
% 0.26/2.08
% 0.26/2.08 thf(fact_0_hf__ext,axiom,
% 0.26/2.08 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.26/2.08 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/2.08 ! [X: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( hF_Mirabelle_hmem @ X @ A )
% 0.26/2.08 = ( hF_Mirabelle_hmem @ X @ B ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_1_hemptyE,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_2_hmem__hempty,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_3_hf__cases,axiom,
% 0.26/2.08 ! [Y2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( Y2 != zero_z189798548lle_hf )
% 0.26/2.08 => ~ ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( Y2
% 0.26/2.08 = ( hF_Mirabelle_hinsert @ A3 @ B2 ) )
% 0.26/2.08 => ( hF_Mirabelle_hmem @ A3 @ B2 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_4_hmem__hinsert,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B3 @ C ) )
% 0.26/2.08 = ( ( A2 = B3 )
% 0.26/2.08 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_5_hmem__def,axiom,
% 0.26/2.08 ( hF_Mirabelle_hmem
% 0.26/2.08 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_6_zero__reorient,axiom,
% 0.26/2.08 ! [X2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( zero_z189798548lle_hf = X2 )
% 0.26/2.08 = ( X2 = zero_z189798548lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_7_zero__reorient,axiom,
% 0.26/2.08 ! [X2: nat] :
% 0.26/2.08 ( ( zero_zero_nat = X2 )
% 0.26/2.08 = ( X2 = zero_zero_nat ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_8_Abs__hf__0,axiom,
% 0.26/2.08 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.26/2.08 = zero_z189798548lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_9_Zero__hf__def,axiom,
% 0.26/2.08 ( zero_z189798548lle_hf
% 0.26/2.08 = ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_10_push__bit__of__0,axiom,
% 0.26/2.08 ! [N: nat] :
% 0.26/2.08 ( ( semiri2013084963it_nat @ N @ zero_zero_nat )
% 0.26/2.08 = zero_zero_nat ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_11_push__bit__eq__0__iff,axiom,
% 0.26/2.08 ! [N: nat,A2: nat] :
% 0.26/2.08 ( ( ( semiri2013084963it_nat @ N @ A2 )
% 0.26/2.08 = zero_zero_nat )
% 0.26/2.08 = ( A2 = zero_zero_nat ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_12_drop__bit__of__0,axiom,
% 0.26/2.08 ! [N: nat] :
% 0.26/2.08 ( ( semiri2115134414it_nat @ N @ zero_zero_nat )
% 0.26/2.08 = zero_zero_nat ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_13_Nats__0,axiom,
% 0.26/2.08 member_nat @ zero_zero_nat @ semiring_1_Nats_nat ).
% 0.26/2.08
% 0.26/2.08 thf(fact_14_HF__hfset,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.26/2.08 = A2 ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_15_empty__iff,axiom,
% 0.26/2.08 ! [C: set_HF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1490636632lle_hf @ C @ bot_bo2093393035lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_16_empty__iff,axiom,
% 0.26/2.08 ! [C: nat] :
% 0.26/2.08 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_17_empty__iff,axiom,
% 0.26/2.08 ! [C: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_18_all__not__in__conv,axiom,
% 0.26/2.08 ! [A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( ! [X: set_HF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1490636632lle_hf @ X @ A4 ) )
% 0.26/2.08 = ( A4 = bot_bo2093393035lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_19_all__not__in__conv,axiom,
% 0.26/2.08 ! [A4: set_nat] :
% 0.26/2.08 ( ( ! [X: nat] :
% 0.26/2.08 ~ ( member_nat @ X @ A4 ) )
% 0.26/2.08 = ( A4 = bot_bot_set_nat ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_20_all__not__in__conv,axiom,
% 0.26/2.08 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( ! [X: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1367349282lle_hf @ X @ A4 ) )
% 0.26/2.08 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_21_Collect__empty__eq,axiom,
% 0.26/2.08 ! [P: nat > $o] :
% 0.26/2.08 ( ( ( collect_nat @ P )
% 0.26/2.08 = bot_bot_set_nat )
% 0.26/2.08 = ( ! [X: nat] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_22_Collect__empty__eq,axiom,
% 0.26/2.08 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( ( collec1758573718lle_hf @ P )
% 0.26/2.08 = bot_bo2093393035lle_hf )
% 0.26/2.08 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_23_Collect__empty__eq,axiom,
% 0.26/2.08 ! [P: hF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( ( collec2046588256lle_hf @ P )
% 0.26/2.08 = bot_bo53200981lle_hf )
% 0.26/2.08 = ( ! [X: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_24_empty__Collect__eq,axiom,
% 0.26/2.08 ! [P: nat > $o] :
% 0.26/2.08 ( ( bot_bot_set_nat
% 0.26/2.08 = ( collect_nat @ P ) )
% 0.26/2.08 = ( ! [X: nat] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_25_empty__Collect__eq,axiom,
% 0.26/2.08 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( bot_bo2093393035lle_hf
% 0.26/2.08 = ( collec1758573718lle_hf @ P ) )
% 0.26/2.08 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_26_empty__Collect__eq,axiom,
% 0.26/2.08 ! [P: hF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( bot_bo53200981lle_hf
% 0.26/2.08 = ( collec2046588256lle_hf @ P ) )
% 0.26/2.08 = ( ! [X: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( P @ X ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_27_bot__apply,axiom,
% 0.26/2.08 ( bot_bo1263054448e_hf_o
% 0.26/2.08 = ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_28_bot__apply,axiom,
% 0.26/2.08 ( bot_bo554042810e_hf_o
% 0.26/2.08 = ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_29_hinsert__def,axiom,
% 0.26/2.08 ( hF_Mirabelle_hinsert
% 0.26/2.08 = ( ^ [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A @ ( hF_Mirabelle_hfset @ B ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_30_hfset__HF,axiom,
% 0.26/2.08 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.08 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
% 0.26/2.08 = A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_31_zero__natural_Orsp,axiom,
% 0.26/2.08 zero_zero_nat = zero_zero_nat ).
% 0.26/2.08
% 0.26/2.08 thf(fact_32_emptyE,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1490636632lle_hf @ A2 @ bot_bo2093393035lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_33_emptyE,axiom,
% 0.26/2.08 ! [A2: nat] :
% 0.26/2.08 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_34_emptyE,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] :
% 0.26/2.08 ~ ( member1367349282lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_35_equals0D,axiom,
% 0.26/2.08 ! [A4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( A4 = bot_bo2093393035lle_hf )
% 0.26/2.08 => ~ ( member1490636632lle_hf @ A2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_36_equals0D,axiom,
% 0.26/2.08 ! [A4: set_nat,A2: nat] :
% 0.26/2.08 ( ( A4 = bot_bot_set_nat )
% 0.26/2.08 => ~ ( member_nat @ A2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_37_equals0D,axiom,
% 0.26/2.08 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( A4 = bot_bo53200981lle_hf )
% 0.26/2.08 => ~ ( member1367349282lle_hf @ A2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_38_insert__absorb2,axiom,
% 0.26/2.08 ! [X2: nat,A4: set_nat] :
% 0.26/2.08 ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A4 ) )
% 0.26/2.08 = ( insert_nat @ X2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_39_insert__absorb2,axiom,
% 0.26/2.08 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ X2 @ A4 ) )
% 0.26/2.08 = ( insert1636143089lle_hf @ X2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_40_insert__absorb2,axiom,
% 0.26/2.08 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ X2 @ A4 ) )
% 0.26/2.08 = ( insert9649339lle_hf @ X2 @ A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_41_insert__iff,axiom,
% 0.26/2.08 ! [A2: nat,B3: nat,A4: set_nat] :
% 0.26/2.08 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
% 0.26/2.08 = ( ( A2 = B3 )
% 0.26/2.08 | ( member_nat @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_42_insert__iff,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
% 0.26/2.08 = ( ( A2 = B3 )
% 0.26/2.08 | ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_43_insert__iff,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
% 0.26/2.08 = ( ( A2 = B3 )
% 0.26/2.08 | ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_44_insertCI,axiom,
% 0.26/2.08 ! [A2: nat,B4: set_nat,B3: nat] :
% 0.26/2.08 ( ( ~ ( member_nat @ A2 @ B4 )
% 0.26/2.08 => ( A2 = B3 ) )
% 0.26/2.08 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_45_insertCI,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( ~ ( member1490636632lle_hf @ A2 @ B4 )
% 0.26/2.08 => ( A2 = B3 ) )
% 0.26/2.08 => ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_46_insertCI,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
% 0.26/2.08 => ( A2 = B3 ) )
% 0.26/2.08 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_47_singletonI,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_48_singletonI,axiom,
% 0.26/2.08 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_49_singletonI,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_50_mk__disjoint__insert,axiom,
% 0.26/2.08 ! [A2: nat,A4: set_nat] :
% 0.26/2.08 ( ( member_nat @ A2 @ A4 )
% 0.26/2.08 => ? [B5: set_nat] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert_nat @ A2 @ B5 ) )
% 0.26/2.08 & ~ ( member_nat @ A2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_51_mk__disjoint__insert,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ A4 )
% 0.26/2.08 => ? [B5: set_se933006839lle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert1636143089lle_hf @ A2 @ B5 ) )
% 0.26/2.08 & ~ ( member1490636632lle_hf @ A2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_52_mk__disjoint__insert,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.26/2.08 => ? [B5: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert9649339lle_hf @ A2 @ B5 ) )
% 0.26/2.08 & ~ ( member1367349282lle_hf @ A2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_53_insert__commute,axiom,
% 0.26/2.08 ! [X2: nat,Y2: nat,A4: set_nat] :
% 0.26/2.08 ( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A4 ) )
% 0.26/2.08 = ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_54_insert__commute,axiom,
% 0.26/2.08 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( insert1636143089lle_hf @ X2 @ ( insert1636143089lle_hf @ Y2 @ A4 ) )
% 0.26/2.08 = ( insert1636143089lle_hf @ Y2 @ ( insert1636143089lle_hf @ X2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_55_insert__commute,axiom,
% 0.26/2.08 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( insert9649339lle_hf @ X2 @ ( insert9649339lle_hf @ Y2 @ A4 ) )
% 0.26/2.08 = ( insert9649339lle_hf @ Y2 @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_56_insert__eq__iff,axiom,
% 0.26/2.08 ! [A2: nat,A4: set_nat,B3: nat,B4: set_nat] :
% 0.26/2.08 ( ~ ( member_nat @ A2 @ A4 )
% 0.26/2.08 => ( ~ ( member_nat @ B3 @ B4 )
% 0.26/2.08 => ( ( ( insert_nat @ A2 @ A4 )
% 0.26/2.08 = ( insert_nat @ B3 @ B4 ) )
% 0.26/2.08 = ( ( ( A2 = B3 )
% 0.26/2.08 => ( A4 = B4 ) )
% 0.26/2.08 & ( ( A2 != B3 )
% 0.26/2.08 => ? [C2: set_nat] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert_nat @ B3 @ C2 ) )
% 0.26/2.08 & ~ ( member_nat @ B3 @ C2 )
% 0.26/2.08 & ( B4
% 0.26/2.08 = ( insert_nat @ A2 @ C2 ) )
% 0.26/2.08 & ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_57_insert__eq__iff,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] :
% 0.26/2.08 ( ~ ( member1490636632lle_hf @ A2 @ A4 )
% 0.26/2.08 => ( ~ ( member1490636632lle_hf @ B3 @ B4 )
% 0.26/2.08 => ( ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.26/2.08 = ( insert1636143089lle_hf @ B3 @ B4 ) )
% 0.26/2.08 = ( ( ( A2 = B3 )
% 0.26/2.08 => ( A4 = B4 ) )
% 0.26/2.08 & ( ( A2 != B3 )
% 0.26/2.08 => ? [C2: set_se933006839lle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert1636143089lle_hf @ B3 @ C2 ) )
% 0.26/2.08 & ~ ( member1490636632lle_hf @ B3 @ C2 )
% 0.26/2.08 & ( B4
% 0.26/2.08 = ( insert1636143089lle_hf @ A2 @ C2 ) )
% 0.26/2.08 & ~ ( member1490636632lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_58_insert__eq__iff,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ~ ( member1367349282lle_hf @ A2 @ A4 )
% 0.26/2.08 => ( ~ ( member1367349282lle_hf @ B3 @ B4 )
% 0.26/2.08 => ( ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.26/2.08 = ( insert9649339lle_hf @ B3 @ B4 ) )
% 0.26/2.08 = ( ( ( A2 = B3 )
% 0.26/2.08 => ( A4 = B4 ) )
% 0.26/2.08 & ( ( A2 != B3 )
% 0.26/2.08 => ? [C2: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert9649339lle_hf @ B3 @ C2 ) )
% 0.26/2.08 & ~ ( member1367349282lle_hf @ B3 @ C2 )
% 0.26/2.08 & ( B4
% 0.26/2.08 = ( insert9649339lle_hf @ A2 @ C2 ) )
% 0.26/2.08 & ~ ( member1367349282lle_hf @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_59_insert__absorb,axiom,
% 0.26/2.08 ! [A2: nat,A4: set_nat] :
% 0.26/2.08 ( ( member_nat @ A2 @ A4 )
% 0.26/2.08 => ( ( insert_nat @ A2 @ A4 )
% 0.26/2.08 = A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_60_insert__absorb,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ A4 )
% 0.26/2.08 => ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.26/2.08 = A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_61_insert__absorb,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.26/2.08 => ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.26/2.08 = A4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_62_insert__ident,axiom,
% 0.26/2.08 ! [X2: nat,A4: set_nat,B4: set_nat] :
% 0.26/2.08 ( ~ ( member_nat @ X2 @ A4 )
% 0.26/2.08 => ( ~ ( member_nat @ X2 @ B4 )
% 0.26/2.08 => ( ( ( insert_nat @ X2 @ A4 )
% 0.26/2.08 = ( insert_nat @ X2 @ B4 ) )
% 0.26/2.08 = ( A4 = B4 ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_63_insert__ident,axiom,
% 0.26/2.08 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.26/2.08 ( ~ ( member1490636632lle_hf @ X2 @ A4 )
% 0.26/2.08 => ( ~ ( member1490636632lle_hf @ X2 @ B4 )
% 0.26/2.08 => ( ( ( insert1636143089lle_hf @ X2 @ A4 )
% 0.26/2.08 = ( insert1636143089lle_hf @ X2 @ B4 ) )
% 0.26/2.08 = ( A4 = B4 ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_64_insert__ident,axiom,
% 0.26/2.08 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.08 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.26/2.08 => ( ( ( insert9649339lle_hf @ X2 @ A4 )
% 0.26/2.08 = ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.26/2.08 = ( A4 = B4 ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_65_Set_Oset__insert,axiom,
% 0.26/2.08 ! [X2: nat,A4: set_nat] :
% 0.26/2.08 ( ( member_nat @ X2 @ A4 )
% 0.26/2.08 => ~ ! [B5: set_nat] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert_nat @ X2 @ B5 ) )
% 0.26/2.08 => ( member_nat @ X2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_66_Set_Oset__insert,axiom,
% 0.26/2.08 ! [X2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.26/2.08 => ~ ! [B5: set_se933006839lle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert1636143089lle_hf @ X2 @ B5 ) )
% 0.26/2.08 => ( member1490636632lle_hf @ X2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_67_Set_Oset__insert,axiom,
% 0.26/2.08 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.08 => ~ ! [B5: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( A4
% 0.26/2.08 = ( insert9649339lle_hf @ X2 @ B5 ) )
% 0.26/2.08 => ( member1367349282lle_hf @ X2 @ B5 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_68_insertI2,axiom,
% 0.26/2.08 ! [A2: nat,B4: set_nat,B3: nat] :
% 0.26/2.08 ( ( member_nat @ A2 @ B4 )
% 0.26/2.08 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_69_insertI2,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf,B3: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ B4 )
% 0.26/2.08 => ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_70_insertI2,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.26/2.08 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_71_insertI1,axiom,
% 0.26/2.08 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_72_insertI1,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B4: set_se933006839lle_hf] : ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ A2 @ B4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_73_insertI1,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_74_insertE,axiom,
% 0.26/2.08 ! [A2: nat,B3: nat,A4: set_nat] :
% 0.26/2.08 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A4 ) )
% 0.26/2.08 => ( ( A2 != B3 )
% 0.26/2.08 => ( member_nat @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_75_insertE,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ A4 ) )
% 0.26/2.08 => ( ( A2 != B3 )
% 0.26/2.08 => ( member1490636632lle_hf @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_76_insertE,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ A4 ) )
% 0.26/2.08 => ( ( A2 != B3 )
% 0.26/2.08 => ( member1367349282lle_hf @ A2 @ A4 ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_77_finite__cases,axiom,
% 0.26/2.08 ! [F: set_se933006839lle_hf] :
% 0.26/2.08 ( ( finite1450550360lle_hf @ F )
% 0.26/2.08 => ( ( F != bot_bo2093393035lle_hf )
% 0.26/2.08 => ~ ! [A5: set_se933006839lle_hf,X3: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( F
% 0.26/2.08 = ( insert1636143089lle_hf @ X3 @ A5 ) )
% 0.26/2.08 => ( ~ ( member1490636632lle_hf @ X3 @ A5 )
% 0.26/2.08 => ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_78_finite__cases,axiom,
% 0.26/2.08 ! [F: set_nat] :
% 0.26/2.08 ( ( finite_finite_nat @ F )
% 0.26/2.08 => ( ( F != bot_bot_set_nat )
% 0.26/2.08 => ~ ! [A5: set_nat,X3: nat] :
% 0.26/2.08 ( ( F
% 0.26/2.08 = ( insert_nat @ X3 @ A5 ) )
% 0.26/2.08 => ( ~ ( member_nat @ X3 @ A5 )
% 0.26/2.08 => ~ ( finite_finite_nat @ A5 ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_79_finite__cases,axiom,
% 0.26/2.08 ! [F: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( finite586181922lle_hf @ F )
% 0.26/2.08 => ( ( F != bot_bo53200981lle_hf )
% 0.26/2.08 => ~ ! [A5: set_HF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( F
% 0.26/2.08 = ( insert9649339lle_hf @ X3 @ A5 ) )
% 0.26/2.08 => ( ~ ( member1367349282lle_hf @ X3 @ A5 )
% 0.26/2.08 => ~ ( finite586181922lle_hf @ A5 ) ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_80_bot__set__def,axiom,
% 0.26/2.08 ( bot_bot_set_nat
% 0.26/2.08 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_81_bot__set__def,axiom,
% 0.26/2.08 ( bot_bo2093393035lle_hf
% 0.26/2.08 = ( collec1758573718lle_hf @ bot_bo554042810e_hf_o ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_82_bot__set__def,axiom,
% 0.26/2.08 ( bot_bo53200981lle_hf
% 0.26/2.08 = ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_83_singleton__inject,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf )
% 0.26/2.08 = ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
% 0.26/2.08 => ( A2 = B3 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_84_singleton__inject,axiom,
% 0.26/2.08 ! [A2: nat,B3: nat] :
% 0.26/2.08 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 0.26/2.08 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.26/2.08 => ( A2 = B3 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_85_singleton__inject,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.26/2.08 = ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.26/2.08 => ( A2 = B3 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_86_insert__not__empty,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( insert1636143089lle_hf @ A2 @ A4 )
% 0.26/2.08 != bot_bo2093393035lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_87_insert__not__empty,axiom,
% 0.26/2.08 ! [A2: nat,A4: set_nat] :
% 0.26/2.08 ( ( insert_nat @ A2 @ A4 )
% 0.26/2.08 != bot_bot_set_nat ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_88_insert__not__empty,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.26/2.08 != bot_bo53200981lle_hf ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_89_doubleton__eq__iff,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf,D: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( ( insert1636143089lle_hf @ A2 @ ( insert1636143089lle_hf @ B3 @ bot_bo2093393035lle_hf ) )
% 0.26/2.08 = ( insert1636143089lle_hf @ C @ ( insert1636143089lle_hf @ D @ bot_bo2093393035lle_hf ) ) )
% 0.26/2.08 = ( ( ( A2 = C )
% 0.26/2.08 & ( B3 = D ) )
% 0.26/2.08 | ( ( A2 = D )
% 0.26/2.08 & ( B3 = C ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_90_doubleton__eq__iff,axiom,
% 0.26/2.08 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.26/2.08 ( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.26/2.08 = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.26/2.08 = ( ( ( A2 = C )
% 0.26/2.08 & ( B3 = D ) )
% 0.26/2.08 | ( ( A2 = D )
% 0.26/2.08 & ( B3 = C ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_91_doubleton__eq__iff,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( ( insert9649339lle_hf @ A2 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.26/2.08 = ( insert9649339lle_hf @ C @ ( insert9649339lle_hf @ D @ bot_bo53200981lle_hf ) ) )
% 0.26/2.08 = ( ( ( A2 = C )
% 0.26/2.08 & ( B3 = D ) )
% 0.26/2.08 | ( ( A2 = D )
% 0.26/2.08 & ( B3 = C ) ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_92_mem__Collect__eq,axiom,
% 0.26/2.08 ! [A2: nat,P: nat > $o] :
% 0.26/2.08 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.26/2.08 = ( P @ A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_93_mem__Collect__eq,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
% 0.26/2.08 = ( P @ A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_94_mem__Collect__eq,axiom,
% 0.26/2.08 ! [A2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ( member1490636632lle_hf @ A2 @ ( collec1758573718lle_hf @ P ) )
% 0.26/2.08 = ( P @ A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_95_Collect__mem__eq,axiom,
% 0.26/2.08 ! [A4: set_nat] :
% 0.26/2.08 ( ( collect_nat
% 0.26/2.08 @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
% 0.26/2.08 = A4 ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_96_Collect__mem__eq,axiom,
% 0.26/2.08 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( collec2046588256lle_hf
% 0.26/2.08 @ ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
% 0.26/2.08 = A4 ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_97_Collect__mem__eq,axiom,
% 0.26/2.08 ! [A4: set_se933006839lle_hf] :
% 0.26/2.08 ( ( collec1758573718lle_hf
% 0.26/2.08 @ ^ [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
% 0.26/2.08 = A4 ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_98_Collect__cong,axiom,
% 0.26/2.08 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ! [X3: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( P @ X3 )
% 0.26/2.08 = ( Q @ X3 ) )
% 0.26/2.08 => ( ( collec2046588256lle_hf @ P )
% 0.26/2.08 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_99_Collect__cong,axiom,
% 0.26/2.08 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.26/2.08 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( P @ X3 )
% 0.26/2.08 = ( Q @ X3 ) )
% 0.26/2.08 => ( ( collec1758573718lle_hf @ P )
% 0.26/2.08 = ( collec1758573718lle_hf @ Q ) ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_100_singleton__iff,axiom,
% 0.26/2.08 ! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
% 0.26/2.08 = ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_101_singleton__iff,axiom,
% 0.26/2.08 ! [B3: nat,A2: nat] :
% 0.26/2.08 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.26/2.08 = ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_102_singleton__iff,axiom,
% 0.26/2.08 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.26/2.08 = ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_103_singletonD,axiom,
% 0.26/2.08 ! [B3: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1490636632lle_hf @ B3 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) )
% 0.26/2.08 => ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_104_singletonD,axiom,
% 0.26/2.08 ! [B3: nat,A2: nat] :
% 0.26/2.08 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.26/2.08 => ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_105_singletonD,axiom,
% 0.26/2.08 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.08 ( ( member1367349282lle_hf @ B3 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.26/2.08 => ( B3 = A2 ) ) ).
% 0.26/2.08
% 0.26/2.08 thf(fact_106_bot__nat__def,axiom,
% 0.26/2.08 bot_bot_nat = zero_zero_nat ).
% 0.26/2.08
% 0.26/2.08 thf(fact_107_finite__hfset,axiom,
% 0.26/2.08 ! [A2: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A2 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_108_bot__fun__def,axiom,
% 0.26/2.09 ( bot_bo1263054448e_hf_o
% 0.26/2.09 = ( ^ [X: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_109_bot__fun__def,axiom,
% 0.26/2.09 ( bot_bo554042810e_hf_o
% 0.26/2.09 = ( ^ [X: set_HF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_110_ex__in__conv,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( ? [X: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X @ A4 ) )
% 0.26/2.09 = ( A4 != bot_bo2093393035lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_111_ex__in__conv,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
% 0.26/2.09 = ( A4 != bot_bot_set_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_112_ex__in__conv,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ? [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ A4 ) )
% 0.26/2.09 = ( A4 != bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_113_equals0I,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ! [Y3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ~ ( member1490636632lle_hf @ Y3 @ A4 )
% 0.26/2.09 => ( A4 = bot_bo2093393035lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_114_equals0I,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ! [Y3: nat] :
% 0.26/2.09 ~ ( member_nat @ Y3 @ A4 )
% 0.26/2.09 => ( A4 = bot_bot_set_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_115_equals0I,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [Y3: hF_Mirabelle_hf] :
% 0.26/2.09 ~ ( member1367349282lle_hf @ Y3 @ A4 )
% 0.26/2.09 => ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_116_finite__insert,axiom,
% 0.26/2.09 ! [A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ ( insert1636143089lle_hf @ A2 @ A4 ) )
% 0.26/2.09 = ( finite1450550360lle_hf @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_117_finite__insert,axiom,
% 0.26/2.09 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.26/2.09 = ( finite586181922lle_hf @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_118_finite__insert,axiom,
% 0.26/2.09 ! [A2: nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) )
% 0.26/2.09 = ( finite_finite_nat @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_119_finite_Ocases,axiom,
% 0.26/2.09 ! [A2: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ A2 )
% 0.26/2.09 => ( ( A2 != bot_bo2093393035lle_hf )
% 0.26/2.09 => ~ ! [A5: set_se933006839lle_hf] :
% 0.26/2.09 ( ? [A3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( A2
% 0.26/2.09 = ( insert1636143089lle_hf @ A3 @ A5 ) )
% 0.26/2.09 => ~ ( finite1450550360lle_hf @ A5 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_120_finite_Ocases,axiom,
% 0.26/2.09 ! [A2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A2 )
% 0.26/2.09 => ( ( A2 != bot_bot_set_nat )
% 0.26/2.09 => ~ ! [A5: set_nat] :
% 0.26/2.09 ( ? [A3: nat] :
% 0.26/2.09 ( A2
% 0.26/2.09 = ( insert_nat @ A3 @ A5 ) )
% 0.26/2.09 => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_121_finite_Ocases,axiom,
% 0.26/2.09 ! [A2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A2 )
% 0.26/2.09 => ( ( A2 != bot_bo53200981lle_hf )
% 0.26/2.09 => ~ ! [A5: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ? [A3: hF_Mirabelle_hf] :
% 0.26/2.09 ( A2
% 0.26/2.09 = ( insert9649339lle_hf @ A3 @ A5 ) )
% 0.26/2.09 => ~ ( finite586181922lle_hf @ A5 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_122_finite_Osimps,axiom,
% 0.26/2.09 ( finite1450550360lle_hf
% 0.26/2.09 = ( ^ [A: set_se933006839lle_hf] :
% 0.26/2.09 ( ( A = bot_bo2093393035lle_hf )
% 0.26/2.09 | ? [A6: set_se933006839lle_hf,B: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( A
% 0.26/2.09 = ( insert1636143089lle_hf @ B @ A6 ) )
% 0.26/2.09 & ( finite1450550360lle_hf @ A6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_123_finite_Osimps,axiom,
% 0.26/2.09 ( finite_finite_nat
% 0.26/2.09 = ( ^ [A: set_nat] :
% 0.26/2.09 ( ( A = bot_bot_set_nat )
% 0.26/2.09 | ? [A6: set_nat,B: nat] :
% 0.26/2.09 ( ( A
% 0.26/2.09 = ( insert_nat @ B @ A6 ) )
% 0.26/2.09 & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_124_finite_Osimps,axiom,
% 0.26/2.09 ( finite586181922lle_hf
% 0.26/2.09 = ( ^ [A: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( A = bot_bo53200981lle_hf )
% 0.26/2.09 | ? [A6: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( A
% 0.26/2.09 = ( insert9649339lle_hf @ B @ A6 ) )
% 0.26/2.09 & ( finite586181922lle_hf @ A6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_125_finite__induct,axiom,
% 0.26/2.09 ! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ F )
% 0.26/2.09 => ( ( P @ bot_bo2093393035lle_hf )
% 0.26/2.09 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ F2 )
% 0.26/2.09 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_126_finite__induct,axiom,
% 0.26/2.09 ! [F: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [X3: nat,F2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F2 )
% 0.26/2.09 => ( ~ ( member_nat @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_127_finite__induct,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F2 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_128_finite_Oinducts,axiom,
% 0.26/2.09 ! [X2: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ X2 )
% 0.26/2.09 => ( ( P @ bot_bo2093393035lle_hf )
% 0.26/2.09 => ( ! [A5: set_se933006839lle_hf,A3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ A5 )
% 0.26/2.09 => ( ( P @ A5 )
% 0.26/2.09 => ( P @ ( insert1636143089lle_hf @ A3 @ A5 ) ) ) )
% 0.26/2.09 => ( P @ X2 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_129_finite_Oinducts,axiom,
% 0.26/2.09 ! [X2: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ X2 )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [A5: set_nat,A3: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A5 )
% 0.26/2.09 => ( ( P @ A5 )
% 0.26/2.09 => ( P @ ( insert_nat @ A3 @ A5 ) ) ) )
% 0.26/2.09 => ( P @ X2 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_130_finite_Oinducts,axiom,
% 0.26/2.09 ! [X2: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ X2 )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [A5: set_HF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A5 )
% 0.26/2.09 => ( ( P @ A5 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ A3 @ A5 ) ) ) )
% 0.26/2.09 => ( P @ X2 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_131_finite__ne__induct,axiom,
% 0.26/2.09 ! [F: set_se933006839lle_hf,P: set_se933006839lle_hf > $o] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ F )
% 0.26/2.09 => ( ( F != bot_bo2093393035lle_hf )
% 0.26/2.09 => ( ! [X3: set_HF_Mirabelle_hf] : ( P @ ( insert1636143089lle_hf @ X3 @ bot_bo2093393035lle_hf ) )
% 0.26/2.09 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ F2 )
% 0.26/2.09 => ( ( F2 != bot_bo2093393035lle_hf )
% 0.26/2.09 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_132_finite__ne__induct,axiom,
% 0.26/2.09 ! [F: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( ( F != bot_bot_set_nat )
% 0.26/2.09 => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.26/2.09 => ( ! [X3: nat,F2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F2 )
% 0.26/2.09 => ( ( F2 != bot_bot_set_nat )
% 0.26/2.09 => ( ~ ( member_nat @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_133_finite__ne__induct,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( ( F != bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F2 )
% 0.26/2.09 => ( ( F2 != bot_bo53200981lle_hf )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_134_infinite__finite__induct,axiom,
% 0.26/2.09 ! [P: set_se933006839lle_hf > $o,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ! [A5: set_se933006839lle_hf] :
% 0.26/2.09 ( ~ ( finite1450550360lle_hf @ A5 )
% 0.26/2.09 => ( P @ A5 ) )
% 0.26/2.09 => ( ( P @ bot_bo2093393035lle_hf )
% 0.26/2.09 => ( ! [X3: set_HF_Mirabelle_hf,F2: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ F2 )
% 0.26/2.09 => ( ~ ( member1490636632lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert1636143089lle_hf @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_135_infinite__finite__induct,axiom,
% 0.26/2.09 ! [P: set_nat > $o,A4: set_nat] :
% 0.26/2.09 ( ! [A5: set_nat] :
% 0.26/2.09 ( ~ ( finite_finite_nat @ A5 )
% 0.26/2.09 => ( P @ A5 ) )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [X3: nat,F2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F2 )
% 0.26/2.09 => ( ~ ( member_nat @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert_nat @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_136_infinite__finite__induct,axiom,
% 0.26/2.09 ! [P: set_HF_Mirabelle_hf > $o,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [A5: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( finite586181922lle_hf @ A5 )
% 0.26/2.09 => ( P @ A5 ) )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F2 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ X3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ X3 @ F2 ) ) ) ) )
% 0.26/2.09 => ( P @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_137_the__elem__eq,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( the_el1104322134lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 = X2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_138_finite_OemptyI,axiom,
% 0.26/2.09 finite_finite_nat @ bot_bot_set_nat ).
% 0.26/2.09
% 0.26/2.09 thf(fact_139_finite_OemptyI,axiom,
% 0.26/2.09 finite586181922lle_hf @ bot_bo53200981lle_hf ).
% 0.26/2.09
% 0.26/2.09 thf(fact_140_infinite__imp__nonempty,axiom,
% 0.26/2.09 ! [S: set_nat] :
% 0.26/2.09 ( ~ ( finite_finite_nat @ S )
% 0.26/2.09 => ( S != bot_bot_set_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_141_infinite__imp__nonempty,axiom,
% 0.26/2.09 ! [S: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( finite586181922lle_hf @ S )
% 0.26/2.09 => ( S != bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_142_finite_OinsertI,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( finite586181922lle_hf @ ( insert9649339lle_hf @ A2 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_143_finite_OinsertI,axiom,
% 0.26/2.09 ! [A4: set_nat,A2: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( finite_finite_nat @ ( insert_nat @ A2 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_144_bot__empty__eq,axiom,
% 0.26/2.09 ( bot_bo1263054448e_hf_o
% 0.26/2.09 = ( ^ [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_145_Collect__empty__eq__bot,axiom,
% 0.26/2.09 ! [P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ( collec1758573718lle_hf @ P )
% 0.26/2.09 = bot_bo2093393035lle_hf )
% 0.26/2.09 = ( P = bot_bo554042810e_hf_o ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_146_Collect__empty__eq__bot,axiom,
% 0.26/2.09 ! [P: hF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ( collec2046588256lle_hf @ P )
% 0.26/2.09 = bot_bo53200981lle_hf )
% 0.26/2.09 = ( P = bot_bo1263054448e_hf_o ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_147_is__singleton__the__elem,axiom,
% 0.26/2.09 ( is_sin1448700567lle_hf
% 0.26/2.09 = ( ^ [A6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( A6
% 0.26/2.09 = ( insert9649339lle_hf @ ( the_el1104322134lle_hf @ A6 ) @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_148_is__singletonI,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf] : ( is_sin1448700567lle_hf @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_149_Set_Ois__empty__def,axiom,
% 0.26/2.09 ( is_emp566801209lle_hf
% 0.26/2.09 = ( ^ [A6: set_HF_Mirabelle_hf] : ( A6 = bot_bo53200981lle_hf ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_150_is__singleton__def,axiom,
% 0.26/2.09 ( is_sin1448700567lle_hf
% 0.26/2.09 = ( ^ [A6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ? [X: hF_Mirabelle_hf] :
% 0.26/2.09 ( A6
% 0.26/2.09 = ( insert9649339lle_hf @ X @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_151_is__singletonE,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( is_sin1448700567lle_hf @ A4 )
% 0.26/2.09 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.26/2.09 ( A4
% 0.26/2.09 != ( insert9649339lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_152_is__singletonI_H,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( A4 != bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ Y3 @ A4 )
% 0.26/2.09 => ( X3 = Y3 ) ) )
% 0.26/2.09 => ( is_sin1448700567lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_153_card__0__eq,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( ( finite_card_nat @ A4 )
% 0.26/2.09 = zero_zero_nat )
% 0.26/2.09 = ( A4 = bot_bot_set_nat ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_154_card__0__eq,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( ( finite1213132899lle_hf @ A4 )
% 0.26/2.09 = zero_zero_nat )
% 0.26/2.09 = ( A4 = bot_bo53200981lle_hf ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_155_finite__subset__induct,axiom,
% 0.26/2.09 ! [F: set_nat,A4: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ F @ A4 )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [A3: nat,F2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F2 )
% 0.26/2.09 => ( ( member_nat @ A3 @ A4 )
% 0.26/2.09 => ( ~ ( member_nat @ A3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_156_finite__subset__induct,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ F @ A4 )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F2 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ A3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_157_finite__subset__induct_H,axiom,
% 0.26/2.09 ! [F: set_nat,A4: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ F @ A4 )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [A3: nat,F2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F2 )
% 0.26/2.09 => ( ( member_nat @ A3 @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ F2 @ A4 )
% 0.26/2.09 => ( ~ ( member_nat @ A3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_158_finite__subset__induct_H,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ F @ A4 )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [A3: hF_Mirabelle_hf,F2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F2 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ F2 @ A4 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ A3 @ F2 )
% 0.26/2.09 => ( ( P @ F2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ A3 @ F2 ) ) ) ) ) ) )
% 0.26/2.09 => ( P @ F ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_159_inj__on__HF,axiom,
% 0.26/2.09 inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_160_order__refl,axiom,
% 0.26/2.09 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_161_subsetI,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [X3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ X3 @ B4 ) )
% 0.26/2.09 => ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_162_le__zero__eq,axiom,
% 0.26/2.09 ! [N: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.26/2.09 = ( N = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_163_subset__empty,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf )
% 0.26/2.09 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_164_empty__subsetI,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A4 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_165_insert__subset,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
% 0.26/2.09 = ( ( member1367349282lle_hf @ X2 @ B4 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_166_singleton__insert__inj__eq_H,axiom,
% 0.26/2.09 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( insert9649339lle_hf @ A2 @ A4 )
% 0.26/2.09 = ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 = ( ( A2 = B3 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_167_singleton__insert__inj__eq,axiom,
% 0.26/2.09 ! [B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf )
% 0.26/2.09 = ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.26/2.09 = ( ( A2 = B3 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_168_card_Oempty,axiom,
% 0.26/2.09 ( ( finite1213132899lle_hf @ bot_bo53200981lle_hf )
% 0.26/2.09 = zero_zero_nat ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_169_card_Oinfinite,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( finite1213132899lle_hf @ A4 )
% 0.26/2.09 = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_170_card_Oinfinite,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ~ ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( finite_card_nat @ A4 )
% 0.26/2.09 = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_171_dual__order_Oantisym,axiom,
% 0.26/2.09 ! [B3: nat,A2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( A2 = B3 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_172_dual__order_Oeq__iff,axiom,
% 0.26/2.09 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.26/2.09 = ( ^ [A: nat,B: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ B @ A )
% 0.26/2.09 & ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_173_dual__order_Otrans,axiom,
% 0.26/2.09 ! [B3: nat,A2: nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.26/2.09 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_174_linorder__wlog,axiom,
% 0.26/2.09 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.26/2.09 ( ! [A3: nat,B2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.26/2.09 => ( P @ A3 @ B2 ) )
% 0.26/2.09 => ( ! [A3: nat,B2: nat] :
% 0.26/2.09 ( ( P @ B2 @ A3 )
% 0.26/2.09 => ( P @ A3 @ B2 ) )
% 0.26/2.09 => ( P @ A2 @ B3 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_175_dual__order_Orefl,axiom,
% 0.26/2.09 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_176_order__trans,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.26/2.09 => ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_177_order__class_Oorder_Oantisym,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.26/2.09 => ( A2 = B3 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_178_ord__le__eq__trans,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( ( B3 = C )
% 0.26/2.09 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_179_ord__eq__le__trans,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat,C: nat] :
% 0.26/2.09 ( ( A2 = B3 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.26/2.09 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_180_order__class_Oorder_Oeq__iff,axiom,
% 0.26/2.09 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.26/2.09 = ( ^ [A: nat,B: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A @ B )
% 0.26/2.09 & ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_181_antisym__conv,axiom,
% 0.26/2.09 ! [Y2: nat,X2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 = ( X2 = Y2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_182_le__cases3,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/2.09 ( ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
% 0.26/2.09 => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
% 0.26/2.09 => ( ( ( ord_less_eq_nat @ X2 @ Z2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
% 0.26/2.09 => ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
% 0.26/2.09 => ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
% 0.26/2.09 => ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
% 0.26/2.09 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_183_order_Otrans,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.26/2.09 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_184_le__cases,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat] :
% 0.26/2.09 ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_185_eq__refl,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat] :
% 0.26/2.09 ( ( X2 = Y2 )
% 0.26/2.09 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_186_linear,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_187_antisym,axiom,
% 0.26/2.09 ! [X2: nat,Y2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.26/2.09 => ( X2 = Y2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_188_eq__iff,axiom,
% 0.26/2.09 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.26/2.09 = ( ^ [X: nat,Y4: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.26/2.09 & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_189_ord__le__eq__subst,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( ( ( F3 @ B3 )
% 0.26/2.09 = C )
% 0.26/2.09 => ( ! [X3: nat,Y3: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_190_ord__eq__le__subst,axiom,
% 0.26/2.09 ! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
% 0.26/2.09 ( ( A2
% 0.26/2.09 = ( F3 @ B3 ) )
% 0.26/2.09 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.26/2.09 => ( ! [X3: nat,Y3: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_191_order__subst2,axiom,
% 0.26/2.09 ! [A2: nat,B3: nat,F3: nat > nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( F3 @ B3 ) @ C )
% 0.26/2.09 => ( ! [X3: nat,Y3: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ A2 ) @ C ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_192_order__subst1,axiom,
% 0.26/2.09 ! [A2: nat,F3: nat > nat,B3: nat,C: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ ( F3 @ B3 ) )
% 0.26/2.09 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.26/2.09 => ( ! [X3: nat,Y3: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ A2 @ ( F3 @ C ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_193_Collect__mono__iff,axiom,
% 0.26/2.09 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) )
% 0.26/2.09 = ( ! [X: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( P @ X )
% 0.26/2.09 => ( Q @ X ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_194_Collect__mono,axiom,
% 0.26/2.09 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( P @ X3 )
% 0.26/2.09 => ( Q @ X3 ) )
% 0.26/2.09 => ( ord_le2016357975lle_hf @ ( collec1758573718lle_hf @ P ) @ ( collec1758573718lle_hf @ Q ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_195_subset__iff,axiom,
% 0.26/2.09 ( ord_le432112161lle_hf
% 0.26/2.09 = ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ! [T: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ T @ A6 )
% 0.26/2.09 => ( member1367349282lle_hf @ T @ B6 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_196_subset__eq,axiom,
% 0.26/2.09 ( ord_le432112161lle_hf
% 0.26/2.09 = ( ^ [A6: set_HF_Mirabelle_hf,B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ! [X: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X @ A6 )
% 0.26/2.09 => ( member1367349282lle_hf @ X @ B6 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_197_subsetD,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ C @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_198_in__mono,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ X2 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_199_card__subset__eq,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( ( finite1213132899lle_hf @ A4 )
% 0.26/2.09 = ( finite1213132899lle_hf @ B4 ) )
% 0.26/2.09 => ( A4 = B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_200_card__subset__eq,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.26/2.09 => ( ( ( finite_card_nat @ A4 )
% 0.26/2.09 = ( finite_card_nat @ B4 ) )
% 0.26/2.09 => ( A4 = B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_201_infinite__arbitrarily__large,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,N: nat] :
% 0.26/2.09 ( ~ ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ? [B5: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B5 )
% 0.26/2.09 & ( ( finite1213132899lle_hf @ B5 )
% 0.26/2.09 = N )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B5 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_202_infinite__arbitrarily__large,axiom,
% 0.26/2.09 ! [A4: set_nat,N: nat] :
% 0.26/2.09 ( ~ ( finite_finite_nat @ A4 )
% 0.26/2.09 => ? [B5: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B5 )
% 0.26/2.09 & ( ( finite_card_nat @ B5 )
% 0.26/2.09 = N )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_203_zero__le,axiom,
% 0.26/2.09 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_204_le__numeral__extra_I3_J,axiom,
% 0.26/2.09 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.26/2.09
% 0.26/2.09 thf(fact_205_bot_Oextremum,axiom,
% 0.26/2.09 ! [A2: set_HF_Mirabelle_hf] : ( ord_le432112161lle_hf @ bot_bo53200981lle_hf @ A2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_206_bot_Oextremum,axiom,
% 0.26/2.09 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_207_bot_Oextremum__unique,axiom,
% 0.26/2.09 ! [A2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.26/2.09 = ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_208_bot_Oextremum__unique,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.26/2.09 = ( A2 = bot_bot_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_209_bot_Oextremum__uniqueI,axiom,
% 0.26/2.09 ! [A2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_210_bot_Oextremum__uniqueI,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.26/2.09 => ( A2 = bot_bot_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_211_finite__has__maximal2,axiom,
% 0.26/2.09 ! [A4: set_nat,A2: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( member_nat @ A2 @ A4 )
% 0.26/2.09 => ? [X3: nat] :
% 0.26/2.09 ( ( member_nat @ X3 @ A4 )
% 0.26/2.09 & ( ord_less_eq_nat @ A2 @ X3 )
% 0.26/2.09 & ! [Xa: nat] :
% 0.26/2.09 ( ( member_nat @ Xa @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ X3 @ Xa )
% 0.26/2.09 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_212_finite__has__minimal2,axiom,
% 0.26/2.09 ! [A4: set_nat,A2: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( member_nat @ A2 @ A4 )
% 0.26/2.09 => ? [X3: nat] :
% 0.26/2.09 ( ( member_nat @ X3 @ A4 )
% 0.26/2.09 & ( ord_less_eq_nat @ X3 @ A2 )
% 0.26/2.09 & ! [Xa: nat] :
% 0.26/2.09 ( ( member_nat @ Xa @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ Xa @ X3 )
% 0.26/2.09 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_213_finite__subset,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_214_finite__subset,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.26/2.09 => ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_215_infinite__super,axiom,
% 0.26/2.09 ! [S: set_HF_Mirabelle_hf,T2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ S @ T2 )
% 0.26/2.09 => ( ~ ( finite586181922lle_hf @ S )
% 0.26/2.09 => ~ ( finite586181922lle_hf @ T2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_216_infinite__super,axiom,
% 0.26/2.09 ! [S: set_nat,T2: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.26/2.09 => ( ~ ( finite_finite_nat @ S )
% 0.26/2.09 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_217_rev__finite__subset,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_218_rev__finite__subset,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.26/2.09 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_219_insert__mono,axiom,
% 0.26/2.09 ! [C3: set_HF_Mirabelle_hf,D2: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ C3 @ D2 )
% 0.26/2.09 => ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ A2 @ C3 ) @ ( insert9649339lle_hf @ A2 @ D2 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_220_subset__insert,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.26/2.09 = ( ord_le432112161lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_221_subset__insertI,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] : ( ord_le432112161lle_hf @ B4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_222_subset__insertI2,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ B3 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_223_finite__has__minimal,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( A4 != bot_bot_set_nat )
% 0.26/2.09 => ? [X3: nat] :
% 0.26/2.09 ( ( member_nat @ X3 @ A4 )
% 0.26/2.09 & ! [Xa: nat] :
% 0.26/2.09 ( ( member_nat @ Xa @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ Xa @ X3 )
% 0.26/2.09 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_224_finite__has__maximal,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( A4 != bot_bot_set_nat )
% 0.26/2.09 => ? [X3: nat] :
% 0.26/2.09 ( ( member_nat @ X3 @ A4 )
% 0.26/2.09 & ! [Xa: nat] :
% 0.26/2.09 ( ( member_nat @ Xa @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ X3 @ Xa )
% 0.26/2.09 => ( X3 = Xa ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_225_subset__singleton__iff,axiom,
% 0.26/2.09 ! [X4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ X4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 = ( ( X4 = bot_bo53200981lle_hf )
% 0.26/2.09 | ( X4
% 0.26/2.09 = ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_226_subset__singletonD,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 => ( ( A4 = bot_bo53200981lle_hf )
% 0.26/2.09 | ( A4
% 0.26/2.09 = ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_227_card__eq__0__iff,axiom,
% 0.26/2.09 ! [A4: set_nat] :
% 0.26/2.09 ( ( ( finite_card_nat @ A4 )
% 0.26/2.09 = zero_zero_nat )
% 0.26/2.09 = ( ( A4 = bot_bot_set_nat )
% 0.26/2.09 | ~ ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_228_card__eq__0__iff,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( finite1213132899lle_hf @ A4 )
% 0.26/2.09 = zero_zero_nat )
% 0.26/2.09 = ( ( A4 = bot_bo53200981lle_hf )
% 0.26/2.09 | ~ ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_229_inj__on__empty,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] : ( inj_on811196232lle_hf @ F3 @ bot_bo2093393035lle_hf ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_230_finite__ranking__induct,axiom,
% 0.26/2.09 ! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
% 0.26/2.09 ( ( finite_finite_nat @ S )
% 0.26/2.09 => ( ( P @ bot_bot_set_nat )
% 0.26/2.09 => ( ! [X3: nat,S2: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ S2 )
% 0.26/2.09 => ( ! [Y5: nat] :
% 0.26/2.09 ( ( member_nat @ Y5 @ S2 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
% 0.26/2.09 => ( ( P @ S2 )
% 0.26/2.09 => ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
% 0.26/2.09 => ( P @ S ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_231_finite__ranking__induct,axiom,
% 0.26/2.09 ! [S: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o,F3: hF_Mirabelle_hf > nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ S )
% 0.26/2.09 => ( ( P @ bot_bo53200981lle_hf )
% 0.26/2.09 => ( ! [X3: hF_Mirabelle_hf,S2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ S2 )
% 0.26/2.09 => ( ! [Y5: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ Y5 @ S2 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X3 ) ) )
% 0.26/2.09 => ( ( P @ S2 )
% 0.26/2.09 => ( P @ ( insert9649339lle_hf @ X3 @ S2 ) ) ) ) )
% 0.26/2.09 => ( P @ S ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_232_inj__on__subset,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ B4 @ A4 )
% 0.26/2.09 => ( inj_on811196232lle_hf @ F3 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_233_subset__inj__on,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ B4 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_234_insert__subsetI,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ X4 @ A4 )
% 0.26/2.09 => ( ord_le432112161lle_hf @ ( insert9649339lle_hf @ X2 @ X4 ) @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_235_subset__emptyI,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [X3: hF_Mirabelle_hf] :
% 0.26/2.09 ~ ( member1367349282lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( ord_le432112161lle_hf @ A4 @ bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_236_bot__nat__0_Oextremum,axiom,
% 0.26/2.09 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_237_le0,axiom,
% 0.26/2.09 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_238_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.26/2.09 => ( A2 = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_239_bot__nat__0_Oextremum__unique,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.26/2.09 = ( A2 = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_240_le__0__eq,axiom,
% 0.26/2.09 ! [N: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.26/2.09 = ( N = zero_zero_nat ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_241_less__eq__nat_Osimps_I1_J,axiom,
% 0.26/2.09 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_242_finite__if__finite__subsets__card__bdd,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,C3: nat] :
% 0.26/2.09 ( ! [G: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ G @ F )
% 0.26/2.09 => ( ( finite586181922lle_hf @ G )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G ) @ C3 ) ) )
% 0.26/2.09 => ( ( finite586181922lle_hf @ F )
% 0.26/2.09 & ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F ) @ C3 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_243_finite__if__finite__subsets__card__bdd,axiom,
% 0.26/2.09 ! [F: set_nat,C3: nat] :
% 0.26/2.09 ( ! [G: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ G @ F )
% 0.26/2.09 => ( ( finite_finite_nat @ G )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
% 0.26/2.09 => ( ( finite_finite_nat @ F )
% 0.26/2.09 & ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C3 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_244_card__seteq,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ B4 ) @ ( finite1213132899lle_hf @ A4 ) )
% 0.26/2.09 => ( A4 = B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_245_card__seteq,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
% 0.26/2.09 => ( A4 = B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_246_card__mono,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_247_card__mono,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_248_card__insert__le,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_249_card__insert__le,axiom,
% 0.26/2.09 ! [A4: set_nat,X2: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_250_inj__on__inverseI,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf,G2: hF_Mirabelle_hf > set_HF_Mirabelle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ! [X3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( ( G2 @ ( F3 @ X3 ) )
% 0.26/2.09 = X3 ) )
% 0.26/2.09 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_251_inj__on__contraD,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( X2 != Y2 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.26/2.09 => ( ( F3 @ X2 )
% 0.26/2.09 != ( F3 @ Y2 ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_252_inj__on__eq__iff,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.26/2.09 => ( ( ( F3 @ X2 )
% 0.26/2.09 = ( F3 @ Y2 ) )
% 0.26/2.09 = ( X2 = Y2 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_253_inj__on__cong,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,G2: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ! [A3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ A3 @ A4 )
% 0.26/2.09 => ( ( F3 @ A3 )
% 0.26/2.09 = ( G2 @ A3 ) ) )
% 0.26/2.09 => ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 = ( inj_on811196232lle_hf @ G2 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_254_inj__on__def,axiom,
% 0.26/2.09 ( inj_on811196232lle_hf
% 0.26/2.09 = ( ^ [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A6: set_se933006839lle_hf] :
% 0.26/2.09 ! [X: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ X @ A6 )
% 0.26/2.09 => ! [Y4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ Y4 @ A6 )
% 0.26/2.09 => ( ( ( F4 @ X )
% 0.26/2.09 = ( F4 @ Y4 ) )
% 0.26/2.09 => ( X = Y4 ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_255_inj__onI,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf,F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ! [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ Y3 @ A4 )
% 0.26/2.09 => ( ( ( F3 @ X3 )
% 0.26/2.09 = ( F3 @ Y3 ) )
% 0.26/2.09 => ( X3 = Y3 ) ) ) )
% 0.26/2.09 => ( inj_on811196232lle_hf @ F3 @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_256_inj__onD,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( ( F3 @ X2 )
% 0.26/2.09 = ( F3 @ Y2 ) )
% 0.26/2.09 => ( ( member1490636632lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ Y2 @ A4 )
% 0.26/2.09 => ( X2 = Y2 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_257_obtain__subset__with__card__n,axiom,
% 0.26/2.09 ! [N: nat,S: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_less_eq_nat @ N @ ( finite1213132899lle_hf @ S ) )
% 0.26/2.09 => ~ ! [T3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ T3 @ S )
% 0.26/2.09 => ( ( ( finite1213132899lle_hf @ T3 )
% 0.26/2.09 = N )
% 0.26/2.09 => ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_258_obtain__subset__with__card__n,axiom,
% 0.26/2.09 ! [N: nat,S: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 0.26/2.09 => ~ ! [T3: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ T3 @ S )
% 0.26/2.09 => ( ( ( finite_card_nat @ T3 )
% 0.26/2.09 = N )
% 0.26/2.09 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_259_card__le__if__inj__on__rel,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ! [A3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.26/2.09 => ? [B7: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ B7 @ B4 )
% 0.26/2.09 & ( R @ A3 @ B7 ) ) )
% 0.26/2.09 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ B2 @ B4 )
% 0.26/2.09 => ( ( R @ A1 @ B2 )
% 0.26/2.09 => ( ( R @ A22 @ B2 )
% 0.26/2.09 => ( A1 = A22 ) ) ) ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_260_card__le__if__inj__on__rel,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_HF_Mirabelle_hf,R: hF_Mirabelle_hf > nat > $o] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ! [A3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ A3 @ A4 )
% 0.26/2.09 => ? [B7: nat] :
% 0.26/2.09 ( ( member_nat @ B7 @ B4 )
% 0.26/2.09 & ( R @ A3 @ B7 ) ) )
% 0.26/2.09 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B2: nat] :
% 0.26/2.09 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.26/2.09 => ( ( member_nat @ B2 @ B4 )
% 0.26/2.09 => ( ( R @ A1 @ B2 )
% 0.26/2.09 => ( ( R @ A22 @ B2 )
% 0.26/2.09 => ( A1 = A22 ) ) ) ) ) )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_261_arg__min__least,axiom,
% 0.26/2.09 ! [S: set_nat,Y2: nat,F3: nat > nat] :
% 0.26/2.09 ( ( finite_finite_nat @ S )
% 0.26/2.09 => ( ( S != bot_bot_set_nat )
% 0.26/2.09 => ( ( member_nat @ Y2 @ S )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ ( lattic1974000059at_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_262_arg__min__least,axiom,
% 0.26/2.09 ! [S: set_HF_Mirabelle_hf,Y2: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ S )
% 0.26/2.09 => ( ( S != bot_bo53200981lle_hf )
% 0.26/2.09 => ( ( member1367349282lle_hf @ Y2 @ S )
% 0.26/2.09 => ( ord_less_eq_nat @ ( F3 @ ( lattic710307446hf_nat @ F3 @ S ) ) @ ( F3 @ Y2 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_263_bounded__Max__nat,axiom,
% 0.26/2.09 ! [P: nat > $o,X2: nat,M: nat] :
% 0.26/2.09 ( ( P @ X2 )
% 0.26/2.09 => ( ! [X3: nat] :
% 0.26/2.09 ( ( P @ X3 )
% 0.26/2.09 => ( ord_less_eq_nat @ X3 @ M ) )
% 0.26/2.09 => ~ ! [M2: nat] :
% 0.26/2.09 ( ( P @ M2 )
% 0.26/2.09 => ~ ! [X5: nat] :
% 0.26/2.09 ( ( P @ X5 )
% 0.26/2.09 => ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_264_Nat_Oex__has__greatest__nat,axiom,
% 0.26/2.09 ! [P: nat > $o,K: nat,B3: nat] :
% 0.26/2.09 ( ( P @ K )
% 0.26/2.09 => ( ! [Y3: nat] :
% 0.26/2.09 ( ( P @ Y3 )
% 0.26/2.09 => ( ord_less_eq_nat @ Y3 @ B3 ) )
% 0.26/2.09 => ? [X3: nat] :
% 0.26/2.09 ( ( P @ X3 )
% 0.26/2.09 & ! [Y5: nat] :
% 0.26/2.09 ( ( P @ Y5 )
% 0.26/2.09 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_265_nat__le__linear,axiom,
% 0.26/2.09 ! [M3: nat,N: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ M3 @ N )
% 0.26/2.09 | ( ord_less_eq_nat @ N @ M3 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_266_le__antisym,axiom,
% 0.26/2.09 ! [M3: nat,N: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ M3 @ N )
% 0.26/2.09 => ( ( ord_less_eq_nat @ N @ M3 )
% 0.26/2.09 => ( M3 = N ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_267_eq__imp__le,axiom,
% 0.26/2.09 ! [M3: nat,N: nat] :
% 0.26/2.09 ( ( M3 = N )
% 0.26/2.09 => ( ord_less_eq_nat @ M3 @ N ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_268_le__trans,axiom,
% 0.26/2.09 ! [I: nat,J: nat,K: nat] :
% 0.26/2.09 ( ( ord_less_eq_nat @ I @ J )
% 0.26/2.09 => ( ( ord_less_eq_nat @ J @ K )
% 0.26/2.09 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_269_le__refl,axiom,
% 0.26/2.09 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_270_finite__nat__set__iff__bounded__le,axiom,
% 0.26/2.09 ( finite_finite_nat
% 0.26/2.09 = ( ^ [N2: set_nat] :
% 0.26/2.09 ? [M4: nat] :
% 0.26/2.09 ! [X: nat] :
% 0.26/2.09 ( ( member_nat @ X @ N2 )
% 0.26/2.09 => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_271_card__Diff1__le,axiom,
% 0.26/2.09 ! [A4: set_nat,X2: nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_272_card__Diff1__le,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) ) @ ( finite1213132899lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_273_card__le__inj,axiom,
% 0.26/2.09 ! [A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite1450550360lle_hf @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.26/2.09 => ? [F5: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F5 @ A4 ) @ B4 )
% 0.26/2.09 & ( inj_on811196232lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_274_card__le__inj,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.26/2.09 => ? [F5: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F5 @ A4 ) @ B4 )
% 0.26/2.09 & ( inj_on755450110lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_275_card__le__inj,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.26/2.09 => ? [F5: hF_Mirabelle_hf > nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F5 @ A4 ) @ B4 )
% 0.26/2.09 & ( inj_on1874279374hf_nat @ F5 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_276_card__le__inj,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.26/2.09 => ? [F5: nat > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F5 @ A4 ) @ B4 )
% 0.26/2.09 & ( inj_on1988990670lle_hf @ F5 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_277_card__le__inj,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.26/2.09 => ? [F5: nat > nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B4 )
% 0.26/2.09 & ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_278_card__inj__on__le,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ B4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_279_image__eqI,axiom,
% 0.26/2.09 ! [B3: hF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( B3
% 0.26/2.09 = ( F3 @ X2 ) )
% 0.26/2.09 => ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ B3 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_280_DiffI,axiom,
% 0.26/2.09 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ C @ A4 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ C @ B4 )
% 0.26/2.09 => ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_281_Diff__iff,axiom,
% 0.26/2.09 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.26/2.09 = ( ( member1367349282lle_hf @ C @ A4 )
% 0.26/2.09 & ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_282_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( minus_minus_nat @ A2 @ A2 )
% 0.26/2.09 = zero_zero_nat ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_283_diff__zero,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 0.26/2.09 = A2 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_284_zero__diff,axiom,
% 0.26/2.09 ! [A2: nat] :
% 0.26/2.09 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 0.26/2.09 = zero_zero_nat ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_285_image__empty,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( image_1743964010lle_hf @ F3 @ bot_bo53200981lle_hf )
% 0.26/2.09 = bot_bo53200981lle_hf ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_286_empty__is__image,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( bot_bo53200981lle_hf
% 0.26/2.09 = ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_287_image__is__empty,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( image_1743964010lle_hf @ F3 @ A4 )
% 0.26/2.09 = bot_bo53200981lle_hf )
% 0.26/2.09 = ( A4 = bot_bo53200981lle_hf ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_288_finite__imageI,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_289_finite__imageI,axiom,
% 0.26/2.09 ! [F: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ F )
% 0.26/2.09 => ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_290_finite__imageI,axiom,
% 0.26/2.09 ! [F: set_nat,H: nat > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_291_finite__imageI,axiom,
% 0.26/2.09 ! [F: set_nat,H: nat > nat] :
% 0.26/2.09 ( ( finite_finite_nat @ F )
% 0.26/2.09 => ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_292_image__insert,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( image_1743964010lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.26/2.09 = ( insert9649339lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_293_insert__image,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( insert9649339lle_hf @ ( F3 @ X2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( image_1743964010lle_hf @ F3 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_294_Diff__empty,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( minus_1450406810lle_hf @ A4 @ bot_bo53200981lle_hf )
% 0.26/2.09 = A4 ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_295_empty__Diff,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A4 )
% 0.26/2.09 = bot_bo53200981lle_hf ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_296_Diff__cancel,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( minus_1450406810lle_hf @ A4 @ A4 )
% 0.26/2.09 = bot_bo53200981lle_hf ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_297_finite__Diff,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_298_finite__Diff,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_299_finite__Diff2,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.26/2.09 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_300_finite__Diff2,axiom,
% 0.26/2.09 ! [B4: set_nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
% 0.26/2.09 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_301_Diff__insert0,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ B4 ) )
% 0.26/2.09 = ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_302_insert__Diff1,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X2 @ B4 )
% 0.26/2.09 => ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ B4 )
% 0.26/2.09 = ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_303_Diff__eq__empty__iff,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( minus_1450406810lle_hf @ A4 @ B4 )
% 0.26/2.09 = bot_bo53200981lle_hf )
% 0.26/2.09 = ( ord_le432112161lle_hf @ A4 @ B4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_304_insert__Diff__single,axiom,
% 0.26/2.09 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
% 0.26/2.09 = ( insert9649339lle_hf @ A2 @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_305_finite__Diff__insert,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) ) )
% 0.26/2.09 = ( finite586181922lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_306_finite__Diff__insert,axiom,
% 0.26/2.09 ! [A4: set_nat,A2: nat,B4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A2 @ B4 ) ) )
% 0.26/2.09 = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_307_inj__on__insert,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ ( insert1636143089lle_hf @ A2 @ A4 ) )
% 0.26/2.09 = ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 & ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ ( insert1636143089lle_hf @ A2 @ bot_bo2093393035lle_hf ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_308_inj__on__insert,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on755450110lle_hf @ F3 @ ( insert9649339lle_hf @ A2 @ A4 ) )
% 0.26/2.09 = ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.26/2.09 & ~ ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_309_inj__on__image__set__diff,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ C3 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ ( minus_500612048lle_hf @ A4 @ B4 ) @ C3 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ B4 @ C3 )
% 0.26/2.09 => ( ( image_899003828lle_hf @ F3 @ ( minus_500612048lle_hf @ A4 @ B4 ) )
% 0.26/2.09 = ( minus_1450406810lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) @ ( image_899003828lle_hf @ F3 @ B4 ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_310_image__subsetI,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [X3: hF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ ( F3 @ X3 ) @ B4 ) )
% 0.26/2.09 => ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) @ B4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_311_finite__surj,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_312_finite__surj,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_nat,F3: hF_Mirabelle_hf > nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.26/2.09 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_313_finite__surj,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_314_finite__surj,axiom,
% 0.26/2.09 ! [A4: set_nat,B4: set_nat,F3: nat > nat] :
% 0.26/2.09 ( ( finite_finite_nat @ A4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.26/2.09 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_315_finite__subset__image,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ? [C4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.26/2.09 & ( finite586181922lle_hf @ C4 )
% 0.26/2.09 & ( B4
% 0.26/2.09 = ( image_1743964010lle_hf @ F3 @ C4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_316_finite__subset__image,axiom,
% 0.26/2.09 ! [B4: set_HF_Mirabelle_hf,F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ? [C4: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.26/2.09 & ( finite_finite_nat @ C4 )
% 0.26/2.09 & ( B4
% 0.26/2.09 = ( image_246164834lle_hf @ F3 @ C4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_317_finite__subset__image,axiom,
% 0.26/2.09 ! [B4: set_nat,F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.26/2.09 => ? [C4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.26/2.09 & ( finite586181922lle_hf @ C4 )
% 0.26/2.09 & ( B4
% 0.26/2.09 = ( image_131453538hf_nat @ F3 @ C4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_318_finite__subset__image,axiom,
% 0.26/2.09 ! [B4: set_nat,F3: nat > nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B4 )
% 0.26/2.09 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.26/2.09 => ? [C4: set_nat] :
% 0.26/2.09 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.26/2.09 & ( finite_finite_nat @ C4 )
% 0.26/2.09 & ( B4
% 0.26/2.09 = ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_319_ex__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ? [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 & ( P @ B6 ) ) )
% 0.26/2.09 = ( ? [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ A4 )
% 0.26/2.09 & ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_320_ex__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ? [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.26/2.09 & ( P @ B6 ) ) )
% 0.26/2.09 = ( ? [B6: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ A4 )
% 0.26/2.09 & ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_321_ex__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.26/2.09 ( ( ? [B6: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.26/2.09 & ( P @ B6 ) ) )
% 0.26/2.09 = ( ? [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ A4 )
% 0.26/2.09 & ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_322_ex__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( ? [B6: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) )
% 0.26/2.09 & ( P @ B6 ) ) )
% 0.26/2.09 = ( ? [B6: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ A4 )
% 0.26/2.09 & ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_323_all__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ! [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ ( image_1743964010lle_hf @ F3 @ A4 ) ) )
% 0.26/2.09 => ( P @ B6 ) ) )
% 0.26/2.09 = ( ! [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ A4 ) )
% 0.26/2.09 => ( P @ ( image_1743964010lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_324_all__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.26/2.09 ( ( ! [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ ( image_246164834lle_hf @ F3 @ A4 ) ) )
% 0.26/2.09 => ( P @ B6 ) ) )
% 0.26/2.09 = ( ! [B6: set_nat] :
% 0.26/2.09 ( ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ A4 ) )
% 0.26/2.09 => ( P @ ( image_246164834lle_hf @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_325_all__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.26/2.09 ( ( ! [B6: set_nat] :
% 0.26/2.09 ( ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ ( image_131453538hf_nat @ F3 @ A4 ) ) )
% 0.26/2.09 => ( P @ B6 ) ) )
% 0.26/2.09 = ( ! [B6: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ( finite586181922lle_hf @ B6 )
% 0.26/2.09 & ( ord_le432112161lle_hf @ B6 @ A4 ) )
% 0.26/2.09 => ( P @ ( image_131453538hf_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_326_all__finite__subset__image,axiom,
% 0.26/2.09 ! [F3: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.26/2.09 ( ( ! [B6: set_nat] :
% 0.26/2.09 ( ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F3 @ A4 ) ) )
% 0.26/2.09 => ( P @ B6 ) ) )
% 0.26/2.09 = ( ! [B6: set_nat] :
% 0.26/2.09 ( ( ( finite_finite_nat @ B6 )
% 0.26/2.09 & ( ord_less_eq_set_nat @ B6 @ A4 ) )
% 0.26/2.09 => ( P @ ( image_nat_nat @ F3 @ B6 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_327_finite__imageD,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_328_finite__imageD,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_329_finite__imageD,axiom,
% 0.26/2.09 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.26/2.09 ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ( ( inj_on1988990670lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_330_finite__imageD,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.26/2.09 => ( ( inj_on1874279374hf_nat @ F3 @ A4 )
% 0.26/2.09 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_331_finite__imageD,axiom,
% 0.26/2.09 ! [F3: nat > nat,A4: set_nat] :
% 0.26/2.09 ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
% 0.26/2.09 => ( ( inj_on_nat_nat @ F3 @ A4 )
% 0.26/2.09 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_332_finite__image__iff,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_333_finite__image__iff,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_334_finite__image__iff,axiom,
% 0.26/2.09 ! [F3: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.26/2.09 ( ( inj_on1988990670lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_335_finite__image__iff,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on1874279374hf_nat @ F3 @ A4 )
% 0.26/2.09 => ( ( finite_finite_nat @ ( image_131453538hf_nat @ F3 @ A4 ) )
% 0.26/2.09 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_336_finite__image__iff,axiom,
% 0.26/2.09 ! [F3: nat > nat,A4: set_nat] :
% 0.26/2.09 ( ( inj_on_nat_nat @ F3 @ A4 )
% 0.26/2.09 => ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A4 ) )
% 0.26/2.09 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_337_inj__on__image__eq__iff,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,C3: set_se933006839lle_hf,A4: set_se933006839lle_hf,B4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ C3 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ A4 @ C3 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ B4 @ C3 )
% 0.26/2.09 => ( ( ( image_899003828lle_hf @ F3 @ A4 )
% 0.26/2.09 = ( image_899003828lle_hf @ F3 @ B4 ) )
% 0.26/2.09 = ( A4 = B4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_338_inj__on__image__mem__iff,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_se933006839lle_hf,A2: set_HF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ B4 )
% 0.26/2.09 => ( ( member1490636632lle_hf @ A2 @ B4 )
% 0.26/2.09 => ( ( ord_le2016357975lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( member1490636632lle_hf @ A2 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_339_inj__on__image__mem__iff,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on755450110lle_hf @ F3 @ B4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.26/2.09 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.26/2.09 => ( ( member1367349282lle_hf @ ( F3 @ A2 ) @ ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( member1367349282lle_hf @ A2 @ A4 ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_340_inj__img__insertE,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.26/2.09 => ( ( ( insert9649339lle_hf @ X2 @ B4 )
% 0.26/2.09 = ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ~ ! [X6: set_HF_Mirabelle_hf,A7: set_se933006839lle_hf] :
% 0.26/2.09 ( ~ ( member1490636632lle_hf @ X6 @ A7 )
% 0.26/2.09 => ( ( A4
% 0.26/2.09 = ( insert1636143089lle_hf @ X6 @ A7 ) )
% 0.26/2.09 => ( ( X2
% 0.26/2.09 = ( F3 @ X6 ) )
% 0.26/2.09 => ( B4
% 0.26/2.09 != ( image_899003828lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_341_inj__img__insertE,axiom,
% 0.26/2.09 ! [F3: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( inj_on755450110lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ~ ( member1367349282lle_hf @ X2 @ B4 )
% 0.26/2.09 => ( ( ( insert9649339lle_hf @ X2 @ B4 )
% 0.26/2.09 = ( image_1743964010lle_hf @ F3 @ A4 ) )
% 0.26/2.09 => ~ ! [X6: hF_Mirabelle_hf,A7: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( member1367349282lle_hf @ X6 @ A7 )
% 0.26/2.09 => ( ( A4
% 0.26/2.09 = ( insert9649339lle_hf @ X6 @ A7 ) )
% 0.26/2.09 => ( ( X2
% 0.26/2.09 = ( F3 @ X6 ) )
% 0.26/2.09 => ( B4
% 0.26/2.09 != ( image_1743964010lle_hf @ F3 @ A7 ) ) ) ) ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_342_card__image,axiom,
% 0.26/2.09 ! [F3: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.26/2.09 ( ( inj_on811196232lle_hf @ F3 @ A4 )
% 0.26/2.09 => ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F3 @ A4 ) )
% 0.26/2.09 = ( finite90088345lle_hf @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_343_Diff__insert__absorb,axiom,
% 0.26/2.09 ! [X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ~ ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 => ( ( minus_1450406810lle_hf @ ( insert9649339lle_hf @ X2 @ A4 ) @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) )
% 0.26/2.09 = A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_344_Diff__insert2,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.26/2.09 = ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_345_insert__Diff,axiom,
% 0.26/2.09 ! [A2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ A2 @ A4 )
% 0.26/2.09 => ( ( insert9649339lle_hf @ A2 @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) )
% 0.26/2.09 = A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_346_Diff__insert,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.26/2.09 = ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A4 @ B4 ) @ ( insert9649339lle_hf @ A2 @ bot_bo53200981lle_hf ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_347_subset__Diff__insert,axiom,
% 0.26/2.09 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,X2: hF_Mirabelle_hf,C3: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ ( insert9649339lle_hf @ X2 @ C3 ) ) )
% 0.26/2.09 = ( ( ord_le432112161lle_hf @ A4 @ ( minus_1450406810lle_hf @ B4 @ C3 ) )
% 0.26/2.09 & ~ ( member1367349282lle_hf @ X2 @ A4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_348_in__image__insert__iff,axiom,
% 0.26/2.09 ! [B4: set_se933006839lle_hf,X2: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ! [C4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1490636632lle_hf @ C4 @ B4 )
% 0.26/2.09 => ~ ( member1367349282lle_hf @ X2 @ C4 ) )
% 0.26/2.09 => ( ( member1490636632lle_hf @ A4 @ ( image_1514960916lle_hf @ ( insert9649339lle_hf @ X2 ) @ B4 ) )
% 0.26/2.09 = ( ( member1367349282lle_hf @ X2 @ A4 )
% 0.26/2.09 & ( member1490636632lle_hf @ ( minus_1450406810lle_hf @ A4 @ ( insert9649339lle_hf @ X2 @ bot_bo53200981lle_hf ) ) @ B4 ) ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_349_DiffE,axiom,
% 0.26/2.09 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.26/2.09 => ~ ( ( member1367349282lle_hf @ C @ A4 )
% 0.26/2.09 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(fact_350_DiffD1,axiom,
% 0.26/2.09 ! [C: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.26/2.09 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A4 @ B4 ) )
% 0.26/2.09 => ( member1367349282lle_hf @ C @ A4 ) ) ).
% 0.26/2.09
% 0.26/2.09 thf(conj_0,conjecture,
% 0.26/2.09 ( ( z = zero_z189798548lle_hf )
% 0.26/2.09 = ( ! [X: hF_Mirabelle_hf] :
% 0.26/2.09 ~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
% 0.26/2.09 ------------------------
% 4.05/2.13 % (20481)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/2Mi)
% 4.05/2.13 % (20480)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/27Mi)
% 4.05/2.13 % (20482)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/2Mi)
% 4.05/2.13 % (20484)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/18Mi)
% 4.05/2.13 % (20478)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/183Mi)
% 4.05/2.13 % (20479)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/4Mi)
% 4.05/2.13 % (20483)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/275Mi)
% 4.05/2.13 % (20481)Instruction limit reached!
% 4.05/2.13 % (20481)------------------------------
% 4.05/2.13 % (20481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.13 % (20481)Termination reason: Unknown
% 4.05/2.13 % (20481)Termination phase: shuffling
% 4.05/2.13 % (20482)Instruction limit reached!
% 4.05/2.13 % (20482)------------------------------
% 4.05/2.13 % (20482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.13
% 4.05/2.13 % (20481)Memory used [KB]: 1407
% 4.05/2.13 % (20481)Time elapsed: 0.004 s
% 4.05/2.13 % (20481)Instructions burned: 3 (million)
% 4.05/2.13 % (20481)------------------------------
% 4.05/2.13 % (20481)------------------------------
% 4.05/2.13 % (20482)Termination reason: Unknown
% 4.05/2.13 % (20482)Termination phase: shuffling
% 4.05/2.13
% 4.05/2.13 % (20482)Memory used [KB]: 1407
% 4.05/2.13 % (20482)Time elapsed: 0.004 s
% 4.05/2.13 % (20482)Instructions burned: 3 (million)
% 4.05/2.13 % (20482)------------------------------
% 4.05/2.13 % (20482)------------------------------
% 4.05/2.14 % (20479)Instruction limit reached!
% 4.05/2.14 % (20479)------------------------------
% 4.05/2.14 % (20479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.14 % (20479)Termination reason: Unknown
% 4.05/2.14 % (20479)Termination phase: shuffling
% 4.05/2.14
% 4.05/2.14 % (20479)Memory used [KB]: 1535
% 4.05/2.14 % (20479)Time elapsed: 0.005 s
% 4.05/2.14 % (20479)Instructions burned: 5 (million)
% 4.05/2.14 % (20479)------------------------------
% 4.05/2.14 % (20479)------------------------------
% 4.05/2.14 % (20484)Instruction limit reached!
% 4.05/2.14 % (20484)------------------------------
% 4.05/2.14 % (20484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.14 % (20484)Termination reason: Unknown
% 4.05/2.14 % (20484)Termination phase: shuffling
% 4.05/2.14
% 4.05/2.14 % (20484)Memory used [KB]: 1791
% 4.05/2.14 % (20484)Time elapsed: 0.012 s
% 4.05/2.14 % (20484)Instructions burned: 18 (million)
% 4.05/2.14 % (20484)------------------------------
% 4.05/2.14 % (20484)------------------------------
% 4.05/2.15 % (20480)Instruction limit reached!
% 4.05/2.15 % (20480)------------------------------
% 4.05/2.15 % (20480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.15 % (20480)Termination reason: Unknown
% 4.05/2.15 % (20480)Termination phase: Property scanning
% 4.05/2.15
% 4.05/2.15 % (20480)Memory used [KB]: 1918
% 4.05/2.15 % (20480)Time elapsed: 0.018 s
% 4.05/2.15 % (20480)Instructions burned: 28 (million)
% 4.05/2.15 % (20480)------------------------------
% 4.05/2.15 % (20480)------------------------------
% 4.05/2.15 % (20485)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/3Mi)
% 4.05/2.15 % (20486)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/37Mi)
% 4.05/2.15 % (20487)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/15Mi)
% 4.05/2.16 % (20485)Instruction limit reached!
% 4.05/2.16 % (20485)------------------------------
% 4.05/2.16 % (20485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.16 % (20485)Termination reason: Unknown
% 4.05/2.16 % (20485)Termination phase: shuffling
% 4.05/2.16
% 4.05/2.16 % (20485)Memory used [KB]: 1407
% 4.05/2.16 % (20485)Time elapsed: 0.004 s
% 4.05/2.16 % (20485)Instructions burned: 3 (million)
% 4.05/2.16 % (20485)------------------------------
% 4.05/2.16 % (20485)------------------------------
% 4.05/2.16 % (20483)Refutation not found, incomplete strategy
% 4.05/2.16 % (20483)------------------------------
% 4.05/2.16 % (20483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.16 % (20483)Termination reason: Refutation not found, incomplete strategy
% 4.05/2.16
% 4.05/2.16
% 4.05/2.16 % (20483)Memory used [KB]: 6524
% 4.05/2.16 % (20483)Time elapsed: 0.026 s
% 4.05/2.16 % (20483)Instructions burned: 42 (million)
% 4.05/2.16 % (20483)------------------------------
% 4.05/2.16 % (20483)------------------------------
% 4.05/2.16 % (20487)Instruction limit reached!
% 4.05/2.16 % (20487)------------------------------
% 4.05/2.16 % (20487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.16 % (20487)Termination reason: Unknown
% 4.05/2.16 % (20487)Termination phase: shuffling
% 4.05/2.16
% 4.05/2.16 % (20487)Memory used [KB]: 1663
% 4.05/2.16 % (20487)Time elapsed: 0.011 s
% 4.05/2.16 % (20487)Instructions burned: 16 (million)
% 4.05/2.16 % (20487)------------------------------
% 4.05/2.16 % (20487)------------------------------
% 4.05/2.17 % (20488)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/3Mi)
% 4.05/2.17 % (20489)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/1041Mi)
% 4.05/2.17 % (20488)Instruction limit reached!
% 4.05/2.17 % (20488)------------------------------
% 4.05/2.17 % (20488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.17 % (20488)Termination reason: Unknown
% 4.05/2.17 % (20488)Termination phase: shuffling
% 4.05/2.17
% 4.05/2.17 % (20488)Memory used [KB]: 1407
% 4.05/2.17 % (20488)Time elapsed: 0.004 s
% 4.05/2.17 % (20488)Instructions burned: 3 (million)
% 4.05/2.17 % (20488)------------------------------
% 4.05/2.17 % (20488)------------------------------
% 4.05/2.17 % (20486)Instruction limit reached!
% 4.05/2.17 % (20486)------------------------------
% 4.05/2.17 % (20486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.17 % (20486)Termination reason: Unknown
% 4.05/2.17 % (20486)Termination phase: shuffling
% 4.05/2.17
% 4.05/2.17 % (20486)Memory used [KB]: 2174
% 4.05/2.17 % (20486)Time elapsed: 0.022 s
% 4.05/2.17 % (20486)Instructions burned: 38 (million)
% 4.05/2.17 % (20486)------------------------------
% 4.05/2.17 % (20486)------------------------------
% 4.05/2.18 % (20490)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/7Mi)
% 4.05/2.18 % (20491)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/16Mi)
% 4.05/2.18 % (20490)Instruction limit reached!
% 4.05/2.18 % (20490)------------------------------
% 4.05/2.18 % (20490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.18 % (20490)Termination reason: Unknown
% 4.05/2.18 % (20490)Termination phase: shuffling
% 4.05/2.18
% 4.05/2.18 % (20490)Memory used [KB]: 1535
% 4.05/2.18 % (20490)Time elapsed: 0.007 s
% 4.05/2.18 % (20490)Instructions burned: 8 (million)
% 4.05/2.18 % (20490)------------------------------
% 4.05/2.18 % (20490)------------------------------
% 4.05/2.18 % (20492)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/3Mi)
% 4.05/2.19 % (20492)Instruction limit reached!
% 4.05/2.19 % (20492)------------------------------
% 4.05/2.19 % (20492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.19 % (20492)Termination reason: Unknown
% 4.05/2.19 % (20492)Termination phase: shuffling
% 4.05/2.19
% 4.05/2.19 % (20492)Memory used [KB]: 1535
% 4.05/2.19 % (20492)Time elapsed: 0.004 s
% 4.05/2.19 % (20492)Instructions burned: 4 (million)
% 4.05/2.19 % (20492)------------------------------
% 4.05/2.19 % (20492)------------------------------
% 4.05/2.19 % (20491)Instruction limit reached!
% 4.05/2.19 % (20491)------------------------------
% 4.05/2.19 % (20491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.19 % (20491)Termination reason: Unknown
% 4.05/2.19 % (20491)Termination phase: shuffling
% 4.05/2.19
% 4.05/2.19 % (20491)Memory used [KB]: 1791
% 4.05/2.19 % (20491)Time elapsed: 0.011 s
% 4.05/2.19 % (20491)Instructions burned: 16 (million)
% 4.05/2.19 % (20491)------------------------------
% 4.05/2.19 % (20491)------------------------------
% 4.05/2.19 % (20493)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/3Mi)
% 4.05/2.19 % (20493)Instruction limit reached!
% 4.05/2.19 % (20493)------------------------------
% 4.05/2.19 % (20493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.19 % (20493)Termination reason: Unknown
% 4.05/2.19 % (20493)Termination phase: shuffling
% 4.05/2.19
% 4.05/2.19 % (20493)Memory used [KB]: 1535
% 4.05/2.19 % (20493)Time elapsed: 0.004 s
% 4.05/2.19 % (20493)Instructions burned: 4 (million)
% 4.05/2.19 % (20493)------------------------------
% 4.05/2.19 % (20493)------------------------------
% 4.05/2.19 % (20494)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/7Mi)
% 4.05/2.20 % (20494)Instruction limit reached!
% 4.05/2.20 % (20494)------------------------------
% 4.05/2.20 % (20494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.20 % (20494)Termination reason: Unknown
% 4.05/2.20 % (20494)Termination phase: shuffling
% 4.05/2.20
% 4.05/2.20 % (20494)Memory used [KB]: 1535
% 4.05/2.20 % (20494)Time elapsed: 0.007 s
% 4.05/2.20 % (20494)Instructions burned: 7 (million)
% 4.05/2.20 % (20494)------------------------------
% 4.05/2.20 % (20494)------------------------------
% 4.05/2.20 % (20495)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/3Mi)
% 4.05/2.20 % (20495)Instruction limit reached!
% 4.05/2.20 % (20495)------------------------------
% 4.05/2.20 % (20495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.20 % (20495)Termination reason: Unknown
% 4.05/2.20 % (20495)Termination phase: shuffling
% 4.05/2.20
% 4.05/2.20 % (20495)Memory used [KB]: 1535
% 4.05/2.20 % (20495)Time elapsed: 0.004 s
% 4.05/2.20 % (20495)Instructions burned: 4 (million)
% 4.05/2.20 % (20495)------------------------------
% 4.05/2.20 % (20495)------------------------------
% 4.05/2.20 % (20496)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/4Mi)
% 4.05/2.20 % (20497)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/18Mi)
% 4.05/2.21 % (20496)Instruction limit reached!
% 4.05/2.21 % (20496)------------------------------
% 4.05/2.21 % (20496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.21 % (20496)Termination reason: Unknown
% 4.05/2.21 % (20496)Termination phase: shuffling
% 4.05/2.21
% 4.05/2.21 % (20496)Memory used [KB]: 1535
% 4.05/2.21 % (20496)Time elapsed: 0.004 s
% 4.05/2.21 % (20496)Instructions burned: 4 (million)
% 4.05/2.21 % (20496)------------------------------
% 4.05/2.21 % (20496)------------------------------
% 4.05/2.21 % (20498)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/710Mi)
% 4.05/2.22 % (20497)Instruction limit reached!
% 4.05/2.22 % (20497)------------------------------
% 4.05/2.22 % (20497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.22 % (20497)Termination reason: Unknown
% 4.05/2.22 % (20497)Termination phase: shuffling
% 4.05/2.22
% 4.05/2.22 % (20497)Memory used [KB]: 1918
% 4.05/2.22 % (20497)Time elapsed: 0.012 s
% 4.05/2.22 % (20497)Instructions burned: 19 (million)
% 4.05/2.22 % (20497)------------------------------
% 4.05/2.22 % (20497)------------------------------
% 4.05/2.22 % (20499)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/6Mi)
% 4.05/2.22 % (20500)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_20378 for (2999ds/902Mi)
% 4.05/2.22 % (20499)Instruction limit reached!
% 4.05/2.22 % (20499)------------------------------
% 4.05/2.22 % (20499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.22 % (20499)Termination reason: Unknown
% 4.05/2.22 % (20499)Termination phase: shuffling
% 4.05/2.22
% 4.05/2.22 % (20499)Memory used [KB]: 1535
% 4.05/2.22 % (20499)Time elapsed: 0.006 s
% 4.05/2.22 % (20499)Instructions burned: 6 (million)
% 4.05/2.22 % (20499)------------------------------
% 4.05/2.22 % (20499)------------------------------
% 4.05/2.23 % (20501)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_20378 for (2998ds/21Mi)
% 4.05/2.23 % (20478)Instruction limit reached!
% 4.05/2.23 % (20478)------------------------------
% 4.05/2.23 % (20478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.23 % (20478)Termination reason: Unknown
% 4.05/2.23 % (20478)Termination phase: Saturation
% 4.05/2.23
% 4.05/2.23 % (20478)Memory used [KB]: 7547
% 4.05/2.23 % (20478)Time elapsed: 0.102 s
% 4.05/2.23 % (20478)Instructions burned: 184 (million)
% 4.05/2.23 % (20478)------------------------------
% 4.05/2.23 % (20478)------------------------------
% 4.05/2.24 % (20502)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_20378 for (2998ds/5Mi)
% 4.05/2.24 % (20501)Instruction limit reached!
% 4.05/2.24 % (20501)------------------------------
% 4.05/2.24 % (20501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.24 % (20501)Termination reason: Unknown
% 4.05/2.24 % (20501)Termination phase: shuffling
% 4.05/2.24
% 4.05/2.24 % (20501)Memory used [KB]: 1918
% 4.05/2.24 % (20501)Time elapsed: 0.014 s
% 4.05/2.24 % (20502)Instruction limit reached!
% 4.05/2.24 % (20502)------------------------------
% 4.05/2.24 % (20502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.24 % (20502)Termination reason: Unknown
% 4.05/2.24 % (20502)Termination phase: shuffling
% 4.05/2.24
% 4.05/2.24 % (20502)Memory used [KB]: 1535
% 4.05/2.24 % (20502)Time elapsed: 0.005 s
% 4.05/2.24 % (20502)Instructions burned: 6 (million)
% 4.05/2.24 % (20502)------------------------------
% 4.05/2.24 % (20502)------------------------------
% 4.05/2.24 % (20501)Instructions burned: 21 (million)
% 4.05/2.24 % (20501)------------------------------
% 4.05/2.24 % (20501)------------------------------
% 4.05/2.24 % (20498)First to succeed.
% 4.05/2.24 % (20503)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_20378 for (2998ds/6Mi)
% 4.05/2.24 % (20498)Refutation found. Thanks to Tanya!
% 4.05/2.24 % SZS status Theorem for DTF2THF_20378
% 4.05/2.24 % SZS output start Proof for DTF2THF_20378
% 4.05/2.24 thf(type_def_5, type, set_HF_Mirabelle_hf: $tType).
% 4.05/2.24 thf(type_def_6, type, nat: $tType).
% 4.05/2.24 thf(type_def_8, type, set_nat: $tType).
% 4.05/2.24 thf(type_def_9, type, set_se933006839lle_hf: $tType).
% 4.05/2.24 thf(type_def_10, type, hF_Mirabelle_hf: $tType).
% 4.05/2.24 thf(func_def_0, type, set_se933006839lle_hf: $tType).
% 4.05/2.24 thf(func_def_1, type, set_HF_Mirabelle_hf: $tType).
% 4.05/2.24 thf(func_def_2, type, hF_Mirabelle_hf: $tType).
% 4.05/2.24 thf(func_def_3, type, set_nat: $tType).
% 4.05/2.24 thf(func_def_4, type, nat: $tType).
% 4.05/2.24 thf(func_def_5, type, finite1213132899lle_hf: set_HF_Mirabelle_hf > nat).
% 4.05/2.24 thf(func_def_6, type, finite_card_nat: set_nat > nat).
% 4.05/2.24 thf(func_def_7, type, finite90088345lle_hf: set_se933006839lle_hf > nat).
% 4.05/2.24 thf(func_def_8, type, finite586181922lle_hf: set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_9, type, finite_finite_nat: set_nat > $o).
% 4.05/2.24 thf(func_def_10, type, finite1450550360lle_hf: set_se933006839lle_hf > $o).
% 4.05/2.24 thf(func_def_11, type, inj_on755450110lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf) > set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_12, type, inj_on1874279374hf_nat: (hF_Mirabelle_hf > nat) > set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_13, type, inj_on1988990670lle_hf: (nat > hF_Mirabelle_hf) > set_nat > $o).
% 4.05/2.24 thf(func_def_14, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 4.05/2.24 thf(func_def_15, type, inj_on811196232lle_hf: (set_HF_Mirabelle_hf > hF_Mirabelle_hf) > set_se933006839lle_hf > $o).
% 4.05/2.24 thf(func_def_16, type, minus_minus_nat: nat > nat > nat).
% 4.05/2.24 thf(func_def_17, type, minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_18, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 4.05/2.24 thf(func_def_19, type, minus_500612048lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > set_se933006839lle_hf).
% 4.05/2.24 thf(func_def_20, type, zero_z189798548lle_hf: hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_21, type, zero_zero_nat: nat).
% 4.05/2.24 thf(func_def_22, type, hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_23, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_24, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_25, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_26, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_27, type, lattic710307446hf_nat: (hF_Mirabelle_hf > nat) > set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_28, type, lattic1974000059at_nat: (nat > nat) > set_nat > nat).
% 4.05/2.24 thf(func_def_29, type, semiring_1_Nats_nat: set_nat).
% 4.05/2.24 thf(func_def_30, type, bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_31, type, bot_bot_nat_o: nat > $o).
% 4.05/2.24 thf(func_def_32, type, bot_bo554042810e_hf_o: set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_34, type, bot_bot_nat: nat).
% 4.05/2.24 thf(func_def_35, type, bot_bo53200981lle_hf: set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_36, type, bot_bot_set_nat: set_nat).
% 4.05/2.24 thf(func_def_37, type, bot_bo2093393035lle_hf: set_se933006839lle_hf).
% 4.05/2.24 thf(func_def_38, type, ord_less_eq_nat: nat > nat > $o).
% 4.05/2.24 thf(func_def_39, type, ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_40, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 4.05/2.24 thf(func_def_41, type, ord_le2016357975lle_hf: set_se933006839lle_hf > set_se933006839lle_hf > $o).
% 4.05/2.24 thf(func_def_42, type, semiri2115134414it_nat: nat > nat > nat).
% 4.05/2.24 thf(func_def_43, type, semiri2013084963it_nat: nat > nat > nat).
% 4.05/2.24 thf(func_def_44, type, collec2046588256lle_hf: (hF_Mirabelle_hf > $o) > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_45, type, collect_nat: (nat > $o) > set_nat).
% 4.05/2.24 thf(func_def_46, type, collec1758573718lle_hf: (set_HF_Mirabelle_hf > $o) > set_se933006839lle_hf).
% 4.05/2.24 thf(func_def_47, type, image_1743964010lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_48, type, image_131453538hf_nat: (hF_Mirabelle_hf > nat) > set_HF_Mirabelle_hf > set_nat).
% 4.05/2.24 thf(func_def_49, type, image_246164834lle_hf: (nat > hF_Mirabelle_hf) > set_nat > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_50, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 4.05/2.24 thf(func_def_51, type, image_899003828lle_hf: (set_HF_Mirabelle_hf > hF_Mirabelle_hf) > set_se933006839lle_hf > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_52, type, image_1514960916lle_hf: (set_HF_Mirabelle_hf > set_HF_Mirabelle_hf) > set_se933006839lle_hf > set_se933006839lle_hf).
% 4.05/2.24 thf(func_def_53, type, insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_54, type, insert_nat: nat > set_nat > set_nat).
% 4.05/2.24 thf(func_def_55, type, insert1636143089lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > set_se933006839lle_hf).
% 4.05/2.24 thf(func_def_56, type, is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_57, type, is_sin1448700567lle_hf: set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_58, type, the_el1104322134lle_hf: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_59, type, member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 4.05/2.24 thf(func_def_60, type, member_nat: nat > set_nat > $o).
% 4.05/2.24 thf(func_def_61, type, member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o).
% 4.05/2.24 thf(func_def_62, type, z: hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_77, type, sK0: hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_78, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(func_def_79, type, sK2: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 4.05/2.24 thf(f1050,plain,(
% 4.05/2.24 $false),
% 4.05/2.24 inference(avatar_sat_refutation,[status(thm)],[f1030,f1035,f1040,f1043,f1049])).
% 4.05/2.24 thf(f1049,plain,(
% 4.05/2.24 ~spl3_1 | spl3_2),
% 4.05/2.24 inference(avatar_contradiction_clause,[status(thm)],[f1048])).
% 4.05/2.24 thf(f1048,plain,(
% 4.05/2.24 $false | (~spl3_1 | spl3_2)),
% 4.05/2.24 inference(subsumption_resolution,[status(thm)],[f1047,f1028])).
% 4.05/2.24 thf(f1028,plain,(
% 4.05/2.24 ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) != z) | spl3_2),
% 4.05/2.24 inference(avatar_component_clause,[status(thm)],[f1027])).
% 4.05/2.24 thf(f1027,definition,(
% 4.05/2.24 spl3_2 <=> ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z)),
% 4.05/2.24 introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition])).
% 4.05/2.24 thf(f1047,plain,(
% 4.05/2.24 ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z) | ~spl3_1),
% 4.05/2.24 inference(trivial_inequality_removal,[status(thm)],[f1045])).
% 4.05/2.24 thf(f1045,plain,(
% 4.05/2.24 ($true != $true) | ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z) | ~spl3_1),
% 4.05/2.24 inference(superposition,[status(thm)],[f1025,f1044])).
% 4.05/2.24 thf(f1044,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (($true = (member1367349282lle_hf @ (sK2 @ X0) @ (hF_Mirabelle_hfset @ X0))) | ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = X0)) )),
% 4.05/2.24 inference(superposition,[status(thm)],[f1021,f1005])).
% 4.05/2.24 thf(f1005,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ (sK2 @ X0) @ (sK1 @ X0)) = X0) | ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = X0)) )),
% 4.05/2.24 inference(definition_unfolding,[status(thm)],[f992,f985])).
% 4.05/2.24 thf(f985,plain,(
% 4.05/2.24 (zero_z189798548lle_hf = (hF_Mirabelle_HF @ bot_bo53200981lle_hf))),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f10])).
% 4.05/2.24 thf(f10,axiom,(
% 4.05/2.24 (zero_z189798548lle_hf = (hF_Mirabelle_HF @ bot_bo53200981lle_hf))),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_9_Zero__hf__def)).
% 4.05/2.24 thf(f992,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf = X0) | ((hF_Mirabelle_hinsert @ (sK2 @ X0) @ (sK1 @ X0)) = X0)) )),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f980])).
% 4.05/2.24 thf(f980,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf = X0) | (((hF_Mirabelle_hinsert @ (sK2 @ X0) @ (sK1 @ X0)) = X0) & ((hF_Mirabelle_hmem @ (sK2 @ X0) @ (sK1 @ X0)) != $true)))),
% 4.05/2.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2])],[f973,f979])).
% 4.05/2.24 thf(f979,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : (? [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X2 @ X1) = X0) & ($true != (hF_Mirabelle_hmem @ X2 @ X1))) => (((hF_Mirabelle_hinsert @ (sK2 @ X0) @ (sK1 @ X0)) = X0) & ((hF_Mirabelle_hmem @ (sK2 @ X0) @ (sK1 @ X0)) != $true)))),
% 4.05/2.24 introduced(definition,[],[choice_axiom])).
% 4.05/2.24 thf(f973,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf = X0) | ? [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X2 @ X1) = X0) & ($true != (hF_Mirabelle_hmem @ X2 @ X1))))),
% 4.05/2.24 inference(ennf_transformation,[status(thm)],[f620])).
% 4.05/2.24 thf(f620,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf != X0) => ~! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X2 @ X1) = X0) => ($true = (hF_Mirabelle_hmem @ X2 @ X1))))),
% 4.05/2.24 inference(fool_elimination,[status(thm)],[f619])).
% 4.05/2.24 thf(f619,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf != X0) => ~! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X2 @ X1) = X0) => (hF_Mirabelle_hmem @ X2 @ X1)))),
% 4.05/2.24 inference(rectify,[status(thm)],[f4])).
% 4.05/2.24 thf(f4,axiom,(
% 4.05/2.24 ! [X6 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf != X6) => ~! [X8 : hF_Mirabelle_hf,X7 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X7 @ X8) = X6) => (hF_Mirabelle_hmem @ X7 @ X8)))),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_3_hf__cases)).
% 4.05/2.24 thf(f1021,plain,(
% 4.05/2.24 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf] : (($true = (member1367349282lle_hf @ X2 @ (hF_Mirabelle_hfset @ (hF_Mirabelle_hinsert @ X2 @ X0))))) )),
% 4.05/2.24 inference(beta_eta_normalization,[status(thm)],[f1013])).
% 4.05/2.24 thf(f1013,plain,(
% 4.05/2.24 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf] : (($true = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X2 @ (hF_Mirabelle_hinsert @ X2 @ X0)))) )),
% 4.05/2.24 inference(equality_resolution,[status(thm)],[f1008])).
% 4.05/2.24 thf(f1008,plain,(
% 4.05/2.24 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0))) | (X1 != X2)) )),
% 4.05/2.24 inference(definition_unfolding,[status(thm)],[f996,f997])).
% 4.05/2.24 thf(f997,plain,(
% 4.05/2.24 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f420])).
% 4.05/2.24 thf(f420,plain,(
% 4.05/2.24 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 4.05/2.24 inference(fool_elimination,[status(thm)],[f419])).
% 4.05/2.24 thf(f419,plain,(
% 4.05/2.24 (hF_Mirabelle_hmem = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ X1))))),
% 4.05/2.24 inference(rectify,[status(thm)],[f6])).
% 4.05/2.24 thf(f6,axiom,(
% 4.05/2.24 (hF_Mirabelle_hmem = (^[X2 : hF_Mirabelle_hf, X3 : hF_Mirabelle_hf] : (member1367349282lle_hf @ X2 @ (hF_Mirabelle_hfset @ X3))))),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_5_hmem__def)).
% 4.05/2.24 thf(f996,plain,(
% 4.05/2.24 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0))) | (X1 != X2)) )),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f982])).
% 4.05/2.24 thf(f982,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((($true = (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0))) | ((X1 != X2) & ($true != (hF_Mirabelle_hmem @ X2 @ X0)))) & ((X1 = X2) | ($true = (hF_Mirabelle_hmem @ X2 @ X0)) | ($true != (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0)))))),
% 4.05/2.24 inference(flattening,[status(thm)],[f981])).
% 4.05/2.24 thf(f981,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((($true = (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0))) | ((X1 != X2) & ($true != (hF_Mirabelle_hmem @ X2 @ X0)))) & (((X1 = X2) | ($true = (hF_Mirabelle_hmem @ X2 @ X0))) | ($true != (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0)))))),
% 4.05/2.24 inference(nnf_transformation,[status(thm)],[f598])).
% 4.05/2.24 thf(f598,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0))) <=> ((X1 = X2) | ($true = (hF_Mirabelle_hmem @ X2 @ X0))))),
% 4.05/2.24 inference(fool_elimination,[status(thm)],[f597])).
% 4.05/2.24 thf(f597,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X2 @ (hF_Mirabelle_hinsert @ X1 @ X0)) = (hF_Mirabelle_hmem @ X2 @ X0) | (X1 = X2))),
% 4.05/2.24 inference(rectify,[status(thm)],[f5])).
% 4.05/2.24 thf(f5,axiom,(
% 4.05/2.24 ! [X10 : hF_Mirabelle_hf,X9 : hF_Mirabelle_hf,X5 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X5 @ (hF_Mirabelle_hinsert @ X9 @ X10)) = (hF_Mirabelle_hmem @ X5 @ X10) | (X5 = X9))),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_4_hmem__hinsert)).
% 4.05/2.24 thf(f1025,plain,(
% 4.05/2.24 ( ! [X1 : hF_Mirabelle_hf] : (($true != (member1367349282lle_hf @ X1 @ (hF_Mirabelle_hfset @ z)))) ) | ~spl3_1),
% 4.05/2.24 inference(avatar_component_clause,[status(thm)],[f1024])).
% 4.05/2.24 thf(f1024,definition,(
% 4.05/2.24 spl3_1 <=> ! [X1 : hF_Mirabelle_hf] : ($true != (member1367349282lle_hf @ X1 @ (hF_Mirabelle_hfset @ z)))),
% 4.05/2.24 introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition])).
% 4.05/2.24 thf(f1043,plain,(
% 4.05/2.24 ~spl3_1 | ~spl3_3),
% 4.05/2.24 inference(avatar_contradiction_clause,[status(thm)],[f1042])).
% 4.05/2.24 thf(f1042,plain,(
% 4.05/2.24 $false | (~spl3_1 | ~spl3_3)),
% 4.05/2.24 inference(trivial_inequality_removal,[status(thm)],[f1041])).
% 4.05/2.24 thf(f1041,plain,(
% 4.05/2.24 ($true != $true) | (~spl3_1 | ~spl3_3)),
% 4.05/2.24 inference(superposition,[status(thm)],[f1025,f1034])).
% 4.05/2.24 thf(f1034,plain,(
% 4.05/2.24 ($true = (member1367349282lle_hf @ sK0 @ (hF_Mirabelle_hfset @ z))) | ~spl3_3),
% 4.05/2.24 inference(avatar_component_clause,[status(thm)],[f1032])).
% 4.05/2.24 thf(f1032,definition,(
% 4.05/2.24 spl3_3 <=> ($true = (member1367349282lle_hf @ sK0 @ (hF_Mirabelle_hfset @ z)))),
% 4.05/2.24 introduced(definition,[new_symbols(naming,[spl3_3])],[avatar_definition])).
% 4.05/2.24 thf(f1040,plain,(
% 4.05/2.24 spl3_1 | ~spl3_2),
% 4.05/2.24 inference(avatar_split_clause,[status(thm)],[f1039,f1027,f1024])).
% 4.05/2.24 thf(f1039,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (($true != (member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ z)))) ) | ~spl3_2),
% 4.05/2.24 inference(backward_demodulation,[status(thm)],[f1022,f1029])).
% 4.05/2.24 thf(f1029,plain,(
% 4.05/2.24 ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z) | ~spl3_2),
% 4.05/2.24 inference(avatar_component_clause,[status(thm)],[f1027])).
% 4.05/2.24 thf(f1022,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (((member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ (hF_Mirabelle_HF @ bot_bo53200981lle_hf))) != $true)) )),
% 4.05/2.24 inference(beta_eta_normalization,[status(thm)],[f1000])).
% 4.05/2.24 thf(f1000,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X0 @ (hF_Mirabelle_HF @ bot_bo53200981lle_hf)))) )),
% 4.05/2.24 inference(definition_unfolding,[status(thm)],[f986,f997,f985])).
% 4.05/2.24 thf(f986,plain,(
% 4.05/2.24 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf) != $true)) )),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f970])).
% 4.05/2.24 thf(f970,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf) != $true)),
% 4.05/2.24 inference(flattening,[status(thm)],[f552])).
% 4.05/2.24 thf(f552,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ~((hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf) = $true)),
% 4.05/2.24 inference(fool_elimination,[status(thm)],[f551])).
% 4.05/2.24 thf(f551,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X0 @ zero_z189798548lle_hf)),
% 4.05/2.24 inference(rectify,[status(thm)],[f3])).
% 4.05/2.24 thf(f3,axiom,(
% 4.05/2.24 ! [X5 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X5 @ zero_z189798548lle_hf)),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',fact_2_hmem__hempty)).
% 4.05/2.24 thf(f1035,plain,(
% 4.05/2.24 spl3_3 | ~spl3_2),
% 4.05/2.24 inference(avatar_split_clause,[status(thm)],[f1016,f1027,f1032])).
% 4.05/2.24 thf(f1016,plain,(
% 4.05/2.24 ($true = (member1367349282lle_hf @ sK0 @ (hF_Mirabelle_hfset @ z))) | ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) != z)),
% 4.05/2.24 inference(beta_eta_normalization,[status(thm)],[f998])).
% 4.05/2.24 thf(f998,plain,(
% 4.05/2.24 ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) != z) | (((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ sK0 @ z) = $true)),
% 4.05/2.24 inference(definition_unfolding,[status(thm)],[f984,f985,f997])).
% 4.05/2.24 thf(f984,plain,(
% 4.05/2.24 (zero_z189798548lle_hf != z) | ($true = (hF_Mirabelle_hmem @ sK0 @ z))),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f977])).
% 4.05/2.24 thf(f977,plain,(
% 4.05/2.24 ((zero_z189798548lle_hf != z) | ($true = (hF_Mirabelle_hmem @ sK0 @ z))) & ((zero_z189798548lle_hf = z) | ! [X1 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X1 @ z)))),
% 4.05/2.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f975,f976])).
% 4.05/2.24 thf(f976,plain,(
% 4.05/2.24 ? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ z)) => ($true = (hF_Mirabelle_hmem @ sK0 @ z))),
% 4.05/2.24 introduced(definition,[],[choice_axiom])).
% 4.05/2.24 thf(f975,plain,(
% 4.05/2.24 ((zero_z189798548lle_hf != z) | ? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ z))) & ((zero_z189798548lle_hf = z) | ! [X1 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X1 @ z)))),
% 4.05/2.24 inference(rectify,[status(thm)],[f974])).
% 4.05/2.24 thf(f974,plain,(
% 4.05/2.24 ((zero_z189798548lle_hf != z) | ? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ z))) & ((zero_z189798548lle_hf = z) | ! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ z)))),
% 4.05/2.24 inference(nnf_transformation,[status(thm)],[f972])).
% 4.05/2.24 thf(f972,plain,(
% 4.05/2.24 ! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ z)) <~> (zero_z189798548lle_hf = z)),
% 4.05/2.24 inference(ennf_transformation,[status(thm)],[f969])).
% 4.05/2.24 thf(f969,plain,(
% 4.05/2.24 ~(! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ z)) <=> (zero_z189798548lle_hf = z))),
% 4.05/2.24 inference(flattening,[status(thm)],[f870])).
% 4.05/2.24 thf(f870,plain,(
% 4.05/2.24 ~(! [X0 : hF_Mirabelle_hf] : ~($true = (hF_Mirabelle_hmem @ X0 @ z)) <=> (zero_z189798548lle_hf = z))),
% 4.05/2.24 inference(fool_elimination,[status(thm)],[f869])).
% 4.05/2.24 thf(f869,plain,(
% 4.05/2.24 ~(! [X0 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X0 @ z) = (zero_z189798548lle_hf = z))),
% 4.05/2.24 inference(rectify,[status(thm)],[f353])).
% 4.05/2.24 thf(f353,negated_conjecture,(
% 4.05/2.24 ~(! [X4 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X4 @ z) = (zero_z189798548lle_hf = z))),
% 4.05/2.24 inference(negated_conjecture,[status(cth)],[f352])).
% 4.05/2.24 thf(f352,conjecture,(
% 4.05/2.24 (! [X4 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X4 @ z) = (zero_z189798548lle_hf = z))),
% 4.05/2.24 file('/export/starexec/sandbox/tmp/tmp.xkGKwgmzX0/DTF2THF_20378.p',conj_0)).
% 4.05/2.24 thf(f1030,plain,(
% 4.05/2.24 spl3_1 | spl3_2),
% 4.05/2.24 inference(avatar_split_clause,[status(thm)],[f1018,f1027,f1024])).
% 4.05/2.24 thf(f1018,plain,(
% 4.05/2.24 ( ! [X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z) | ($true != (member1367349282lle_hf @ X1 @ (hF_Mirabelle_hfset @ z)))) )),
% 4.05/2.24 inference(beta_eta_normalization,[status(thm)],[f999])).
% 4.05/2.24 thf(f999,plain,(
% 4.05/2.24 ( ! [X1 : hF_Mirabelle_hf] : (($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X1 @ z)) | ((hF_Mirabelle_HF @ bot_bo53200981lle_hf) = z)) )),
% 4.05/2.24 inference(definition_unfolding,[status(thm)],[f983,f985,f997])).
% 4.05/2.24 thf(f983,plain,(
% 4.05/2.24 ( ! [X1 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf = z) | ($true != (hF_Mirabelle_hmem @ X1 @ z))) )),
% 4.05/2.24 inference(cnf_transformation,[status(thm)],[f977])).
% 4.05/2.24 % SZS output end Proof for DTF2THF_20378
% 4.05/2.24 % (20498)------------------------------
% 4.05/2.24 % (20498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.24 % (20498)Termination reason: Refutation
% 4.05/2.24
% 4.05/2.24 % (20498)Memory used [KB]: 6524
% 4.05/2.24 % (20498)Time elapsed: 0.032 s
% 4.05/2.24 % (20498)Instructions burned: 48 (million)
% 4.05/2.24 % (20498)------------------------------
% 4.05/2.24 % (20498)------------------------------
% 4.05/2.24 % (20477)Success in time 0.135 s
%------------------------------------------------------------------------------