%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP076^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n017.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 08:55:21 AM UTC 2026
% Result : Theorem 2.70s 1.66s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 5
% Syntax : Number of formulae : 56 ( 37 unt; 0 typ; 0 def)
% Number of atoms : 340 ( 124 equ; 0 cnn)
% Maximal formula atoms : 8 ( 6 avg)
% Number of connectives : 623 ( 27 ~; 30 |; 20 &; 474 @)
% ( 1 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 52 usr; 12 con; 0-3 aty)
% ( 32 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 120 ( 44 ^; 62 !; 14 ?; 120 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
hF_Mirabelle_hf: $tType ).
thf(type_def_8,type,
set_HF_Mirabelle_hf: $tType ).
thf(type_def_9,type,
set_nat: $tType ).
thf(func_def_0,type,
set_HF_Mirabelle_hf: $tType ).
thf(func_def_1,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_2,type,
set_nat: $tType ).
thf(func_def_3,type,
nat: $tType ).
thf(func_def_4,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_5,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(func_def_6,type,
zero_zero_nat: nat ).
thf(func_def_7,type,
hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_8,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_9,type,
hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_10,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_11,type,
hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_12,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_13,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_14,type,
hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
thf(func_def_15,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(func_def_16,type,
hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
thf(func_def_17,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_18,type,
hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_19,type,
hF_Mir199975595nction: hF_Mirabelle_hf > $o ).
thf(func_def_20,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_21,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_22,type,
hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_23,type,
hF_Mir434065167lation: hF_Mirabelle_hf > $o ).
thf(func_def_24,type,
hF_Mir1653039215strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_25,type,
hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_26,type,
hF_Mir137170979_hpair: hF_Mirabelle_hf > $o ).
thf(func_def_27,type,
inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(func_def_28,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_29,type,
inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_30,type,
inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_31,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_32,type,
sup_su1199008216e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(func_def_33,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_34,type,
sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_35,type,
sup_su1790843629lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_36,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_37,type,
suc: nat > nat ).
thf(func_def_38,type,
top_to22270292e_hf_o: hF_Mirabelle_hf > $o ).
thf(func_def_39,type,
top_top_nat_o: nat > $o ).
thf(func_def_41,type,
top_to489427057lle_hf: set_HF_Mirabelle_hf ).
thf(func_def_42,type,
top_top_set_nat: set_nat ).
thf(func_def_43,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(func_def_44,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_45,type,
insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_46,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_47,type,
type_d1794767497hf_nat: ( hF_Mirabelle_hf > nat ) > ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
thf(func_def_48,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_49,type,
member_nat: nat > set_nat > $o ).
thf(func_def_50,type,
r: hF_Mirabelle_hf ).
thf(func_def_51,type,
x: hF_Mirabelle_hf ).
thf(func_def_65,type,
sK0: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_66,type,
sK1: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_68,type,
sK3: hF_Mirabelle_hf ).
thf(func_def_69,type,
sK4: hF_Mirabelle_hf ).
thf(func_def_70,type,
sK5: hF_Mirabelle_hf ).
thf(f800,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f799,f753]) ).
thf(f753,plain,
sK4 != sK5,
inference(equality_proxy_clausification,[status(thm)],[f751]) ).
thf(f751,plain,
( ( sK4 = sK5 )
= $false ),
inference(binary_proxy_clausification,[status(thm)],[f750]) ).
thf(f750,plain,
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK5 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( sK4 = sK5 ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f749]) ).
thf(f749,plain,
( ( ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y0 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( sK4 = Y0 ) )
@ sK5 )
= $false ),
inference(sigma_clausification,[status(thm)],[f747]) ).
thf(f747,plain,
( $false
= ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y0 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( sK4 = Y0 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f746]) ).
thf(f746,plain,
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK4 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y0 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( sK4 = Y0 ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f745]) ).
thf(f745,plain,
( ( ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y0 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( Y0 = Y1 ) ) ) )
@ sK4 )
= $false ),
inference(sigma_clausification,[status(thm)],[f744]) ).
thf(f744,plain,
( ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y0 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ Y1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( Y0 = Y1 ) ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f743]) ).
thf(f743,plain,
( ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( Y1 = Y2 ) ) ) ) )
@ sK3 )
= $false ),
inference(sigma_clausification,[status(thm)],[f742]) ).
thf(f742,plain,
( $true
!= ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f739]) ).
thf(f739,plain,
( ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y2 ) @ Y0 )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y3 ) @ Y0 )
=> ( Y2 = Y3 ) ) ) ) ) )
@ ( hF_Mir1653039215strict @ r @ x ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f731,f737]) ).
thf(f737,plain,
( hF_Mir199975595nction
= ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y2 ) @ Y0 )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y3 ) @ Y0 )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f558]) ).
thf(f558,plain,
( hF_Mir199975595nction
= ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y2 ) @ Y0 )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y3 ) @ Y0 )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f557]) ).
thf(f557,plain,
( ( ^ [X0: hF_Mirabelle_hf] :
! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X1 ) @ X0 )
=> ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X3 ) @ X0 )
=> ( X1 = X3 ) ) ) )
= hF_Mir199975595nction ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( ^ [X2: hF_Mirabelle_hf] :
! [X4: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X3 @ X4 ) @ X2 )
=> ! [X5: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X3 @ X5 ) @ X2 )
=> ( X4 = X5 ) ) ) )
= hF_Mir199975595nction ),
file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',fact_1_hfunction__def) ).
thf(f731,plain,
( ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f725]) ).
thf(f725,plain,
( ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) )
!= $true ),
inference(flattening,[status(thm)],[f580]) ).
thf(f580,plain,
( ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f579]) ).
thf(f579,plain,
~ ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
~ ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ),
file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',conj_1) ).
thf(f799,plain,
sK4 = sK5,
inference(trivial_inequality_removal,[status(thm)],[f795]) ).
thf(f795,plain,
( ( $false = $true )
| ( sK4 = sK5 ) ),
inference(superposition,[status(thm)],[f781,f771]) ).
thf(f771,plain,
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK5 ) @ r )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f769]) ).
thf(f769,plain,
( ( $true != $true )
| ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK5 ) @ r )
= $true ) ),
inference(superposition,[status(thm)],[f733,f752]) ).
thf(f752,plain,
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK5 ) @ ( hF_Mir1653039215strict @ r @ x ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f750]) ).
thf(f733,plain,
! [X2: hF_Mirabelle_hf,X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
!= $true )
| ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f730]) ).
thf(f730,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( ( ( hF_Mirabelle_hmem @ ( sK1 @ X2 @ X1 ) @ X2 )
= $true )
& ( ( hF_Mirabelle_hpair @ ( sK1 @ X2 @ X1 ) @ ( sK0 @ X2 @ X1 ) )
= X1 )
& ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) )
| ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
!= $true ) )
& ( ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
= $true )
| ! [X5: hF_Mirabelle_hf,X6: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X6 @ X2 )
!= $true )
| ( ( hF_Mirabelle_hpair @ X6 @ X5 )
!= X1 ) )
| ( ( hF_Mirabelle_hmem @ X1 @ X0 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f728,f729]) ).
thf(f729,plain,
! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
& ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 ) )
=> ( ( ( hF_Mirabelle_hmem @ ( sK1 @ X2 @ X1 ) @ X2 )
= $true )
& ( ( hF_Mirabelle_hpair @ ( sK1 @ X2 @ X1 ) @ ( sK0 @ X2 @ X1 ) )
= X1 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f728,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
& ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 ) )
& ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) )
| ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
!= $true ) )
& ( ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
= $true )
| ! [X5: hF_Mirabelle_hf,X6: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X6 @ X2 )
!= $true )
| ( ( hF_Mirabelle_hpair @ X6 @ X5 )
!= X1 ) )
| ( ( hF_Mirabelle_hmem @ X1 @ X0 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f727]) ).
thf(f727,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
& ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 ) )
& ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) )
| ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
!= $true ) )
& ( ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
= $true )
| ! [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
| ( ( hF_Mirabelle_hpair @ X4 @ X3 )
!= X1 ) )
| ( ( hF_Mirabelle_hmem @ X1 @ X0 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f726]) ).
thf(f726,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
& ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 ) )
& ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) )
| ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
!= $true ) )
& ( ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
= $true )
| ! [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
| ( ( hF_Mirabelle_hpair @ X4 @ X3 )
!= X1 ) )
| ( ( hF_Mirabelle_hmem @ X1 @ X0 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f474]) ).
thf(f474,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X4 @ X2 ) )
& ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 ) )
& ( ( hF_Mirabelle_hmem @ X1 @ X0 )
= $true ) )
<=> ( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f473]) ).
thf(f473,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X1 @ X0 )
& ? [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( ( ( hF_Mirabelle_hpair @ X4 @ X3 )
= X1 )
& ( hF_Mirabelle_hmem @ X4 @ X2 ) )
= ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X0 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: hF_Mirabelle_hf,X6: hF_Mirabelle_hf,X7: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X6 @ X0 )
& ? [X4: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( ( ( hF_Mirabelle_hpair @ X3 @ X4 )
= X6 )
& ( hF_Mirabelle_hmem @ X3 @ X7 ) )
= ( hF_Mirabelle_hmem @ X6 @ ( hF_Mir1653039215strict @ X0 @ X7 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',fact_2_hrestrict__iff) ).
thf(f781,plain,
! [X0: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ X0 ) @ r )
= $false )
| ( sK4 = X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f772]) ).
thf(f772,plain,
! [X0: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ X0 ) @ r )
= $false )
| ( sK4 = X0 )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f763,f765]) ).
thf(f765,plain,
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK4 ) @ r )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f764]) ).
thf(f764,plain,
( ( $true != $true )
| ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK4 ) @ r )
= $true ) ),
inference(superposition,[status(thm)],[f733,f748]) ).
thf(f748,plain,
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ sK3 @ sK4 ) @ ( hF_Mir1653039215strict @ r @ x ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f746]) ).
thf(f763,plain,
! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( X2 = X3 )
| ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ r )
= $false )
| ( $false
= ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f762]) ).
thf(f762,plain,
! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ r )
= $false )
| ( ( X2 = X3 )
= $true )
| ( $false
= ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f761]) ).
thf(f761,plain,
! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $false
= ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r ) )
| ( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ r )
=> ( X2 = X3 ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f760]) ).
thf(f760,plain,
! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $false
= ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r ) )
| ( ( ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y0 ) @ r )
=> ( X2 = Y0 ) )
@ X3 )
= $true ) ),
inference(pi_clausification,[status(thm)],[f759]) ).
thf(f759,plain,
! [X2: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $false
= ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r ) )
| ( $true
= ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y0 ) @ r )
=> ( X2 = Y0 ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f758]) ).
thf(f758,plain,
! [X2: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y0 ) @ r )
=> ( X2 = Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f757]) ).
thf(f757,plain,
! [X2: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y0 ) @ r )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y1 ) @ r )
=> ( Y0 = Y1 ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f756]) ).
thf(f756,plain,
! [X1: hF_Mirabelle_hf] :
( ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y0 ) @ r )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ Y1 ) @ r )
=> ( Y0 = Y1 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f755]) ).
thf(f755,plain,
! [X1: hF_Mirabelle_hf] :
( ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y1 ) @ r )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y2 ) @ r )
=> ( Y1 = Y2 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f754]) ).
thf(f754,plain,
( $true
= ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y1 ) @ r )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y0 @ Y2 ) @ r )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f740]) ).
thf(f740,plain,
( ( ^ [Y0: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y2 ) @ Y0 )
=> ( !! @ hF_Mirabelle_hf
@ ^ [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ Y1 @ Y3 ) @ Y0 )
=> ( Y2 = Y3 ) ) ) ) ) )
@ r )
= $true ),
inference(definition_unfolding,[status(thm)],[f736,f737]) ).
thf(f736,plain,
( ( hF_Mir199975595nction @ r )
= $true ),
inference(cnf_transformation,[status(thm)],[f588]) ).
thf(f588,plain,
( ( hF_Mir199975595nction @ r )
= $true ),
inference(fool_elimination,[status(thm)],[f587]) ).
thf(f587,plain,
hF_Mir199975595nction @ r,
inference(rectify,[status(thm)],[f355]) ).
thf(f355,axiom,
hF_Mir199975595nction @ r,
file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP076^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n017.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:23:41 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP076^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer HF problem prob_850__5339526_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : HF/prob_850__5339526_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.56 v9.1.0, 0.75 v9.0.0, 0.60 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 408 ( 258 unt; 52 typ; 0 def)
% 0.29/0.45 % Number of atoms : 781 ( 477 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 5 ( 2 avg)
% 0.29/0.45 % Number of connectives : 2218 ( 71 ~; 9 |; 60 &;1909 @)
% 0.29/0.45 % ( 0 <=>; 169 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 13 ( 4 avg)
% 0.29/0.45 % Number of types : 5 ( 4 usr)
% 0.29/0.45 % Number of type conns : 179 ( 179 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 49 ( 48 usr; 7 con; 0-3 aty)
% 0.29/0.45 % Number of variables : 871 ( 62 ^; 784 !; 25 ?; 871 :)
% 0.29/0.45 % SPC : TH0_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:38:21.238
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (4)
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.29/0.45 set_HF_Mirabelle_hf: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 hF_Mirabelle_hf: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.45 set_nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (48)
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.29/0.45 plus_plus_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.45 zero_zero_nat: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.29/0.45 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.29/0.45 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 0.29/0.45 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.29/0.45 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.29/0.45 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.29/0.45 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.29/0.45 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohcard,type,
% 0.29/0.45 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.29/0.45 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_ORep__hf,type,
% 0.29/0.45 hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.29/0.45 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 0.29/0.45 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohfunction,type,
% 0.29/0.45 hF_Mir199975595nction: hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.29/0.45 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.29/0.45 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 0.29/0.45 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohrelation,type,
% 0.29/0.45 hF_Mir434065167lation: hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohrestrict,type,
% 0.29/0.45 hF_Mir1653039215strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 0.29/0.45 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ois__hpair,type,
% 0.29/0.45 hF_Mir137170979_hpair: hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.29/0.45 inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.29/0.45 inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.45 inf_inf_set_nat: set_nat > set_nat > set_nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.29/0.45 sup_su1199008216e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.29/0.45 sup_su1790843629lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.45 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_OSuc,type,
% 0.29/0.45 suc: nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.29/0.45 top_to22270292e_hf_o: hF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 top_top_nat_o: nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
% 0.29/0.45 top_top_o: $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.29/0.45 top_to489427057lle_hf: set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.45 top_top_set_nat: set_nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.29/0.45 collect_nat: ( nat > $o ) > set_nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.29/0.45 insert_nat: nat > set_nat > set_nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Typedef_Otype__definition_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.29/0.45 type_d1794767497hf_nat: ( hF_Mirabelle_hf > nat ) > ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.29/0.45 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_001t__Nat__Onat,type,
% 0.29/0.45 member_nat: nat > set_nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_r,type,
% 0.29/0.45 r: hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x,type,
% 0.29/0.45 x: hF_Mirabelle_hf ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (354)
% 0.29/0.45 thf(fact_0_hrelation__restr,axiom,
% 0.29/0.45 ! [R: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( hF_Mir434065167lation @ ( hF_Mir1653039215strict @ R @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % hrelation_restr
% 0.29/0.45 thf(fact_1_hfunction__def,axiom,
% 0.29/0.45 ( hF_Mir199975595nction
% 0.29/0.45 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.29/0.45 ! [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y ) @ R2 )
% 0.29/0.45 => ! [Y2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) @ R2 )
% 0.29/0.45 => ( Y = Y2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hfunction_def
% 0.29/0.45 thf(fact_2_hrestrict__iff,axiom,
% 0.29/0.45 ! [Z: hF_Mirabelle_hf,R: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ Z @ ( hF_Mir1653039215strict @ R @ A ) )
% 0.29/0.45 = ( ( hF_Mirabelle_hmem @ Z @ R )
% 0.29/0.45 & ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( Z
% 0.29/0.45 = ( hF_Mirabelle_hpair @ X2 @ Y ) )
% 0.29/0.45 & ( hF_Mirabelle_hmem @ X2 @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hrestrict_iff
% 0.29/0.45 thf(fact_3_hpair__iff,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.29/0.45 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.29/0.45 = ( ( A2 = A3 )
% 0.29/0.45 & ( B = B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_iff
% 0.29/0.45 thf(fact_4_Rep__hf__inject,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( ( hF_Mirabelle_Rep_hf @ X )
% 0.29/0.45 = ( hF_Mirabelle_Rep_hf @ Y3 ) )
% 0.29/0.45 = ( X = Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Rep_hf_inject
% 0.29/0.45 thf(fact_5_hf__equalityI,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ! [X3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X3 @ A2 )
% 0.29/0.45 = ( hF_Mirabelle_hmem @ X3 @ B ) )
% 0.29/0.45 => ( A2 = B ) ) ).
% 0.29/0.45
% 0.29/0.45 % hf_equalityI
% 0.29/0.45 thf(fact_6_hf__ext,axiom,
% 0.29/0.45 ( ( ^ [Y4: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] : ( Y4 = Z2 ) )
% 0.29/0.45 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [X2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X2 @ A4 )
% 0.29/0.45 = ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hf_ext
% 0.29/0.45 thf(fact_7_hmem__ne,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.29/0.45 => ( X != Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_ne
% 0.29/0.45 thf(fact_8_replacement,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U @ X )
% 0.29/0.45 => ( ( R3 @ U @ V )
% 0.29/0.45 => ( ( R3 @ U @ V2 )
% 0.29/0.45 => ( V2 = V ) ) ) )
% 0.29/0.45 => ? [Z3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [V3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.29/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.29/0.45 & ( R3 @ U2 @ V3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % replacement
% 0.29/0.45 thf(fact_9_binary__union,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ? [Z3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.29/0.45 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.29/0.45 | ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % binary_union
% 0.29/0.45 thf(fact_10_hmem__not__sym,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.29/0.45 & ( hF_Mirabelle_hmem @ Y3 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_not_sym
% 0.29/0.45 thf(fact_11_hpair__inject,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.29/0.45 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.29/0.45 => ~ ( ( A2 = A3 )
% 0.29/0.45 => ( B != B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_inject
% 0.29/0.45 thf(fact_12_union__of__set,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] :
% 0.29/0.45 ? [Z3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.29/0.45 = ( ? [Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ Y @ X )
% 0.29/0.45 & ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % union_of_set
% 0.29/0.45 thf(fact_13_comprehension,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ? [Z3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.29/0.45 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.29/0.45 & ( P @ U3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % comprehension
% 0.29/0.45 thf(fact_14_hmem__not__refl,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_not_refl
% 0.29/0.45 thf(fact_15_hpair__neq__fst,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.29/0.45 != A2 ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_neq_fst
% 0.29/0.45 thf(fact_16_hpair__neq__snd,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.29/0.45 != B ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_neq_snd
% 0.29/0.45 thf(fact_17_replacement__fun,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ? [Z3: hF_Mirabelle_hf] :
% 0.29/0.45 ! [V3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.29/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.29/0.45 & ( V3
% 0.29/0.45 = ( F @ U2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % replacement_fun
% 0.29/0.45 thf(fact_18_hfst__conv,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.29/0.45 = A2 ) ).
% 0.29/0.45
% 0.29/0.45 % hfst_conv
% 0.29/0.45 thf(fact_19_hsnd__conv,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.29/0.45 = B ) ).
% 0.29/0.45
% 0.29/0.45 % hsnd_conv
% 0.29/0.45 thf(fact_20_hrelation__def,axiom,
% 0.29/0.45 ( hF_Mir434065167lation
% 0.29/0.45 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.29/0.45 ! [Z4: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ Z4 @ R2 )
% 0.29/0.45 => ( hF_Mir137170979_hpair @ Z4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hrelation_def
% 0.29/0.45 thf(fact_21_is__hpair__def,axiom,
% 0.29/0.45 ( hF_Mir137170979_hpair
% 0.29/0.45 = ( ^ [Z4: hF_Mirabelle_hf] :
% 0.29/0.45 ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( Z4
% 0.29/0.45 = ( hF_Mirabelle_hpair @ X2 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % is_hpair_def
% 0.29/0.45 thf(fact_22_PrimReplace__iff,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.29/0.45 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U @ A )
% 0.29/0.45 => ( ( R3 @ U @ V )
% 0.29/0.45 => ( ( R3 @ U @ V2 )
% 0.29/0.45 => ( V2 = V ) ) ) )
% 0.29/0.45 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A @ R3 ) )
% 0.29/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.29/0.45 & ( R3 @ U2 @ V4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % PrimReplace_iff
% 0.29/0.45 thf(fact_23_HUnion__iff,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A ) )
% 0.29/0.45 = ( ? [Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ Y @ A )
% 0.29/0.45 & ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HUnion_iff
% 0.29/0.45 thf(fact_24_Replace__iff,axiom,
% 0.29/0.45 ! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A @ R3 ) )
% 0.29/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.29/0.45 & ( R3 @ U2 @ V4 )
% 0.29/0.45 & ! [Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( R3 @ U2 @ Y )
% 0.29/0.45 => ( Y = V4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Replace_iff
% 0.29/0.45 thf(fact_25_HCollect__iff,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A ) )
% 0.29/0.45 = ( ( P @ X )
% 0.29/0.45 & ( hF_Mirabelle_hmem @ X @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HCollect_iff
% 0.29/0.45 thf(fact_26_RepFun__iff,axiom,
% 0.29/0.45 ! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A @ F ) )
% 0.29/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.29/0.45 & ( V4
% 0.29/0.45 = ( F @ U2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % RepFun_iff
% 0.29/0.45 thf(fact_27_Rep__hf__inverse,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_Abs_hf @ ( hF_Mirabelle_Rep_hf @ X ) )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % Rep_hf_inverse
% 0.29/0.45 thf(fact_28_RepFun__cong,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ( ( A = B4 )
% 0.29/0.45 => ( ! [X3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.29/0.45 => ( ( F @ X3 )
% 0.29/0.45 = ( G @ X3 ) ) )
% 0.29/0.45 => ( ( hF_Mirabelle_RepFun @ A @ F )
% 0.29/0.45 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % RepFun_cong
% 0.29/0.45 thf(fact_29_Replace__cong,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( A = B4 )
% 0.29/0.45 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.29/0.45 => ( ( P @ X3 @ Y5 )
% 0.29/0.45 = ( Q @ X3 @ Y5 ) ) )
% 0.29/0.45 => ( ( hF_Mirabelle_Replace @ A @ P )
% 0.29/0.45 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Replace_cong
% 0.29/0.45 thf(fact_30_hmem__Sup__ne,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.29/0.45 => ( ( hF_Mirabelle_HUnion @ X )
% 0.29/0.45 != Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_Sup_ne
% 0.29/0.45 thf(fact_31_Abs__hf__inverse,axiom,
% 0.29/0.45 ! [Y3: nat] :
% 0.29/0.45 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.29/0.45 => ( ( hF_Mirabelle_Rep_hf @ ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.29/0.45 = Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Abs_hf_inverse
% 0.29/0.45 thf(fact_32_zero__notin__hpair,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X @ Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_notin_hpair
% 0.29/0.45 thf(fact_33_hmem__def,axiom,
% 0.29/0.45 ( hF_Mirabelle_hmem
% 0.29/0.45 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_def
% 0.29/0.45 thf(fact_34_HCollect__hempty,axiom,
% 0.29/0.45 ! [P: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 0.29/0.45 = zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % HCollect_hempty
% 0.29/0.45 thf(fact_35_HUnion__hempty,axiom,
% 0.29/0.45 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.29/0.45 = zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % HUnion_hempty
% 0.29/0.45 thf(fact_36_Replace__0,axiom,
% 0.29/0.45 ! [R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R3 )
% 0.29/0.45 = zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % Replace_0
% 0.29/0.45 thf(fact_37_RepFun__0,axiom,
% 0.29/0.45 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.29/0.45 = zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % RepFun_0
% 0.29/0.45 thf(fact_38_RepFun__hinsert,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % RepFun_hinsert
% 0.29/0.45 thf(fact_39_hrelation__hunion,axiom,
% 0.29/0.45 ! [F: hF_Mirabelle_hf,G: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mir434065167lation @ ( sup_su638957495lle_hf @ F @ G ) )
% 0.29/0.45 = ( ( hF_Mir434065167lation @ F )
% 0.29/0.45 & ( hF_Mir434065167lation @ G ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hrelation_hunion
% 0.29/0.45 thf(fact_40_hmem__hinsert,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.29/0.45 = ( ( A2 = B )
% 0.29/0.45 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_hinsert
% 0.29/0.45 thf(fact_41_singleton__eq__iff,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( ( hF_Mirabelle_hinsert @ A2 @ zero_z189798548lle_hf )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.29/0.45 = ( A2 = B ) ) ).
% 0.29/0.45
% 0.29/0.45 % singleton_eq_iff
% 0.29/0.45 thf(fact_42_hunion__iff,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A2 @ B ) )
% 0.29/0.45 = ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.29/0.45 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hunion_iff
% 0.29/0.45 thf(fact_43_hunion__hempty__left,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % hunion_hempty_left
% 0.29/0.45 thf(fact_44_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A2: nat,P: nat > $o] :
% 0.29/0.45 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.29/0.45 = ( P @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
% 0.29/0.45 = ( P @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: set_nat] :
% 0.29/0.45 ( ( collect_nat
% 0.29/0.45 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_47_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( collec2046588256lle_hf
% 0.29/0.45 @ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A ) )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_48_Collect__cong,axiom,
% 0.29/0.45 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ! [X3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 = ( Q @ X3 ) )
% 0.29/0.45 => ( ( collec2046588256lle_hf @ P )
% 0.29/0.45 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_cong
% 0.29/0.45 thf(fact_49_Collect__cong,axiom,
% 0.29/0.45 ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.45 ( ! [X3: nat] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 = ( Q @ X3 ) )
% 0.29/0.45 => ( ( collect_nat @ P )
% 0.29/0.45 = ( collect_nat @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_cong
% 0.29/0.45 thf(fact_50_hunion__hempty__right,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ A @ zero_z189798548lle_hf )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % hunion_hempty_right
% 0.29/0.45 thf(fact_51_RepFun__hunion,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A @ B4 ) @ F )
% 0.29/0.45 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % RepFun_hunion
% 0.29/0.45 thf(fact_52_HUnion__hunion,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A @ B4 ) )
% 0.29/0.45 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HUnion_hunion
% 0.29/0.45 thf(fact_53_Replace__hunion,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A @ B4 ) @ R3 )
% 0.29/0.45 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A @ R3 ) @ ( hF_Mirabelle_Replace @ B4 @ R3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Replace_hunion
% 0.29/0.45 thf(fact_54_HUnion__hinsert,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
% 0.29/0.45 = ( sup_su638957495lle_hf @ A2 @ ( hF_Mirabelle_HUnion @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HUnion_hinsert
% 0.29/0.45 thf(fact_55_hf__induct__ax,axiom,
% 0.29/0.45 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ zero_z189798548lle_hf )
% 0.29/0.45 => ( ! [X3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 => ! [Y5: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ Y5 )
% 0.29/0.45 => ( P @ ( hF_Mirabelle_hinsert @ Y5 @ X3 ) ) ) )
% 0.29/0.45 => ( P @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hf_induct_ax
% 0.29/0.45 thf(fact_56_hinsert__eq__sup,axiom,
% 0.29/0.45 ( hF_Mirabelle_hinsert
% 0.29/0.45 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hinsert_eq_sup
% 0.29/0.45 thf(fact_57_hinsert__commute,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y3 @ Z ) )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ Y3 @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hinsert_commute
% 0.29/0.45 thf(fact_58_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 0.29/0.45 = ( ( ( A2 = C )
% 0.29/0.45 & ( B = D ) )
% 0.29/0.45 | ( ( A2 = D )
% 0.29/0.45 & ( B = C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HF_Mirabelle_glliljednj.doubleton_eq_iff
% 0.29/0.45 thf(fact_59_hinsert__nonempty,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hinsert @ A2 @ A )
% 0.29/0.45 != zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % hinsert_nonempty
% 0.29/0.45 thf(fact_60_hunion__hinsert__left,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hunion_hinsert_left
% 0.29/0.45 thf(fact_61_hunion__hinsert__right,axiom,
% 0.29/0.45 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hunion_hinsert_right
% 0.29/0.45 thf(fact_62_hf__cases,axiom,
% 0.29/0.45 ! [Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( Y3 != zero_z189798548lle_hf )
% 0.29/0.45 => ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( Y3
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
% 0.29/0.45 => ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hf_cases
% 0.29/0.45 thf(fact_63_hf__induct,axiom,
% 0.29/0.45 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ zero_z189798548lle_hf )
% 0.29/0.45 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( P @ X3 )
% 0.29/0.45 => ( ( P @ Y5 )
% 0.29/0.45 => ( ~ ( hF_Mirabelle_hmem @ X3 @ Y5 )
% 0.29/0.45 => ( P @ ( hF_Mirabelle_hinsert @ X3 @ Y5 ) ) ) ) )
% 0.29/0.45 => ( P @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hf_induct
% 0.29/0.45 thf(fact_64_hpair__def,axiom,
% 0.29/0.45 ( hF_Mirabelle_hpair
% 0.29/0.45 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_def
% 0.29/0.45 thf(fact_65_hpair__def_H,axiom,
% 0.29/0.45 ( hF_Mirabelle_hpair
% 0.29/0.45 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_def'
% 0.29/0.45 thf(fact_66_hinsert__iff,axiom,
% 0.29/0.45 ! [Z: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( Z
% 0.29/0.45 = ( hF_Mirabelle_hinsert @ Y3 @ X ) )
% 0.29/0.45 = ( ! [U2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.29/0.45 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.29/0.45 | ( U2 = Y3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hinsert_iff
% 0.29/0.45 thf(fact_67_hemptyE,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % hemptyE
% 0.29/0.45 thf(fact_68_hempty__iff,axiom,
% 0.29/0.45 ! [Z: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( Z = zero_z189798548lle_hf )
% 0.29/0.45 = ( ! [X2: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( hF_Mirabelle_hmem @ X2 @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % hempty_iff
% 0.29/0.45 thf(fact_69_hmem__hempty,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % hmem_hempty
% 0.29/0.45 thf(fact_70_hpair__nonzero,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hpair @ X @ Y3 )
% 0.29/0.45 != zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % hpair_nonzero
% 0.29/0.45 thf(fact_71_Rep__hf,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] : ( member_nat @ ( hF_Mirabelle_Rep_hf @ X ) @ top_top_set_nat ) ).
% 0.29/0.45
% 0.29/0.45 % Rep_hf
% 0.29/0.45 thf(fact_72_Rep__hf__cases,axiom,
% 0.29/0.45 ! [Y3: nat] :
% 0.29/0.45 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.29/0.45 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.29/0.45 ( Y3
% 0.29/0.45 != ( hF_Mirabelle_Rep_hf @ X3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Rep_hf_cases
% 0.29/0.45 thf(fact_73_Rep__hf__induct,axiom,
% 0.29/0.45 ! [Y3: nat,P: nat > $o] :
% 0.29/0.45 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.29/0.45 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( hF_Mirabelle_Rep_hf @ X3 ) )
% 0.29/0.45 => ( P @ Y3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Rep_hf_induct
% 0.29/0.45 thf(fact_74_Abs__hf__cases,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] :
% 0.29/0.45 ~ ! [Y5: nat] :
% 0.29/0.45 ( ( X
% 0.29/0.45 = ( hF_Mirabelle_Abs_hf @ Y5 ) )
% 0.29/0.45 => ~ ( member_nat @ Y5 @ top_top_set_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % Abs_hf_cases
% 0.29/0.45 thf(fact_75_Abs__hf__induct,axiom,
% 0.29/0.45 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ! [Y5: nat] :
% 0.29/0.45 ( ( member_nat @ Y5 @ top_top_set_nat )
% 0.29/0.45 => ( P @ ( hF_Mirabelle_Abs_hf @ Y5 ) ) )
% 0.29/0.45 => ( P @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % Abs_hf_induct
% 0.29/0.45 thf(fact_76_Abs__hf__inject,axiom,
% 0.29/0.45 ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( member_nat @ X @ top_top_set_nat )
% 0.29/0.45 => ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.29/0.45 => ( ( ( hF_Mirabelle_Abs_hf @ X )
% 0.29/0.45 = ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.29/0.45 = ( X = Y3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Abs_hf_inject
% 0.29/0.45 thf(fact_77_hrelation__0,axiom,
% 0.29/0.45 hF_Mir434065167lation @ zero_z189798548lle_hf ).
% 0.29/0.45
% 0.29/0.45 % hrelation_0
% 0.29/0.45 thf(fact_78_sup__top__left,axiom,
% 0.29/0.45 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ top_to489427057lle_hf @ X )
% 0.29/0.45 = top_to489427057lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_left
% 0.29/0.45 thf(fact_79_sup__top__left,axiom,
% 0.29/0.45 ! [X: nat > $o] :
% 0.29/0.45 ( ( sup_sup_nat_o @ top_top_nat_o @ X )
% 0.29/0.45 = top_top_nat_o ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_left
% 0.29/0.45 thf(fact_80_sup__top__left,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( sup_su1199008216e_hf_o @ top_to22270292e_hf_o @ X )
% 0.29/0.45 = top_to22270292e_hf_o ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_left
% 0.29/0.45 thf(fact_81_sup__top__left,axiom,
% 0.29/0.45 ! [X: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ top_top_set_nat @ X )
% 0.29/0.45 = top_top_set_nat ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_left
% 0.29/0.45 thf(fact_82_sup__top__right,axiom,
% 0.29/0.45 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ X @ top_to489427057lle_hf )
% 0.29/0.45 = top_to489427057lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_right
% 0.29/0.45 thf(fact_83_sup__top__right,axiom,
% 0.29/0.45 ! [X: nat > $o] :
% 0.29/0.45 ( ( sup_sup_nat_o @ X @ top_top_nat_o )
% 0.29/0.45 = top_top_nat_o ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_right
% 0.29/0.45 thf(fact_84_sup__top__right,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf > $o] :
% 0.29/0.45 ( ( sup_su1199008216e_hf_o @ X @ top_to22270292e_hf_o )
% 0.29/0.45 = top_to22270292e_hf_o ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_right
% 0.29/0.45 thf(fact_85_sup__top__right,axiom,
% 0.29/0.45 ! [X: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ X @ top_top_set_nat )
% 0.29/0.45 = top_top_set_nat ) ).
% 0.29/0.45
% 0.29/0.45 % sup_top_right
% 0.29/0.45 thf(fact_86_HInter__iff,axiom,
% 0.29/0.45 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( A != zero_z189798548lle_hf )
% 0.29/0.45 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A ) )
% 0.29/0.45 = ( ! [Y: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( hF_Mirabelle_hmem @ Y @ A )
% 0.29/0.45 => ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % HInter_iff
% 0.29/0.45 thf(fact_87_sup_Oidem,axiom,
% 0.29/0.45 ! [A2: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ A2 @ A2 )
% 0.29/0.45 = A2 ) ).
% 0.29/0.45
% 0.29/0.45 % sup.idem
% 0.29/0.45 thf(fact_88_sup_Oidem,axiom,
% 0.29/0.45 ! [A2: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ A2 @ A2 )
% 0.29/0.45 = A2 ) ).
% 0.29/0.45
% 0.29/0.45 % sup.idem
% 0.29/0.45 thf(fact_89_sup_Oidem,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ A2 @ A2 )
% 0.29/0.45 = A2 ) ).
% 0.29/0.45
% 0.29/0.45 % sup.idem
% 0.29/0.45 thf(fact_90_sup__idem,axiom,
% 0.29/0.45 ! [X: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ X @ X )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % sup_idem
% 0.29/0.45 thf(fact_91_sup__idem,axiom,
% 0.29/0.45 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ X @ X )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % sup_idem
% 0.29/0.45 thf(fact_92_sup__idem,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ X @ X )
% 0.29/0.45 = X ) ).
% 0.29/0.45
% 0.29/0.45 % sup_idem
% 0.29/0.45 thf(fact_93_sup_Oleft__idem,axiom,
% 0.29/0.45 ! [A2: set_nat,B: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
% 0.29/0.45 = ( sup_sup_set_nat @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.left_idem
% 0.29/0.45 thf(fact_94_sup_Oleft__idem,axiom,
% 0.29/0.45 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A2 @ B ) )
% 0.29/0.45 = ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.left_idem
% 0.29/0.45 thf(fact_95_sup_Oleft__idem,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ A2 @ B ) )
% 0.29/0.45 = ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.left_idem
% 0.29/0.45 thf(fact_96_sup__left__idem,axiom,
% 0.29/0.45 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.29/0.45 = ( sup_sup_set_nat @ X @ Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_idem
% 0.29/0.45 thf(fact_97_sup__left__idem,axiom,
% 0.29/0.45 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.29/0.45 = ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_idem
% 0.29/0.45 thf(fact_98_sup__left__idem,axiom,
% 0.29/0.45 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.29/0.45 = ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_idem
% 0.29/0.45 thf(fact_99_sup_Oright__idem,axiom,
% 0.29/0.45 ! [A2: set_nat,B: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ B )
% 0.29/0.45 = ( sup_sup_set_nat @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.right_idem
% 0.29/0.45 thf(fact_100_sup_Oright__idem,axiom,
% 0.29/0.45 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ B )
% 0.29/0.45 = ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.right_idem
% 0.29/0.45 thf(fact_101_sup_Oright__idem,axiom,
% 0.29/0.45 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ B )
% 0.29/0.45 = ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup.right_idem
% 0.29/0.45 thf(fact_102_HInter__hempty,axiom,
% 0.29/0.45 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 0.29/0.45 = zero_z189798548lle_hf ) ).
% 0.29/0.45
% 0.29/0.45 % HInter_hempty
% 0.29/0.45 thf(fact_103_sup__left__commute,axiom,
% 0.29/0.45 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.45 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
% 0.29/0.45 = ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % sup_left_commute
% 0.29/0.45 thf(fact_104_sup__left__commute,axiom,
% 0.29/0.45 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.45 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_105_sup__left__commute,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_left_commute
% 0.29/0.46 thf(fact_106_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) )
% 0.29/0.46 = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_107_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ B @ ( sup_su1790843629lle_hf @ A2 @ C ) )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_108_sup_Oleft__commute,axiom,
% 0.29/0.46 ! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A2 @ C ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.left_commute
% 0.29/0.46 thf(fact_109_sup__commute,axiom,
% 0.29/0.46 ( sup_sup_set_nat
% 0.29/0.46 = ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_110_sup__commute,axiom,
% 0.29/0.46 ( sup_su1790843629lle_hf
% 0.29/0.46 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_111_sup__commute,axiom,
% 0.29/0.46 ( sup_su638957495lle_hf
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_commute
% 0.29/0.46 thf(fact_112_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_sup_set_nat
% 0.29/0.46 = ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_113_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_su1790843629lle_hf
% 0.29/0.46 = ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_114_sup_Ocommute,axiom,
% 0.29/0.46 ( sup_su638957495lle_hf
% 0.29/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.commute
% 0.29/0.46 thf(fact_115_sup__assoc,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_116_sup__assoc,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_117_sup__assoc,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_assoc
% 0.29/0.46 thf(fact_118_sup_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
% 0.29/0.46 = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_119_sup_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ C )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_120_sup_Oassoc,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ C )
% 0.29/0.46 = ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup.assoc
% 0.29/0.46 thf(fact_121_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( sup_sup_set_nat @ K @ B ) )
% 0.29/0.46 => ( ( sup_sup_set_nat @ A2 @ B4 )
% 0.29/0.46 = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_122_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( sup_su1790843629lle_hf @ K @ B ) )
% 0.29/0.46 => ( ( sup_su1790843629lle_hf @ A2 @ B4 )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_123_boolean__algebra__cancel_Osup2,axiom,
% 0.29/0.46 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( sup_su638957495lle_hf @ K @ B ) )
% 0.29/0.46 => ( ( sup_su638957495lle_hf @ A2 @ B4 )
% 0.29/0.46 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup2
% 0.29/0.46 thf(fact_124_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( sup_sup_set_nat @ K @ A2 ) )
% 0.29/0.46 => ( ( sup_sup_set_nat @ A @ B )
% 0.29/0.46 = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_125_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( sup_su1790843629lle_hf @ K @ A2 ) )
% 0.29/0.46 => ( ( sup_su1790843629lle_hf @ A @ B )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_126_boolean__algebra__cancel_Osup1,axiom,
% 0.29/0.46 ! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( sup_su638957495lle_hf @ K @ A2 ) )
% 0.29/0.46 => ( ( sup_su638957495lle_hf @ A @ B )
% 0.29/0.46 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.sup1
% 0.29/0.46 thf(fact_127_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_sup_set_nat
% 0.29/0.46 = ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_128_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_su1790843629lle_hf
% 0.29/0.46 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_129_inf__sup__aci_I5_J,axiom,
% 0.29/0.46 ( sup_su638957495lle_hf
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(5)
% 0.29/0.46 thf(fact_130_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_131_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_132_inf__sup__aci_I6_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(6)
% 0.29/0.46 thf(fact_133_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_134_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_135_inf__sup__aci_I7_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(7)
% 0.29/0.46 thf(fact_136_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.29/0.46 = ( sup_sup_set_nat @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_137_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_138_inf__sup__aci_I8_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(8)
% 0.29/0.46 thf(fact_139_iso__tuple__UNIV__I,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % iso_tuple_UNIV_I
% 0.29/0.46 thf(fact_140_iso__tuple__UNIV__I,axiom,
% 0.29/0.46 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % iso_tuple_UNIV_I
% 0.29/0.46 thf(fact_141_UNIV__I,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_I
% 0.29/0.46 thf(fact_142_UNIV__I,axiom,
% 0.29/0.46 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_I
% 0.29/0.46 thf(fact_143_top__apply,axiom,
% 0.29/0.46 ( top_top_nat_o
% 0.29/0.46 = ( ^ [X2: nat] : top_top_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_apply
% 0.29/0.46 thf(fact_144_top__apply,axiom,
% 0.29/0.46 ( top_to22270292e_hf_o
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf] : top_top_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_apply
% 0.29/0.46 thf(fact_145_HInter__hinsert,axiom,
% 0.29/0.46 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( A != zero_z189798548lle_hf )
% 0.29/0.46 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A2 @ ( hF_Mirabelle_HInter @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % HInter_hinsert
% 0.29/0.46 thf(fact_146_type__definition__hf,axiom,
% 0.29/0.46 type_d1794767497hf_nat @ hF_Mirabelle_Rep_hf @ hF_Mirabelle_Abs_hf @ top_top_set_nat ).
% 0.29/0.46
% 0.29/0.46 % type_definition_hf
% 0.29/0.46 thf(fact_147_inf__right__idem,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Y3 )
% 0.29/0.46 = ( inf_inf_set_nat @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_148_inf__right__idem,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Y3 )
% 0.29/0.46 = ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_149_inf__right__idem,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Y3 )
% 0.29/0.46 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_right_idem
% 0.29/0.46 thf(fact_150_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
% 0.29/0.46 = ( inf_inf_set_nat @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_151_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ B )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_152_inf_Oright__idem,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ B )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.right_idem
% 0.29/0.46 thf(fact_153_inf__left__idem,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
% 0.29/0.46 = ( inf_inf_set_nat @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_154_inf__left__idem,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_155_inf__left__idem,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_idem
% 0.29/0.46 thf(fact_156_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
% 0.29/0.46 = ( inf_inf_set_nat @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_157_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_158_inf_Oleft__idem,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ A2 @ B ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_idem
% 0.29/0.46 thf(fact_159_inf__idem,axiom,
% 0.29/0.46 ! [X: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ X @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_160_inf__idem,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ X @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_161_inf__idem,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_idem
% 0.29/0.46 thf(fact_162_inf_Oidem,axiom,
% 0.29/0.46 ! [A2: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ A2 @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf.idem
% 0.29/0.46 thf(fact_163_inf_Oidem,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ A2 @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf.idem
% 0.29/0.46 thf(fact_164_inf_Oidem,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ A2 @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf.idem
% 0.29/0.46 thf(fact_165_inf__top_Oright__neutral,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ A2 @ top_to489427057lle_hf )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.right_neutral
% 0.29/0.46 thf(fact_166_inf__top_Oright__neutral,axiom,
% 0.29/0.46 ! [A2: nat > $o] :
% 0.29/0.46 ( ( inf_inf_nat_o @ A2 @ top_top_nat_o )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.right_neutral
% 0.29/0.46 thf(fact_167_inf__top_Oright__neutral,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( inf_in307783154e_hf_o @ A2 @ top_to22270292e_hf_o )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.right_neutral
% 0.29/0.46 thf(fact_168_inf__top_Oright__neutral,axiom,
% 0.29/0.46 ! [A2: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.right_neutral
% 0.29/0.46 thf(fact_169_inf__top_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( top_to489427057lle_hf
% 0.29/0.46 = ( inf_in923488851lle_hf @ A2 @ B ) )
% 0.29/0.46 = ( ( A2 = top_to489427057lle_hf )
% 0.29/0.46 & ( B = top_to489427057lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.neutr_eq_iff
% 0.29/0.46 thf(fact_170_inf__top_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A2: nat > $o,B: nat > $o] :
% 0.29/0.46 ( ( top_top_nat_o
% 0.29/0.46 = ( inf_inf_nat_o @ A2 @ B ) )
% 0.29/0.46 = ( ( A2 = top_top_nat_o )
% 0.29/0.46 & ( B = top_top_nat_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.neutr_eq_iff
% 0.29/0.46 thf(fact_171_inf__top_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( top_to22270292e_hf_o
% 0.29/0.46 = ( inf_in307783154e_hf_o @ A2 @ B ) )
% 0.29/0.46 = ( ( A2 = top_to22270292e_hf_o )
% 0.29/0.46 & ( B = top_to22270292e_hf_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.neutr_eq_iff
% 0.29/0.46 thf(fact_172_inf__top_Oneutr__eq__iff,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( top_top_set_nat
% 0.29/0.46 = ( inf_inf_set_nat @ A2 @ B ) )
% 0.29/0.46 = ( ( A2 = top_top_set_nat )
% 0.29/0.46 & ( B = top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.neutr_eq_iff
% 0.29/0.46 thf(fact_173_inf__top_Oleft__neutral,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.left_neutral
% 0.29/0.46 thf(fact_174_inf__top_Oleft__neutral,axiom,
% 0.29/0.46 ! [A2: nat > $o] :
% 0.29/0.46 ( ( inf_inf_nat_o @ top_top_nat_o @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.left_neutral
% 0.29/0.46 thf(fact_175_inf__top_Oleft__neutral,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.left_neutral
% 0.29/0.46 thf(fact_176_inf__top_Oleft__neutral,axiom,
% 0.29/0.46 ! [A2: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.left_neutral
% 0.29/0.46 thf(fact_177_inf__top_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( inf_in923488851lle_hf @ A2 @ B )
% 0.29/0.46 = top_to489427057lle_hf )
% 0.29/0.46 = ( ( A2 = top_to489427057lle_hf )
% 0.29/0.46 & ( B = top_to489427057lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.eq_neutr_iff
% 0.29/0.46 thf(fact_178_inf__top_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A2: nat > $o,B: nat > $o] :
% 0.29/0.46 ( ( ( inf_inf_nat_o @ A2 @ B )
% 0.29/0.46 = top_top_nat_o )
% 0.29/0.46 = ( ( A2 = top_top_nat_o )
% 0.29/0.46 & ( B = top_top_nat_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.eq_neutr_iff
% 0.29/0.46 thf(fact_179_inf__top_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( ( inf_in307783154e_hf_o @ A2 @ B )
% 0.29/0.46 = top_to22270292e_hf_o )
% 0.29/0.46 = ( ( A2 = top_to22270292e_hf_o )
% 0.29/0.46 & ( B = top_to22270292e_hf_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.eq_neutr_iff
% 0.29/0.46 thf(fact_180_inf__top_Oeq__neutr__iff,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( ( inf_inf_set_nat @ A2 @ B )
% 0.29/0.46 = top_top_set_nat )
% 0.29/0.46 = ( ( A2 = top_top_set_nat )
% 0.29/0.46 & ( B = top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top.eq_neutr_iff
% 0.29/0.46 thf(fact_181_top__eq__inf__iff,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( top_to489427057lle_hf
% 0.29/0.46 = ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( ( X = top_to489427057lle_hf )
% 0.29/0.46 & ( Y3 = top_to489427057lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_eq_inf_iff
% 0.29/0.46 thf(fact_182_top__eq__inf__iff,axiom,
% 0.29/0.46 ! [X: nat > $o,Y3: nat > $o] :
% 0.29/0.46 ( ( top_top_nat_o
% 0.29/0.46 = ( inf_inf_nat_o @ X @ Y3 ) )
% 0.29/0.46 = ( ( X = top_top_nat_o )
% 0.29/0.46 & ( Y3 = top_top_nat_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_eq_inf_iff
% 0.29/0.46 thf(fact_183_top__eq__inf__iff,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( top_to22270292e_hf_o
% 0.29/0.46 = ( inf_in307783154e_hf_o @ X @ Y3 ) )
% 0.29/0.46 = ( ( X = top_to22270292e_hf_o )
% 0.29/0.46 & ( Y3 = top_to22270292e_hf_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_eq_inf_iff
% 0.29/0.46 thf(fact_184_top__eq__inf__iff,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( top_top_set_nat
% 0.29/0.46 = ( inf_inf_set_nat @ X @ Y3 ) )
% 0.29/0.46 = ( ( X = top_top_set_nat )
% 0.29/0.46 & ( Y3 = top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_eq_inf_iff
% 0.29/0.46 thf(fact_185_inf__eq__top__iff,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( inf_in923488851lle_hf @ X @ Y3 )
% 0.29/0.46 = top_to489427057lle_hf )
% 0.29/0.46 = ( ( X = top_to489427057lle_hf )
% 0.29/0.46 & ( Y3 = top_to489427057lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_eq_top_iff
% 0.29/0.46 thf(fact_186_inf__eq__top__iff,axiom,
% 0.29/0.46 ! [X: nat > $o,Y3: nat > $o] :
% 0.29/0.46 ( ( ( inf_inf_nat_o @ X @ Y3 )
% 0.29/0.46 = top_top_nat_o )
% 0.29/0.46 = ( ( X = top_top_nat_o )
% 0.29/0.46 & ( Y3 = top_top_nat_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_eq_top_iff
% 0.29/0.46 thf(fact_187_inf__eq__top__iff,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( ( inf_in307783154e_hf_o @ X @ Y3 )
% 0.29/0.46 = top_to22270292e_hf_o )
% 0.29/0.46 = ( ( X = top_to22270292e_hf_o )
% 0.29/0.46 & ( Y3 = top_to22270292e_hf_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_eq_top_iff
% 0.29/0.46 thf(fact_188_inf__eq__top__iff,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( ( inf_inf_set_nat @ X @ Y3 )
% 0.29/0.46 = top_top_set_nat )
% 0.29/0.46 = ( ( X = top_top_set_nat )
% 0.29/0.46 & ( Y3 = top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_eq_top_iff
% 0.29/0.46 thf(fact_189_inf__top__right,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ X @ top_to489427057lle_hf )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_right
% 0.29/0.46 thf(fact_190_inf__top__right,axiom,
% 0.29/0.46 ! [X: nat > $o] :
% 0.29/0.46 ( ( inf_inf_nat_o @ X @ top_top_nat_o )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_right
% 0.29/0.46 thf(fact_191_inf__top__right,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( inf_in307783154e_hf_o @ X @ top_to22270292e_hf_o )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_right
% 0.29/0.46 thf(fact_192_inf__top__right,axiom,
% 0.29/0.46 ! [X: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ X @ top_top_set_nat )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_right
% 0.29/0.46 thf(fact_193_inf__top__left,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_left
% 0.29/0.46 thf(fact_194_inf__top__left,axiom,
% 0.29/0.46 ! [X: nat > $o] :
% 0.29/0.46 ( ( inf_inf_nat_o @ top_top_nat_o @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_left
% 0.29/0.46 thf(fact_195_inf__top__left,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf > $o] :
% 0.29/0.46 ( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_left
% 0.29/0.46 thf(fact_196_inf__top__left,axiom,
% 0.29/0.46 ! [X: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ top_top_set_nat @ X )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_top_left
% 0.29/0.46 thf(fact_197_inf__sup__absorb,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_absorb
% 0.29/0.46 thf(fact_198_inf__sup__absorb,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_absorb
% 0.29/0.46 thf(fact_199_inf__sup__absorb,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_absorb
% 0.29/0.46 thf(fact_200_sup__inf__absorb,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % sup_inf_absorb
% 0.29/0.46 thf(fact_201_sup__inf__absorb,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % sup_inf_absorb
% 0.29/0.46 thf(fact_202_sup__inf__absorb,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % sup_inf_absorb
% 0.29/0.46 thf(fact_203_hinter__iff,axiom,
% 0.29/0.46 ! [U4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.29/0.46 & ( hF_Mirabelle_hmem @ U4 @ Y3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hinter_iff
% 0.29/0.46 thf(fact_204_hinter__hempty__left,axiom,
% 0.29/0.46 ! [A: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A )
% 0.29/0.46 = zero_z189798548lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % hinter_hempty_left
% 0.29/0.46 thf(fact_205_hinter__hempty__right,axiom,
% 0.29/0.46 ! [A: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ A @ zero_z189798548lle_hf )
% 0.29/0.46 = zero_z189798548lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % hinter_hempty_right
% 0.29/0.46 thf(fact_206_Abs__hf__0,axiom,
% 0.29/0.46 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.29/0.46 = zero_z189798548lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % Abs_hf_0
% 0.29/0.46 thf(fact_207_inf__left__commute,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_208_inf__left__commute,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ Y3 @ ( inf_in923488851lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_209_inf__left__commute,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_left_commute
% 0.29/0.46 thf(fact_210_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
% 0.29/0.46 = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_211_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A2 @ C ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_212_inf_Oleft__commute,axiom,
% 0.29/0.46 ! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A2 @ C ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.left_commute
% 0.29/0.46 thf(fact_213_inf__commute,axiom,
% 0.29/0.46 ( inf_inf_set_nat
% 0.29/0.46 = ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_214_inf__commute,axiom,
% 0.29/0.46 ( inf_in923488851lle_hf
% 0.29/0.46 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_215_inf__commute,axiom,
% 0.29/0.46 ( inf_in956532509lle_hf
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_commute
% 0.29/0.46 thf(fact_216_inf_Ocommute,axiom,
% 0.29/0.46 ( inf_inf_set_nat
% 0.29/0.46 = ( ^ [A4: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_217_inf_Ocommute,axiom,
% 0.29/0.46 ( inf_in923488851lle_hf
% 0.29/0.46 = ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_218_inf_Ocommute,axiom,
% 0.29/0.46 ( inf_in956532509lle_hf
% 0.29/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.commute
% 0.29/0.46 thf(fact_219_inf__assoc,axiom,
% 0.29/0.46 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_220_inf__assoc,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_221_inf__assoc,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_assoc
% 0.29/0.46 thf(fact_222_inf_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
% 0.29/0.46 = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_223_inf_Oassoc,axiom,
% 0.29/0.46 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ C )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_224_inf_Oassoc,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ C )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf.assoc
% 0.29/0.46 thf(fact_225_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( inf_inf_set_nat @ K @ B ) )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A2 @ B4 )
% 0.29/0.46 = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_226_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( inf_in923488851lle_hf @ K @ B ) )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A2 @ B4 )
% 0.29/0.46 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_227_boolean__algebra__cancel_Oinf2,axiom,
% 0.29/0.46 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( inf_in956532509lle_hf @ K @ B ) )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ A2 @ B4 )
% 0.29/0.46 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf2
% 0.29/0.46 thf(fact_228_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( inf_inf_set_nat @ K @ A2 ) )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A @ B )
% 0.29/0.46 = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_229_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( inf_in923488851lle_hf @ K @ A2 ) )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A @ B )
% 0.29/0.46 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_230_boolean__algebra__cancel_Oinf1,axiom,
% 0.29/0.46 ! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( inf_in956532509lle_hf @ K @ A2 ) )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ A @ B )
% 0.29/0.46 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % boolean_algebra_cancel.inf1
% 0.29/0.46 thf(fact_231_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ( inf_inf_set_nat
% 0.29/0.46 = ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_232_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ( inf_in923488851lle_hf
% 0.29/0.46 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_233_inf__sup__aci_I1_J,axiom,
% 0.29/0.46 ( inf_in956532509lle_hf
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(1)
% 0.29/0.46 thf(fact_234_inf__sup__aci_I2_J,axiom,
% 0.29/0.46 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(2)
% 0.29/0.46 thf(fact_235_inf__sup__aci_I2_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
% 0.29/0.46 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(2)
% 0.29/0.46 thf(fact_236_inf__sup__aci_I3_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(3)
% 0.29/0.46 thf(fact_237_inf__sup__aci_I4_J,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_aci(4)
% 0.29/0.46 thf(fact_238_sup__inf__distrib2,axiom,
% 0.29/0.46 ! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ Z ) @ X )
% 0.29/0.46 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_inf_distrib2
% 0.29/0.46 thf(fact_239_sup__inf__distrib1,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sup_inf_distrib1
% 0.29/0.46 thf(fact_240_inf__sup__distrib2,axiom,
% 0.29/0.46 ! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ Z ) @ X )
% 0.29/0.46 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_distrib2
% 0.29/0.46 thf(fact_241_inf__sup__distrib1,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inf_sup_distrib1
% 0.29/0.46 thf(fact_242_distrib__imp2,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su638957495lle_hf @ X3 @ ( inf_in956532509lle_hf @ Y5 @ Z3 ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X3 @ Y5 ) @ ( sup_su638957495lle_hf @ X3 @ Z3 ) ) )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % distrib_imp2
% 0.29/0.46 thf(fact_243_distrib__imp1,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in956532509lle_hf @ X3 @ ( sup_su638957495lle_hf @ Y5 @ Z3 ) )
% 0.29/0.46 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X3 @ Y5 ) @ ( inf_in956532509lle_hf @ X3 @ Z3 ) ) )
% 0.29/0.46 => ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % distrib_imp1
% 0.29/0.46 thf(fact_244_foundation,axiom,
% 0.29/0.46 ! [Z: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( Z != zero_z189798548lle_hf )
% 0.29/0.46 => ? [W: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.29/0.46 & ( ( inf_in956532509lle_hf @ W @ Z )
% 0.29/0.46 = zero_z189798548lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % foundation
% 0.29/0.46 thf(fact_245_hinter__hinsert__right,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.29/0.46 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A ) ) ) )
% 0.29/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.29/0.46 = ( inf_in956532509lle_hf @ B4 @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hinter_hinsert_right
% 0.29/0.46 thf(fact_246_hinter__hinsert__left,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.29/0.46 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A @ B4 ) ) ) )
% 0.29/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.29/0.46 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.29/0.46 = ( inf_in956532509lle_hf @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hinter_hinsert_left
% 0.29/0.46 thf(fact_247_zero__reorient,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( zero_z189798548lle_hf = X )
% 0.29/0.46 = ( X = zero_z189798548lle_hf ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_reorient
% 0.29/0.46 thf(fact_248_zero__reorient,axiom,
% 0.29/0.46 ! [X: nat] :
% 0.29/0.46 ( ( zero_zero_nat = X )
% 0.29/0.46 = ( X = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_reorient
% 0.29/0.46 thf(fact_249_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ! [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A )
% 0.29/0.46 => ( top_to489427057lle_hf = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_250_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A: set_nat] :
% 0.29/0.46 ( ! [X3: nat] : ( member_nat @ X3 @ A )
% 0.29/0.46 => ( top_top_set_nat = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_251_UNIV__witness,axiom,
% 0.29/0.46 ? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ top_to489427057lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_252_UNIV__witness,axiom,
% 0.29/0.46 ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_253_Un__UNIV__right,axiom,
% 0.29/0.46 ! [A: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ A @ top_top_set_nat )
% 0.29/0.46 = top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % Un_UNIV_right
% 0.29/0.46 thf(fact_254_Un__UNIV__left,axiom,
% 0.29/0.46 ! [B4: set_nat] :
% 0.29/0.46 ( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
% 0.29/0.46 = top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % Un_UNIV_left
% 0.29/0.46 thf(fact_255_top__set__def,axiom,
% 0.29/0.46 ( top_top_set_nat
% 0.29/0.46 = ( collect_nat @ top_top_nat_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_set_def
% 0.29/0.46 thf(fact_256_type__copy__ex__RepI,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,F2: nat > $o] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
% 0.29/0.46 => ( ( ? [X4: nat] : ( F2 @ X4 ) )
% 0.29/0.46 = ( ? [B3: hF_Mirabelle_hf] : ( F2 @ ( Rep @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_copy_ex_RepI
% 0.29/0.46 thf(fact_257_type__copy__obj__one__point__absE,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,S: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
% 0.29/0.46 => ~ ! [X3: nat] :
% 0.29/0.46 ( S
% 0.29/0.46 != ( Abs @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_copy_obj_one_point_absE
% 0.29/0.46 thf(fact_258_HF__hfset,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % HF_hfset
% 0.29/0.46 thf(fact_259_type__definition_ORep,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( member_nat @ ( Rep @ X ) @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Rep
% 0.29/0.46 thf(fact_260_type__definition__def,axiom,
% 0.29/0.46 ( type_d1794767497hf_nat
% 0.29/0.46 = ( ^ [Rep2: hF_Mirabelle_hf > nat,Abs2: nat > hF_Mirabelle_hf,A6: set_nat] :
% 0.29/0.46 ( ! [X2: hF_Mirabelle_hf] : ( member_nat @ ( Rep2 @ X2 ) @ A6 )
% 0.29/0.46 & ! [X2: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.29/0.46 = X2 )
% 0.29/0.46 & ! [Y: nat] :
% 0.29/0.46 ( ( member_nat @ Y @ A6 )
% 0.29/0.46 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.29/0.46 = Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition_def
% 0.29/0.46 thf(fact_261_Int__UNIV,axiom,
% 0.29/0.46 ! [A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.29/0.46 = top_top_set_nat )
% 0.29/0.46 = ( ( A = top_top_set_nat )
% 0.29/0.46 & ( B4 = top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_UNIV
% 0.29/0.46 thf(fact_262_UnCI,axiom,
% 0.29/0.46 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.29/0.46 ( ( ~ ( member_nat @ C @ B4 )
% 0.29/0.46 => ( member_nat @ C @ A ) )
% 0.29/0.46 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnCI
% 0.29/0.46 thf(fact_263_UnCI,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( ~ ( member1367349282lle_hf @ C @ B4 )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ A ) )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnCI
% 0.29/0.46 thf(fact_264_Un__iff,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.29/0.46 = ( ( member_nat @ C @ A )
% 0.29/0.46 | ( member_nat @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_iff
% 0.29/0.46 thf(fact_265_Un__iff,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
% 0.29/0.46 = ( ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 | ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_iff
% 0.29/0.46 thf(fact_266_UnE,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.29/0.46 => ( ~ ( member_nat @ C @ A )
% 0.29/0.46 => ( member_nat @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnE
% 0.29/0.46 thf(fact_267_UnE,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
% 0.29/0.46 => ( ~ ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnE
% 0.29/0.46 thf(fact_268_UnI1,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ A )
% 0.29/0.46 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI1
% 0.29/0.46 thf(fact_269_UnI1,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI1
% 0.29/0.46 thf(fact_270_UnI2,axiom,
% 0.29/0.46 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ B4 )
% 0.29/0.46 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI2
% 0.29/0.46 thf(fact_271_UnI2,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ B4 )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UnI2
% 0.29/0.46 thf(fact_272_Int__UNIV__left,axiom,
% 0.29/0.46 ! [B4: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
% 0.29/0.46 = B4 ) ).
% 0.29/0.46
% 0.29/0.46 % Int_UNIV_left
% 0.29/0.46 thf(fact_273_Int__UNIV__right,axiom,
% 0.29/0.46 ! [A: set_nat] :
% 0.29/0.46 ( ( inf_inf_set_nat @ A @ top_top_set_nat )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % Int_UNIV_right
% 0.29/0.46 thf(fact_274_type__definition_ORep__inverse,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( Abs @ ( Rep @ X ) )
% 0.29/0.46 = X ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Rep_inverse
% 0.29/0.46 thf(fact_275_type__definition_OAbs__inverse,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( member_nat @ Y3 @ A )
% 0.29/0.46 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.29/0.46 = Y3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Abs_inverse
% 0.29/0.46 thf(fact_276_type__definition_ORep__inject,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( ( Rep @ X )
% 0.29/0.46 = ( Rep @ Y3 ) )
% 0.29/0.46 = ( X = Y3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Rep_inject
% 0.29/0.46 thf(fact_277_type__definition_ORep__induct,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat,P: nat > $o] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( member_nat @ Y3 @ A )
% 0.29/0.46 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( Rep @ X3 ) )
% 0.29/0.46 => ( P @ Y3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Rep_induct
% 0.29/0.46 thf(fact_278_type__definition_OAbs__inject,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: nat,Y3: nat] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( member_nat @ X @ A )
% 0.29/0.46 => ( ( member_nat @ Y3 @ A )
% 0.29/0.46 => ( ( ( Abs @ X )
% 0.29/0.46 = ( Abs @ Y3 ) )
% 0.29/0.46 = ( X = Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Abs_inject
% 0.29/0.46 thf(fact_279_type__definition_OAbs__induct,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ! [Y5: nat] :
% 0.29/0.46 ( ( member_nat @ Y5 @ A )
% 0.29/0.46 => ( P @ ( Abs @ Y5 ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Abs_induct
% 0.29/0.46 thf(fact_280_type__definition_ORep__cases,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ( ( member_nat @ Y3 @ A )
% 0.29/0.46 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( Rep @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Rep_cases
% 0.29/0.46 thf(fact_281_type__definition_OAbs__cases,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.29/0.46 => ~ ! [Y5: nat] :
% 0.29/0.46 ( ( X
% 0.29/0.46 = ( Abs @ Y5 ) )
% 0.29/0.46 => ~ ( member_nat @ Y5 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.Abs_cases
% 0.29/0.46 thf(fact_282_type__definition_Ointro,axiom,
% 0.29/0.46 ! [Rep: hF_Mirabelle_hf > nat,A: set_nat,Abs: nat > hF_Mirabelle_hf] :
% 0.29/0.46 ( ! [X3: hF_Mirabelle_hf] : ( member_nat @ ( Rep @ X3 ) @ A )
% 0.29/0.46 => ( ! [X3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( Abs @ ( Rep @ X3 ) )
% 0.29/0.46 = X3 )
% 0.29/0.46 => ( ! [Y5: nat] :
% 0.29/0.46 ( ( member_nat @ Y5 @ A )
% 0.29/0.46 => ( ( Rep @ ( Abs @ Y5 ) )
% 0.29/0.46 = Y5 ) )
% 0.29/0.46 => ( type_d1794767497hf_nat @ Rep @ Abs @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition.intro
% 0.29/0.46 thf(fact_283_top__empty__eq,axiom,
% 0.29/0.46 ( top_to22270292e_hf_o
% 0.29/0.46 = ( ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ top_to489427057lle_hf ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_empty_eq
% 0.29/0.46 thf(fact_284_top__empty__eq,axiom,
% 0.29/0.46 ( top_top_nat_o
% 0.29/0.46 = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_empty_eq
% 0.29/0.46 thf(fact_285_hcard__0,axiom,
% 0.29/0.46 ( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
% 0.29/0.46 = zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % hcard_0
% 0.29/0.46 thf(fact_286_hinsert__def,axiom,
% 0.29/0.46 ( hF_Mirabelle_hinsert
% 0.29/0.46 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hinsert_def
% 0.29/0.46 thf(fact_287_insert__absorb2,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ X @ A ) )
% 0.29/0.46 = ( insert9649339lle_hf @ X @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb2
% 0.29/0.46 thf(fact_288_insert__iff,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.29/0.46 = ( ( A2 = B )
% 0.29/0.46 | ( member_nat @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_289_insert__iff,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
% 0.29/0.46 = ( ( A2 = B )
% 0.29/0.46 | ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_290_insertCI,axiom,
% 0.29/0.46 ! [A2: nat,B4: set_nat,B: nat] :
% 0.29/0.46 ( ( ~ ( member_nat @ A2 @ B4 )
% 0.29/0.46 => ( A2 = B ) )
% 0.29/0.46 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_291_insertCI,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
% 0.29/0.46 => ( A2 = B ) )
% 0.29/0.46 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_292_IntI,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ A )
% 0.29/0.46 => ( ( member_nat @ C @ B4 )
% 0.29/0.46 => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntI
% 0.29/0.46 thf(fact_293_IntI,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 => ( ( member1367349282lle_hf @ C @ B4 )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntI
% 0.29/0.46 thf(fact_294_Int__iff,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.29/0.46 = ( ( member_nat @ C @ A )
% 0.29/0.46 & ( member_nat @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_iff
% 0.29/0.46 thf(fact_295_Int__iff,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.29/0.46 = ( ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 & ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_iff
% 0.29/0.46 thf(fact_296_Int__insert__left__if0,axiom,
% 0.29/0.46 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.29/0.46 ( ~ ( member_nat @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if0
% 0.29/0.46 thf(fact_297_Int__insert__left__if0,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ~ ( member1367349282lle_hf @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if0
% 0.29/0.46 thf(fact_298_Int__insert__left__if1,axiom,
% 0.29/0.46 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if1
% 0.29/0.46 thf(fact_299_Int__insert__left__if1,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left_if1
% 0.29/0.46 thf(fact_300_insert__inter__insert,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ A ) @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_inter_insert
% 0.29/0.46 thf(fact_301_Int__insert__right__if0,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ~ ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.29/0.46 = ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if0
% 0.29/0.46 thf(fact_302_Int__insert__right__if0,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if0
% 0.29/0.46 thf(fact_303_Int__insert__right__if1,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.29/0.46 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if1
% 0.29/0.46 thf(fact_304_Int__insert__right__if1,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right_if1
% 0.29/0.46 thf(fact_305_Un__insert__left,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B4 @ C2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_insert_left
% 0.29/0.46 thf(fact_306_Un__insert__right,axiom,
% 0.29/0.46 ! [A: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( sup_su1790843629lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Un_insert_right
% 0.29/0.46 thf(fact_307_insert__UNIV,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( insert9649339lle_hf @ X @ top_to489427057lle_hf )
% 0.29/0.46 = top_to489427057lle_hf ) ).
% 0.29/0.46
% 0.29/0.46 % insert_UNIV
% 0.29/0.46 thf(fact_308_insert__UNIV,axiom,
% 0.29/0.46 ! [X: nat] :
% 0.29/0.46 ( ( insert_nat @ X @ top_top_set_nat )
% 0.29/0.46 = top_top_set_nat ) ).
% 0.29/0.46
% 0.29/0.46 % insert_UNIV
% 0.29/0.46 thf(fact_309_IntE,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.29/0.46 => ~ ( ( member_nat @ C @ A )
% 0.29/0.46 => ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntE
% 0.29/0.46 thf(fact_310_IntE,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.29/0.46 => ~ ( ( member1367349282lle_hf @ C @ A )
% 0.29/0.46 => ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntE
% 0.29/0.46 thf(fact_311_IntD1,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.29/0.46 => ( member_nat @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD1
% 0.29/0.46 thf(fact_312_IntD1,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD1
% 0.29/0.46 thf(fact_313_IntD2,axiom,
% 0.29/0.46 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.29/0.46 => ( member_nat @ C @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD2
% 0.29/0.46 thf(fact_314_IntD2,axiom,
% 0.29/0.46 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.29/0.46 => ( member1367349282lle_hf @ C @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % IntD2
% 0.29/0.46 thf(fact_315_Int__insert__left,axiom,
% 0.29/0.46 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( ( member_nat @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
% 0.29/0.46 & ( ~ ( member_nat @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left
% 0.29/0.46 thf(fact_316_Int__insert__left,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( member1367349282lle_hf @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) )
% 0.29/0.46 & ( ~ ( member1367349282lle_hf @ A2 @ C2 )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.29/0.46 = ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_left
% 0.29/0.46 thf(fact_317_Int__insert__right,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.29/0.46 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
% 0.29/0.46 & ( ~ ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.29/0.46 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right
% 0.29/0.46 thf(fact_318_Int__insert__right,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) )
% 0.29/0.46 & ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.29/0.46 = ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Int_insert_right
% 0.29/0.46 thf(fact_319_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ? [B6: set_nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_nat @ A2 @ B6 ) )
% 0.29/0.46 & ~ ( member_nat @ A2 @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_320_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ? [B6: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ B6 ) )
% 0.29/0.46 & ~ ( member1367349282lle_hf @ A2 @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_321_insert__commute,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ Y3 @ A ) )
% 0.29/0.46 = ( insert9649339lle_hf @ Y3 @ ( insert9649339lle_hf @ X @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_commute
% 0.29/0.46 thf(fact_322_insert__eq__iff,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
% 0.29/0.46 ( ~ ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ~ ( member_nat @ B @ B4 )
% 0.29/0.46 => ( ( ( insert_nat @ A2 @ A )
% 0.29/0.46 = ( insert_nat @ B @ B4 ) )
% 0.29/0.46 = ( ( ( A2 = B )
% 0.29/0.46 => ( A = B4 ) )
% 0.29/0.46 & ( ( A2 != B )
% 0.29/0.46 => ? [C3: set_nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_nat @ B @ C3 ) )
% 0.29/0.46 & ~ ( member_nat @ B @ C3 )
% 0.29/0.46 & ( B4
% 0.29/0.46 = ( insert_nat @ A2 @ C3 ) )
% 0.29/0.46 & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_323_insert__eq__iff,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ~ ( member1367349282lle_hf @ B @ B4 )
% 0.29/0.46 => ( ( ( insert9649339lle_hf @ A2 @ A )
% 0.29/0.46 = ( insert9649339lle_hf @ B @ B4 ) )
% 0.29/0.46 = ( ( ( A2 = B )
% 0.29/0.46 => ( A = B4 ) )
% 0.29/0.46 & ( ( A2 != B )
% 0.29/0.46 => ? [C3: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert9649339lle_hf @ B @ C3 ) )
% 0.29/0.46 & ~ ( member1367349282lle_hf @ B @ C3 )
% 0.29/0.46 & ( B4
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ C3 ) )
% 0.29/0.46 & ~ ( member1367349282lle_hf @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_324_insert__absorb,axiom,
% 0.29/0.46 ! [A2: nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ A )
% 0.29/0.46 => ( ( insert_nat @ A2 @ A )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_325_insert__absorb,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ A )
% 0.29/0.46 => ( ( insert9649339lle_hf @ A2 @ A )
% 0.29/0.46 = A ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_326_insert__ident,axiom,
% 0.29/0.46 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.29/0.46 ( ~ ( member_nat @ X @ A )
% 0.29/0.46 => ( ~ ( member_nat @ X @ B4 )
% 0.29/0.46 => ( ( ( insert_nat @ X @ A )
% 0.29/0.46 = ( insert_nat @ X @ B4 ) )
% 0.29/0.46 = ( A = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_327_insert__ident,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ~ ( member1367349282lle_hf @ X @ A )
% 0.29/0.46 => ( ~ ( member1367349282lle_hf @ X @ B4 )
% 0.29/0.46 => ( ( ( insert9649339lle_hf @ X @ A )
% 0.29/0.46 = ( insert9649339lle_hf @ X @ B4 ) )
% 0.29/0.46 = ( A = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_328_Set_Oset__insert,axiom,
% 0.29/0.46 ! [X: nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ X @ A )
% 0.29/0.46 => ~ ! [B6: set_nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert_nat @ X @ B6 ) )
% 0.29/0.46 => ( member_nat @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_329_Set_Oset__insert,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ X @ A )
% 0.29/0.46 => ~ ! [B6: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( insert9649339lle_hf @ X @ B6 ) )
% 0.29/0.46 => ( member1367349282lle_hf @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.set_insert
% 0.29/0.46 thf(fact_330_insertI2,axiom,
% 0.29/0.46 ! [A2: nat,B4: set_nat,B: nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ B4 )
% 0.29/0.46 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_331_insertI2,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.29/0.46 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI2
% 0.29/0.46 thf(fact_332_insertI1,axiom,
% 0.29/0.46 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_333_insertI1,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertI1
% 0.29/0.46 thf(fact_334_insertE,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,A: set_nat] :
% 0.29/0.46 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.29/0.46 => ( ( A2 != B )
% 0.29/0.46 => ( member_nat @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_335_insertE,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.29/0.46 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
% 0.29/0.46 => ( ( A2 != B )
% 0.29/0.46 => ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertE
% 0.29/0.46 thf(fact_336_hfset__hinsert,axiom,
% 0.29/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ A2 @ B ) )
% 0.29/0.46 = ( insert9649339lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hfset_hinsert
% 0.29/0.46 thf(fact_337_hcard__union__inter,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( plus_plus_nat @ ( hF_Mirabelle_hcard @ ( sup_su638957495lle_hf @ X @ Y3 ) ) @ ( hF_Mirabelle_hcard @ ( inf_in956532509lle_hf @ X @ Y3 ) ) )
% 0.29/0.46 = ( plus_plus_nat @ ( hF_Mirabelle_hcard @ X ) @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hcard_union_inter
% 0.29/0.46 thf(fact_338_hcard__hinsert__if,axiom,
% 0.29/0.46 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.29/0.46 ( ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.29/0.46 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.29/0.46 = ( hF_Mirabelle_hcard @ Y3 ) ) )
% 0.29/0.46 & ( ~ ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.29/0.46 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.29/0.46 = ( suc @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % hcard_hinsert_if
% 0.29/0.46 thf(fact_339_add__left__cancel,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ A2 @ B )
% 0.29/0.46 = ( plus_plus_nat @ A2 @ C ) )
% 0.29/0.46 = ( B = C ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_cancel
% 0.29/0.46 thf(fact_340_add__right__cancel,axiom,
% 0.29/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ B @ A2 )
% 0.29/0.46 = ( plus_plus_nat @ C @ A2 ) )
% 0.29/0.46 = ( B = C ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_cancel
% 0.29/0.46 thf(fact_341_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.46 ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( zero_zero_nat
% 0.29/0.46 = ( plus_plus_nat @ X @ Y3 ) )
% 0.29/0.46 = ( ( X = zero_zero_nat )
% 0.29/0.46 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_eq_add_iff_both_eq_0
% 0.29/0.46 thf(fact_342_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.46 ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ X @ Y3 )
% 0.29/0.46 = zero_zero_nat )
% 0.29/0.46 = ( ( X = zero_zero_nat )
% 0.29/0.46 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_eq_0_iff_both_eq_0
% 0.29/0.46 thf(fact_343_add__cancel__right__right,axiom,
% 0.29/0.46 ! [A2: nat,B: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus_nat @ A2 @ B ) )
% 0.29/0.46 = ( B = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_right_right
% 0.29/0.46 thf(fact_344_add__cancel__right__left,axiom,
% 0.29/0.46 ! [A2: nat,B: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus_nat @ B @ A2 ) )
% 0.29/0.46 = ( B = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_right_left
% 0.29/0.46 thf(fact_345_add__cancel__left__right,axiom,
% 0.29/0.46 ! [A2: nat,B: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ A2 @ B )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( B = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_left_right
% 0.29/0.46 thf(fact_346_add__cancel__left__left,axiom,
% 0.29/0.46 ! [B: nat,A2: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ B @ A2 )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( B = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_cancel_left_left
% 0.29/0.46 thf(fact_347_add_Oright__neutral,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_neutral
% 0.29/0.46 thf(fact_348_add_Oleft__neutral,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_neutral
% 0.29/0.46 thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.29/0.46 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.46 thf(fact_350_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ( plus_plus_nat @ I @ K )
% 0.29/0.46 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(4)
% 0.29/0.46 thf(fact_351_group__cancel_Oadd1,axiom,
% 0.29/0.46 ! [A: nat,K: nat,A2: nat,B: nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( plus_plus_nat @ K @ A2 ) )
% 0.29/0.46 => ( ( plus_plus_nat @ A @ B )
% 0.29/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add1
% 0.29/0.46 thf(fact_352_group__cancel_Oadd2,axiom,
% 0.29/0.46 ! [B4: nat,K: nat,B: nat,A2: nat] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( plus_plus_nat @ K @ B ) )
% 0.29/0.46 => ( ( plus_plus_nat @ A2 @ B4 )
% 0.29/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add2
% 0.29/0.46 thf(fact_353_add_Oassoc,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.29/0.46 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.assoc
% 0.29/0.46
% 0.29/0.46 % Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 hF_Mir199975595nction @ r ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 0.30/1.51 ---- Embedded in THF ---
% 0.30/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.51 %%% Generated on Tue Feb 24 08:23:42 EST 2026
% 0.30/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.51 %%% Logic specification used:
% 0.30/1.51 %%% thf(spec, logic, $$dhol).
% 0.30/1.51
% 0.30/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2DTF3790.p
% See solution above
% 0.30/1.51 ------------------------
% 0.30/1.55 ---- Cleaned THF ---
% 0.30/1.55 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.30/1.55 set_HF_Mirabelle_hf: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 hF_Mirabelle_hf: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/1.55 set_nat: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(ty_n_t__Nat__Onat,type,
% 0.30/1.55 nat: $tType ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.30/1.55 plus_plus_nat: nat > nat > nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.30/1.55 zero_zero_nat: nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.30/1.55 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.30/1.55 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 0.30/1.55 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.30/1.55 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.30/1.55 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.30/1.55 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.30/1.55 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohcard,type,
% 0.30/1.55 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.30/1.55 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_ORep__hf,type,
% 0.30/1.55 hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.30/1.55 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 0.30/1.55 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohfunction,type,
% 0.30/1.55 hF_Mir199975595nction: hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.30/1.55 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.30/1.55 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 0.30/1.55 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohrelation,type,
% 0.30/1.55 hF_Mir434065167lation: hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohrestrict,type,
% 0.30/1.55 hF_Mir1653039215strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 0.30/1.55 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_HF__Mirabelle__glliljednj_Ois__hpair,type,
% 0.30/1.55 hF_Mir137170979_hpair: hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.30/1.55 inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.30/1.55 inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.30/1.55 inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/1.55 inf_inf_set_nat: set_nat > set_nat > set_nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Osup__class_Osup_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.30/1.55 sup_su1199008216e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.30/1.55 sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Osup__class_Osup_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.30/1.55 sup_su1790843629lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/1.55 sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Nat_OSuc,type,
% 0.30/1.55 suc: nat > nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Otop__class_Otop_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.30/1.55 top_to22270292e_hf_o: hF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.30/1.55 top_top_nat_o: nat > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
% 0.30/1.55 top_top_o: $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.30/1.55 top_to489427057lle_hf: set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/1.55 top_top_set_nat: set_nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.30/1.55 collect_nat: ( nat > $o ) > set_nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 0.30/1.55 insert_nat: nat > set_nat > set_nat ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_Typedef_Otype__definition_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.30/1.55 type_d1794767497hf_nat: ( hF_Mirabelle_hf > nat ) > ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.30/1.55 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_c_member_001t__Nat__Onat,type,
% 0.30/1.55 member_nat: nat > set_nat > $o ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_r,type,
% 0.30/1.55 r: hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(sy_v_x,type,
% 0.30/1.55 x: hF_Mirabelle_hf ).
% 0.30/1.55
% 0.30/1.55 thf(fact_0_hrelation__restr,axiom,
% 0.30/1.55 ! [R: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( hF_Mir434065167lation @ ( hF_Mir1653039215strict @ R @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_1_hfunction__def,axiom,
% 0.30/1.55 ( hF_Mir199975595nction
% 0.30/1.55 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.30/1.55 ! [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y ) @ R2 )
% 0.30/1.55 => ! [Y2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) @ R2 )
% 0.30/1.55 => ( Y = Y2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_2_hrestrict__iff,axiom,
% 0.30/1.55 ! [Z: hF_Mirabelle_hf,R: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ Z @ ( hF_Mir1653039215strict @ R @ A ) )
% 0.30/1.55 = ( ( hF_Mirabelle_hmem @ Z @ R )
% 0.30/1.55 & ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( Z
% 0.30/1.55 = ( hF_Mirabelle_hpair @ X2 @ Y ) )
% 0.30/1.55 & ( hF_Mirabelle_hmem @ X2 @ A ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_3_hpair__iff,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.30/1.55 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.30/1.55 = ( ( A2 = A3 )
% 0.30/1.55 & ( B = B2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_4_Rep__hf__inject,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( ( hF_Mirabelle_Rep_hf @ X )
% 0.30/1.55 = ( hF_Mirabelle_Rep_hf @ Y3 ) )
% 0.30/1.55 = ( X = Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_5_hf__equalityI,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ! [X3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X3 @ A2 )
% 0.30/1.55 = ( hF_Mirabelle_hmem @ X3 @ B ) )
% 0.30/1.55 => ( A2 = B ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_6_hf__ext,axiom,
% 0.30/1.55 ( ( ^ [Y4: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] : ( Y4 = Z2 ) )
% 0.30/1.55 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [X2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X2 @ A4 )
% 0.30/1.55 = ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_7_hmem__ne,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.30/1.55 => ( X != Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_8_replacement,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U @ X )
% 0.30/1.55 => ( ( R3 @ U @ V )
% 0.30/1.55 => ( ( R3 @ U @ V2 )
% 0.30/1.55 => ( V2 = V ) ) ) )
% 0.30/1.55 => ? [Z3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [V3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.30/1.55 = ( ? [U2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.30/1.55 & ( R3 @ U2 @ V3 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_9_binary__union,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ? [Z3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [U3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.30/1.55 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.30/1.55 | ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_10_hmem__not__sym,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ~ ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.30/1.55 & ( hF_Mirabelle_hmem @ Y3 @ X ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_11_hpair__inject,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.30/1.55 = ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.30/1.55 => ~ ( ( A2 = A3 )
% 0.30/1.55 => ( B != B2 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_12_union__of__set,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf] :
% 0.30/1.55 ? [Z3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [U3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.30/1.55 = ( ? [Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ Y @ X )
% 0.30/1.55 & ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_13_comprehension,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.30/1.55 ? [Z3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [U3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.30/1.55 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.30/1.55 & ( P @ U3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_14_hmem__not__refl,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf] :
% 0.30/1.55 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_15_hpair__neq__fst,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.30/1.55 != A2 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_16_hpair__neq__snd,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.30/1.55 != B ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_17_replacement__fun,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ? [Z3: hF_Mirabelle_hf] :
% 0.30/1.55 ! [V3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.30/1.55 = ( ? [U2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.30/1.55 & ( V3
% 0.30/1.55 = ( F @ U2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_18_hfst__conv,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.30/1.55 = A2 ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_19_hsnd__conv,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.30/1.55 = B ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_20_hrelation__def,axiom,
% 0.30/1.55 ( hF_Mir434065167lation
% 0.30/1.55 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.30/1.55 ! [Z4: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ Z4 @ R2 )
% 0.30/1.55 => ( hF_Mir137170979_hpair @ Z4 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_21_is__hpair__def,axiom,
% 0.30/1.55 ( hF_Mir137170979_hpair
% 0.30/1.55 = ( ^ [Z4: hF_Mirabelle_hf] :
% 0.30/1.55 ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( Z4
% 0.30/1.55 = ( hF_Mirabelle_hpair @ X2 @ Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_22_PrimReplace__iff,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.30/1.55 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U @ A )
% 0.30/1.55 => ( ( R3 @ U @ V )
% 0.30/1.55 => ( ( R3 @ U @ V2 )
% 0.30/1.55 => ( V2 = V ) ) ) )
% 0.30/1.55 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A @ R3 ) )
% 0.30/1.55 = ( ? [U2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.30/1.55 & ( R3 @ U2 @ V4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_23_HUnion__iff,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A ) )
% 0.30/1.55 = ( ? [Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ Y @ A )
% 0.30/1.55 & ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_24_Replace__iff,axiom,
% 0.30/1.55 ! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A @ R3 ) )
% 0.30/1.55 = ( ? [U2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.30/1.55 & ( R3 @ U2 @ V4 )
% 0.30/1.55 & ! [Y: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( R3 @ U2 @ Y )
% 0.30/1.55 => ( Y = V4 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_25_HCollect__iff,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A ) )
% 0.30/1.55 = ( ( P @ X )
% 0.30/1.55 & ( hF_Mirabelle_hmem @ X @ A ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_26_RepFun__iff,axiom,
% 0.30/1.55 ! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A @ F ) )
% 0.30/1.55 = ( ? [U2: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ U2 @ A )
% 0.30/1.55 & ( V4
% 0.30/1.55 = ( F @ U2 ) ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_27_Rep__hf__inverse,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_Abs_hf @ ( hF_Mirabelle_Rep_hf @ X ) )
% 0.30/1.55 = X ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_28_RepFun__cong,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ( ( A = B4 )
% 0.30/1.55 => ( ! [X3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.30/1.55 => ( ( F @ X3 )
% 0.30/1.55 = ( G @ X3 ) ) )
% 0.30/1.55 => ( ( hF_Mirabelle_RepFun @ A @ F )
% 0.30/1.55 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_29_Replace__cong,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( A = B4 )
% 0.30/1.55 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.30/1.55 => ( ( P @ X3 @ Y5 )
% 0.30/1.55 = ( Q @ X3 @ Y5 ) ) )
% 0.30/1.55 => ( ( hF_Mirabelle_Replace @ A @ P )
% 0.30/1.55 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_30_hmem__Sup__ne,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.30/1.55 => ( ( hF_Mirabelle_HUnion @ X )
% 0.30/1.55 != Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_31_Abs__hf__inverse,axiom,
% 0.30/1.55 ! [Y3: nat] :
% 0.30/1.55 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.30/1.55 => ( ( hF_Mirabelle_Rep_hf @ ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.30/1.55 = Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_32_zero__notin__hpair,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.55 ~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X @ Y3 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_33_hmem__def,axiom,
% 0.30/1.55 ( hF_Mirabelle_hmem
% 0.30/1.55 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_34_HCollect__hempty,axiom,
% 0.30/1.55 ! [P: hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 0.30/1.55 = zero_z189798548lle_hf ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_35_HUnion__hempty,axiom,
% 0.30/1.55 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.30/1.55 = zero_z189798548lle_hf ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_36_Replace__0,axiom,
% 0.30/1.55 ! [R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R3 )
% 0.30/1.55 = zero_z189798548lle_hf ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_37_RepFun__0,axiom,
% 0.30/1.55 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.30/1.55 = zero_z189798548lle_hf ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_38_RepFun__hinsert,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 0.30/1.55 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_39_hrelation__hunion,axiom,
% 0.30/1.55 ! [F: hF_Mirabelle_hf,G: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mir434065167lation @ ( sup_su638957495lle_hf @ F @ G ) )
% 0.30/1.55 = ( ( hF_Mir434065167lation @ F )
% 0.30/1.55 & ( hF_Mir434065167lation @ G ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_40_hmem__hinsert,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.30/1.55 = ( ( A2 = B )
% 0.30/1.55 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_41_singleton__eq__iff,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( ( hF_Mirabelle_hinsert @ A2 @ zero_z189798548lle_hf )
% 0.30/1.55 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.30/1.55 = ( A2 = B ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_42_hunion__iff,axiom,
% 0.30/1.55 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A2 @ B ) )
% 0.30/1.55 = ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.30/1.55 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_43_hunion__hempty__left,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A )
% 0.30/1.55 = A ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_44_mem__Collect__eq,axiom,
% 0.30/1.55 ! [A2: nat,P: nat > $o] :
% 0.30/1.55 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.30/1.55 = ( P @ A2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
% 0.30/1.55 = ( P @ A2 ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/1.55 ! [A: set_nat] :
% 0.30/1.55 ( ( collect_nat
% 0.30/1.55 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 0.30/1.55 = A ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_47_Collect__mem__eq,axiom,
% 0.30/1.55 ! [A: set_HF_Mirabelle_hf] :
% 0.30/1.55 ( ( collec2046588256lle_hf
% 0.30/1.55 @ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A ) )
% 0.30/1.55 = A ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_48_Collect__cong,axiom,
% 0.30/1.55 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ! [X3: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( P @ X3 )
% 0.30/1.55 = ( Q @ X3 ) )
% 0.30/1.55 => ( ( collec2046588256lle_hf @ P )
% 0.30/1.55 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_49_Collect__cong,axiom,
% 0.30/1.55 ! [P: nat > $o,Q: nat > $o] :
% 0.30/1.55 ( ! [X3: nat] :
% 0.30/1.55 ( ( P @ X3 )
% 0.30/1.55 = ( Q @ X3 ) )
% 0.30/1.55 => ( ( collect_nat @ P )
% 0.30/1.55 = ( collect_nat @ Q ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_50_hunion__hempty__right,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( sup_su638957495lle_hf @ A @ zero_z189798548lle_hf )
% 0.30/1.55 = A ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_51_RepFun__hunion,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A @ B4 ) @ F )
% 0.30/1.55 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_52_HUnion__hunion,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A @ B4 ) )
% 0.30/1.55 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_53_Replace__hunion,axiom,
% 0.30/1.55 ! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.30/1.55 ( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A @ B4 ) @ R3 )
% 0.30/1.55 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A @ R3 ) @ ( hF_Mirabelle_Replace @ B4 @ R3 ) ) ) ).
% 0.30/1.55
% 0.30/1.55 thf(fact_54_HUnion__hinsert,axiom,
% 0.30/1.55 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.55 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ A2 @ ( hF_Mirabelle_HUnion @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_55_hf__induct__ax,axiom,
% 0.30/1.56 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( P @ zero_z189798548lle_hf )
% 0.30/1.56 => ( ! [X3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( P @ X3 )
% 0.30/1.56 => ! [Y5: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( P @ Y5 )
% 0.30/1.56 => ( P @ ( hF_Mirabelle_hinsert @ Y5 @ X3 ) ) ) )
% 0.30/1.56 => ( P @ X ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_56_hinsert__eq__sup,axiom,
% 0.30/1.56 ( hF_Mirabelle_hinsert
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_57_hinsert__commute,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y3 @ Z ) )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ Y3 @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_58_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 0.30/1.56 = ( ( ( A2 = C )
% 0.30/1.56 & ( B = D ) )
% 0.30/1.56 | ( ( A2 = D )
% 0.30/1.56 & ( B = C ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_59_hinsert__nonempty,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hinsert @ A2 @ A )
% 0.30/1.56 != zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_60_hunion__hinsert__left,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_61_hunion__hinsert__right,axiom,
% 0.30/1.56 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_62_hf__cases,axiom,
% 0.30/1.56 ! [Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Y3 != zero_z189798548lle_hf )
% 0.30/1.56 => ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Y3
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
% 0.30/1.56 => ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_63_hf__induct,axiom,
% 0.30/1.56 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( P @ zero_z189798548lle_hf )
% 0.30/1.56 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( P @ X3 )
% 0.30/1.56 => ( ( P @ Y5 )
% 0.30/1.56 => ( ~ ( hF_Mirabelle_hmem @ X3 @ Y5 )
% 0.30/1.56 => ( P @ ( hF_Mirabelle_hinsert @ X3 @ Y5 ) ) ) ) )
% 0.30/1.56 => ( P @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_64_hpair__def,axiom,
% 0.30/1.56 ( hF_Mirabelle_hpair
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_65_hpair__def_H,axiom,
% 0.30/1.56 ( hF_Mirabelle_hpair
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_66_hinsert__iff,axiom,
% 0.30/1.56 ! [Z: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Z
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ Y3 @ X ) )
% 0.30/1.56 = ( ! [U2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.30/1.56 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.30/1.56 | ( U2 = Y3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_67_hemptyE,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf] :
% 0.30/1.56 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_68_hempty__iff,axiom,
% 0.30/1.56 ! [Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Z = zero_z189798548lle_hf )
% 0.30/1.56 = ( ! [X2: hF_Mirabelle_hf] :
% 0.30/1.56 ~ ( hF_Mirabelle_hmem @ X2 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_69_hmem__hempty,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf] :
% 0.30/1.56 ~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_70_hpair__nonzero,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hpair @ X @ Y3 )
% 0.30/1.56 != zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_71_Rep__hf,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] : ( member_nat @ ( hF_Mirabelle_Rep_hf @ X ) @ top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_72_Rep__hf__cases,axiom,
% 0.30/1.56 ! [Y3: nat] :
% 0.30/1.56 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.30/1.56 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.30/1.56 ( Y3
% 0.30/1.56 != ( hF_Mirabelle_Rep_hf @ X3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_73_Rep__hf__induct,axiom,
% 0.30/1.56 ! [Y3: nat,P: nat > $o] :
% 0.30/1.56 ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.30/1.56 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( hF_Mirabelle_Rep_hf @ X3 ) )
% 0.30/1.56 => ( P @ Y3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_74_Abs__hf__cases,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] :
% 0.30/1.56 ~ ! [Y5: nat] :
% 0.30/1.56 ( ( X
% 0.30/1.56 = ( hF_Mirabelle_Abs_hf @ Y5 ) )
% 0.30/1.56 => ~ ( member_nat @ Y5 @ top_top_set_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_75_Abs__hf__induct,axiom,
% 0.30/1.56 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ! [Y5: nat] :
% 0.30/1.56 ( ( member_nat @ Y5 @ top_top_set_nat )
% 0.30/1.56 => ( P @ ( hF_Mirabelle_Abs_hf @ Y5 ) ) )
% 0.30/1.56 => ( P @ X ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_76_Abs__hf__inject,axiom,
% 0.30/1.56 ! [X: nat,Y3: nat] :
% 0.30/1.56 ( ( member_nat @ X @ top_top_set_nat )
% 0.30/1.56 => ( ( member_nat @ Y3 @ top_top_set_nat )
% 0.30/1.56 => ( ( ( hF_Mirabelle_Abs_hf @ X )
% 0.30/1.56 = ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.30/1.56 = ( X = Y3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_77_hrelation__0,axiom,
% 0.30/1.56 hF_Mir434065167lation @ zero_z189798548lle_hf ).
% 0.30/1.56
% 0.30/1.56 thf(fact_78_sup__top__left,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ top_to489427057lle_hf @ X )
% 0.30/1.56 = top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_79_sup__top__left,axiom,
% 0.30/1.56 ! [X: nat > $o] :
% 0.30/1.56 ( ( sup_sup_nat_o @ top_top_nat_o @ X )
% 0.30/1.56 = top_top_nat_o ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_80_sup__top__left,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( sup_su1199008216e_hf_o @ top_to22270292e_hf_o @ X )
% 0.30/1.56 = top_to22270292e_hf_o ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_81_sup__top__left,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ top_top_set_nat @ X )
% 0.30/1.56 = top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_82_sup__top__right,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ top_to489427057lle_hf )
% 0.30/1.56 = top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_83_sup__top__right,axiom,
% 0.30/1.56 ! [X: nat > $o] :
% 0.30/1.56 ( ( sup_sup_nat_o @ X @ top_top_nat_o )
% 0.30/1.56 = top_top_nat_o ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_84_sup__top__right,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( sup_su1199008216e_hf_o @ X @ top_to22270292e_hf_o )
% 0.30/1.56 = top_to22270292e_hf_o ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_85_sup__top__right,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ top_top_set_nat )
% 0.30/1.56 = top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_86_HInter__iff,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( A != zero_z189798548lle_hf )
% 0.30/1.56 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A ) )
% 0.30/1.56 = ( ! [Y: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hmem @ Y @ A )
% 0.30/1.56 => ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_87_sup_Oidem,axiom,
% 0.30/1.56 ! [A2: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_88_sup_Oidem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_89_sup_Oidem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_90_sup__idem,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_91_sup__idem,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_92_sup__idem,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_93_sup_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
% 0.30/1.56 = ( sup_sup_set_nat @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_94_sup_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A2 @ B ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_95_sup_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ A2 @ B ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_96_sup__left__idem,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.30/1.56 = ( sup_sup_set_nat @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_97_sup__left__idem,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_98_sup__left__idem,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_99_sup_Oright__idem,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ B )
% 0.30/1.56 = ( sup_sup_set_nat @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_100_sup_Oright__idem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ B )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_101_sup_Oright__idem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ B )
% 0.30/1.56 = ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_102_HInter__hempty,axiom,
% 0.30/1.56 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 0.30/1.56 = zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_103_sup__left__commute,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_104_sup__left__commute,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_105_sup__left__commute,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_106_sup_Oleft__commute,axiom,
% 0.30/1.56 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) )
% 0.30/1.56 = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_107_sup_Oleft__commute,axiom,
% 0.30/1.56 ! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ B @ ( sup_su1790843629lle_hf @ A2 @ C ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_108_sup_Oleft__commute,axiom,
% 0.30/1.56 ! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A2 @ C ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_109_sup__commute,axiom,
% 0.30/1.56 ( sup_sup_set_nat
% 0.30/1.56 = ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_110_sup__commute,axiom,
% 0.30/1.56 ( sup_su1790843629lle_hf
% 0.30/1.56 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_111_sup__commute,axiom,
% 0.30/1.56 ( sup_su638957495lle_hf
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_112_sup_Ocommute,axiom,
% 0.30/1.56 ( sup_sup_set_nat
% 0.30/1.56 = ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_113_sup_Ocommute,axiom,
% 0.30/1.56 ( sup_su1790843629lle_hf
% 0.30/1.56 = ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_114_sup_Ocommute,axiom,
% 0.30/1.56 ( sup_su638957495lle_hf
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_115_sup__assoc,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_116_sup__assoc,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_117_sup__assoc,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_118_sup_Oassoc,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
% 0.30/1.56 = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_119_sup_Oassoc,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ C )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_120_sup_Oassoc,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ C )
% 0.30/1.56 = ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_121_boolean__algebra__cancel_Osup2,axiom,
% 0.30/1.56 ! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( sup_sup_set_nat @ K @ B ) )
% 0.30/1.56 => ( ( sup_sup_set_nat @ A2 @ B4 )
% 0.30/1.56 = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_122_boolean__algebra__cancel_Osup2,axiom,
% 0.30/1.56 ! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( sup_su1790843629lle_hf @ K @ B ) )
% 0.30/1.56 => ( ( sup_su1790843629lle_hf @ A2 @ B4 )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_123_boolean__algebra__cancel_Osup2,axiom,
% 0.30/1.56 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( sup_su638957495lle_hf @ K @ B ) )
% 0.30/1.56 => ( ( sup_su638957495lle_hf @ A2 @ B4 )
% 0.30/1.56 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_124_boolean__algebra__cancel_Osup1,axiom,
% 0.30/1.56 ! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( sup_sup_set_nat @ K @ A2 ) )
% 0.30/1.56 => ( ( sup_sup_set_nat @ A @ B )
% 0.30/1.56 = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_125_boolean__algebra__cancel_Osup1,axiom,
% 0.30/1.56 ! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( sup_su1790843629lle_hf @ K @ A2 ) )
% 0.30/1.56 => ( ( sup_su1790843629lle_hf @ A @ B )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_126_boolean__algebra__cancel_Osup1,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( sup_su638957495lle_hf @ K @ A2 ) )
% 0.30/1.56 => ( ( sup_su638957495lle_hf @ A @ B )
% 0.30/1.56 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_127_inf__sup__aci_I5_J,axiom,
% 0.30/1.56 ( sup_sup_set_nat
% 0.30/1.56 = ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_128_inf__sup__aci_I5_J,axiom,
% 0.30/1.56 ( sup_su1790843629lle_hf
% 0.30/1.56 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_129_inf__sup__aci_I5_J,axiom,
% 0.30/1.56 ( sup_su638957495lle_hf
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_130_inf__sup__aci_I6_J,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_131_inf__sup__aci_I6_J,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_132_inf__sup__aci_I6_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_133_inf__sup__aci_I7_J,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_134_inf__sup__aci_I7_J,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_135_inf__sup__aci_I7_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_136_inf__sup__aci_I8_J,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.30/1.56 = ( sup_sup_set_nat @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_137_inf__sup__aci_I8_J,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_138_inf__sup__aci_I8_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_139_iso__tuple__UNIV__I,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_140_iso__tuple__UNIV__I,axiom,
% 0.30/1.56 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_141_UNIV__I,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_142_UNIV__I,axiom,
% 0.30/1.56 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_143_top__apply,axiom,
% 0.30/1.56 ( top_top_nat_o
% 0.30/1.56 = ( ^ [X2: nat] : top_top_o ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_144_top__apply,axiom,
% 0.30/1.56 ( top_to22270292e_hf_o
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf] : top_top_o ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_145_HInter__hinsert,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( A != zero_z189798548lle_hf )
% 0.30/1.56 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A2 @ ( hF_Mirabelle_HInter @ A ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_146_type__definition__hf,axiom,
% 0.30/1.56 type_d1794767497hf_nat @ hF_Mirabelle_Rep_hf @ hF_Mirabelle_Abs_hf @ top_top_set_nat ).
% 0.30/1.56
% 0.30/1.56 thf(fact_147_inf__right__idem,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Y3 )
% 0.30/1.56 = ( inf_inf_set_nat @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_148_inf__right__idem,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Y3 )
% 0.30/1.56 = ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_149_inf__right__idem,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Y3 )
% 0.30/1.56 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_150_inf_Oright__idem,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
% 0.30/1.56 = ( inf_inf_set_nat @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_151_inf_Oright__idem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ B )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_152_inf_Oright__idem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ B )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_153_inf__left__idem,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
% 0.30/1.56 = ( inf_inf_set_nat @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_154_inf__left__idem,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_155_inf__left__idem,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_156_inf_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
% 0.30/1.56 = ( inf_inf_set_nat @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_157_inf_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_158_inf_Oleft__idem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ A2 @ B ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_159_inf__idem,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_160_inf__idem,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_161_inf__idem,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_162_inf_Oidem,axiom,
% 0.30/1.56 ! [A2: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_163_inf_Oidem,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_164_inf_Oidem,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ A2 @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_165_inf__top_Oright__neutral,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ A2 @ top_to489427057lle_hf )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_166_inf__top_Oright__neutral,axiom,
% 0.30/1.56 ! [A2: nat > $o] :
% 0.30/1.56 ( ( inf_inf_nat_o @ A2 @ top_top_nat_o )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_167_inf__top_Oright__neutral,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( inf_in307783154e_hf_o @ A2 @ top_to22270292e_hf_o )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_168_inf__top_Oright__neutral,axiom,
% 0.30/1.56 ! [A2: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_169_inf__top_Oneutr__eq__iff,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( top_to489427057lle_hf
% 0.30/1.56 = ( inf_in923488851lle_hf @ A2 @ B ) )
% 0.30/1.56 = ( ( A2 = top_to489427057lle_hf )
% 0.30/1.56 & ( B = top_to489427057lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_170_inf__top_Oneutr__eq__iff,axiom,
% 0.30/1.56 ! [A2: nat > $o,B: nat > $o] :
% 0.30/1.56 ( ( top_top_nat_o
% 0.30/1.56 = ( inf_inf_nat_o @ A2 @ B ) )
% 0.30/1.56 = ( ( A2 = top_top_nat_o )
% 0.30/1.56 & ( B = top_top_nat_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_171_inf__top_Oneutr__eq__iff,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( top_to22270292e_hf_o
% 0.30/1.56 = ( inf_in307783154e_hf_o @ A2 @ B ) )
% 0.30/1.56 = ( ( A2 = top_to22270292e_hf_o )
% 0.30/1.56 & ( B = top_to22270292e_hf_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_172_inf__top_Oneutr__eq__iff,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( top_top_set_nat
% 0.30/1.56 = ( inf_inf_set_nat @ A2 @ B ) )
% 0.30/1.56 = ( ( A2 = top_top_set_nat )
% 0.30/1.56 & ( B = top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_173_inf__top_Oleft__neutral,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_174_inf__top_Oleft__neutral,axiom,
% 0.30/1.56 ! [A2: nat > $o] :
% 0.30/1.56 ( ( inf_inf_nat_o @ top_top_nat_o @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_175_inf__top_Oleft__neutral,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_176_inf__top_Oleft__neutral,axiom,
% 0.30/1.56 ! [A2: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_177_inf__top_Oeq__neutr__iff,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( inf_in923488851lle_hf @ A2 @ B )
% 0.30/1.56 = top_to489427057lle_hf )
% 0.30/1.56 = ( ( A2 = top_to489427057lle_hf )
% 0.30/1.56 & ( B = top_to489427057lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_178_inf__top_Oeq__neutr__iff,axiom,
% 0.30/1.56 ! [A2: nat > $o,B: nat > $o] :
% 0.30/1.56 ( ( ( inf_inf_nat_o @ A2 @ B )
% 0.30/1.56 = top_top_nat_o )
% 0.30/1.56 = ( ( A2 = top_top_nat_o )
% 0.30/1.56 & ( B = top_top_nat_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_179_inf__top_Oeq__neutr__iff,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( ( inf_in307783154e_hf_o @ A2 @ B )
% 0.30/1.56 = top_to22270292e_hf_o )
% 0.30/1.56 = ( ( A2 = top_to22270292e_hf_o )
% 0.30/1.56 & ( B = top_to22270292e_hf_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_180_inf__top_Oeq__neutr__iff,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( ( inf_inf_set_nat @ A2 @ B )
% 0.30/1.56 = top_top_set_nat )
% 0.30/1.56 = ( ( A2 = top_top_set_nat )
% 0.30/1.56 & ( B = top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_181_top__eq__inf__iff,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( top_to489427057lle_hf
% 0.30/1.56 = ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( ( X = top_to489427057lle_hf )
% 0.30/1.56 & ( Y3 = top_to489427057lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_182_top__eq__inf__iff,axiom,
% 0.30/1.56 ! [X: nat > $o,Y3: nat > $o] :
% 0.30/1.56 ( ( top_top_nat_o
% 0.30/1.56 = ( inf_inf_nat_o @ X @ Y3 ) )
% 0.30/1.56 = ( ( X = top_top_nat_o )
% 0.30/1.56 & ( Y3 = top_top_nat_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_183_top__eq__inf__iff,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( top_to22270292e_hf_o
% 0.30/1.56 = ( inf_in307783154e_hf_o @ X @ Y3 ) )
% 0.30/1.56 = ( ( X = top_to22270292e_hf_o )
% 0.30/1.56 & ( Y3 = top_to22270292e_hf_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_184_top__eq__inf__iff,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( top_top_set_nat
% 0.30/1.56 = ( inf_inf_set_nat @ X @ Y3 ) )
% 0.30/1.56 = ( ( X = top_top_set_nat )
% 0.30/1.56 & ( Y3 = top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_185_inf__eq__top__iff,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( inf_in923488851lle_hf @ X @ Y3 )
% 0.30/1.56 = top_to489427057lle_hf )
% 0.30/1.56 = ( ( X = top_to489427057lle_hf )
% 0.30/1.56 & ( Y3 = top_to489427057lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_186_inf__eq__top__iff,axiom,
% 0.30/1.56 ! [X: nat > $o,Y3: nat > $o] :
% 0.30/1.56 ( ( ( inf_inf_nat_o @ X @ Y3 )
% 0.30/1.56 = top_top_nat_o )
% 0.30/1.56 = ( ( X = top_top_nat_o )
% 0.30/1.56 & ( Y3 = top_top_nat_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_187_inf__eq__top__iff,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( ( inf_in307783154e_hf_o @ X @ Y3 )
% 0.30/1.56 = top_to22270292e_hf_o )
% 0.30/1.56 = ( ( X = top_to22270292e_hf_o )
% 0.30/1.56 & ( Y3 = top_to22270292e_hf_o ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_188_inf__eq__top__iff,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( ( inf_inf_set_nat @ X @ Y3 )
% 0.30/1.56 = top_top_set_nat )
% 0.30/1.56 = ( ( X = top_top_set_nat )
% 0.30/1.56 & ( Y3 = top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_189_inf__top__right,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ X @ top_to489427057lle_hf )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_190_inf__top__right,axiom,
% 0.30/1.56 ! [X: nat > $o] :
% 0.30/1.56 ( ( inf_inf_nat_o @ X @ top_top_nat_o )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_191_inf__top__right,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( inf_in307783154e_hf_o @ X @ top_to22270292e_hf_o )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_192_inf__top__right,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ X @ top_top_set_nat )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_193_inf__top__left,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_194_inf__top__left,axiom,
% 0.30/1.56 ! [X: nat > $o] :
% 0.30/1.56 ( ( inf_inf_nat_o @ top_top_nat_o @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_195_inf__top__left,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf > $o] :
% 0.30/1.56 ( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_196_inf__top__left,axiom,
% 0.30/1.56 ! [X: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ top_top_set_nat @ X )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_197_inf__sup__absorb,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_198_inf__sup__absorb,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_199_inf__sup__absorb,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_200_sup__inf__absorb,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_201_sup__inf__absorb,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_202_sup__inf__absorb,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.30/1.56 = X ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_203_hinter__iff,axiom,
% 0.30/1.56 ! [U4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.30/1.56 & ( hF_Mirabelle_hmem @ U4 @ Y3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_204_hinter__hempty__left,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A )
% 0.30/1.56 = zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_205_hinter__hempty__right,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ A @ zero_z189798548lle_hf )
% 0.30/1.56 = zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_206_Abs__hf__0,axiom,
% 0.30/1.56 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.30/1.56 = zero_z189798548lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_207_inf__left__commute,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_208_inf__left__commute,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ Y3 @ ( inf_in923488851lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_209_inf__left__commute,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_210_inf_Oleft__commute,axiom,
% 0.30/1.56 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
% 0.30/1.56 = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_211_inf_Oleft__commute,axiom,
% 0.30/1.56 ! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A2 @ C ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_212_inf_Oleft__commute,axiom,
% 0.30/1.56 ! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A2 @ C ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_213_inf__commute,axiom,
% 0.30/1.56 ( inf_inf_set_nat
% 0.30/1.56 = ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_214_inf__commute,axiom,
% 0.30/1.56 ( inf_in923488851lle_hf
% 0.30/1.56 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_215_inf__commute,axiom,
% 0.30/1.56 ( inf_in956532509lle_hf
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_216_inf_Ocommute,axiom,
% 0.30/1.56 ( inf_inf_set_nat
% 0.30/1.56 = ( ^ [A4: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_217_inf_Ocommute,axiom,
% 0.30/1.56 ( inf_in923488851lle_hf
% 0.30/1.56 = ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_218_inf_Ocommute,axiom,
% 0.30/1.56 ( inf_in956532509lle_hf
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_219_inf__assoc,axiom,
% 0.30/1.56 ! [X: set_nat,Y3: set_nat,Z: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_220_inf__assoc,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_221_inf__assoc,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_222_inf_Oassoc,axiom,
% 0.30/1.56 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
% 0.30/1.56 = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_223_inf_Oassoc,axiom,
% 0.30/1.56 ! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ C )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_224_inf_Oassoc,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ C )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_225_boolean__algebra__cancel_Oinf2,axiom,
% 0.30/1.56 ! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( inf_inf_set_nat @ K @ B ) )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A2 @ B4 )
% 0.30/1.56 = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_226_boolean__algebra__cancel_Oinf2,axiom,
% 0.30/1.56 ! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( inf_in923488851lle_hf @ K @ B ) )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A2 @ B4 )
% 0.30/1.56 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_227_boolean__algebra__cancel_Oinf2,axiom,
% 0.30/1.56 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( inf_in956532509lle_hf @ K @ B ) )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ A2 @ B4 )
% 0.30/1.56 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_228_boolean__algebra__cancel_Oinf1,axiom,
% 0.30/1.56 ! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( inf_inf_set_nat @ K @ A2 ) )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A @ B )
% 0.30/1.56 = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_229_boolean__algebra__cancel_Oinf1,axiom,
% 0.30/1.56 ! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( inf_in923488851lle_hf @ K @ A2 ) )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A @ B )
% 0.30/1.56 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_230_boolean__algebra__cancel_Oinf1,axiom,
% 0.30/1.56 ! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( inf_in956532509lle_hf @ K @ A2 ) )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ A @ B )
% 0.30/1.56 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_231_inf__sup__aci_I1_J,axiom,
% 0.30/1.56 ( inf_inf_set_nat
% 0.30/1.56 = ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_232_inf__sup__aci_I1_J,axiom,
% 0.30/1.56 ( inf_in923488851lle_hf
% 0.30/1.56 = ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_233_inf__sup__aci_I1_J,axiom,
% 0.30/1.56 ( inf_in956532509lle_hf
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_234_inf__sup__aci_I2_J,axiom,
% 0.30/1.56 ! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_235_inf__sup__aci_I2_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
% 0.30/1.56 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_236_inf__sup__aci_I3_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_237_inf__sup__aci_I4_J,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_238_sup__inf__distrib2,axiom,
% 0.30/1.56 ! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ Z ) @ X )
% 0.30/1.56 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_239_sup__inf__distrib1,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_240_inf__sup__distrib2,axiom,
% 0.30/1.56 ! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ Z ) @ X )
% 0.30/1.56 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_241_inf__sup__distrib1,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_242_distrib__imp2,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su638957495lle_hf @ X3 @ ( inf_in956532509lle_hf @ Y5 @ Z3 ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X3 @ Y5 ) @ ( sup_su638957495lle_hf @ X3 @ Z3 ) ) )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_243_distrib__imp1,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in956532509lle_hf @ X3 @ ( sup_su638957495lle_hf @ Y5 @ Z3 ) )
% 0.30/1.56 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X3 @ Y5 ) @ ( inf_in956532509lle_hf @ X3 @ Z3 ) ) )
% 0.30/1.56 => ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_244_foundation,axiom,
% 0.30/1.56 ! [Z: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Z != zero_z189798548lle_hf )
% 0.30/1.56 => ? [W: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.30/1.56 & ( ( inf_in956532509lle_hf @ W @ Z )
% 0.30/1.56 = zero_z189798548lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_245_hinter__hinsert__right,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A ) ) ) )
% 0.30/1.56 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
% 0.30/1.56 = ( inf_in956532509lle_hf @ B4 @ A ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_246_hinter__hinsert__left,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.30/1.56 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A @ B4 ) ) ) )
% 0.30/1.56 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.30/1.56 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
% 0.30/1.56 = ( inf_in956532509lle_hf @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_247_zero__reorient,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( zero_z189798548lle_hf = X )
% 0.30/1.56 = ( X = zero_z189798548lle_hf ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_248_zero__reorient,axiom,
% 0.30/1.56 ! [X: nat] :
% 0.30/1.56 ( ( zero_zero_nat = X )
% 0.30/1.56 = ( X = zero_zero_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_249_UNIV__eq__I,axiom,
% 0.30/1.56 ! [A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ! [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A )
% 0.30/1.56 => ( top_to489427057lle_hf = A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_250_UNIV__eq__I,axiom,
% 0.30/1.56 ! [A: set_nat] :
% 0.30/1.56 ( ! [X3: nat] : ( member_nat @ X3 @ A )
% 0.30/1.56 => ( top_top_set_nat = A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_251_UNIV__witness,axiom,
% 0.30/1.56 ? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_252_UNIV__witness,axiom,
% 0.30/1.56 ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_253_Un__UNIV__right,axiom,
% 0.30/1.56 ! [A: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ A @ top_top_set_nat )
% 0.30/1.56 = top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_254_Un__UNIV__left,axiom,
% 0.30/1.56 ! [B4: set_nat] :
% 0.30/1.56 ( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
% 0.30/1.56 = top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_255_top__set__def,axiom,
% 0.30/1.56 ( top_top_set_nat
% 0.30/1.56 = ( collect_nat @ top_top_nat_o ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_256_type__copy__ex__RepI,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,F2: nat > $o] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
% 0.30/1.56 => ( ( ? [X4: nat] : ( F2 @ X4 ) )
% 0.30/1.56 = ( ? [B3: hF_Mirabelle_hf] : ( F2 @ ( Rep @ B3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_257_type__copy__obj__one__point__absE,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,S: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
% 0.30/1.56 => ~ ! [X3: nat] :
% 0.30/1.56 ( S
% 0.30/1.56 != ( Abs @ X3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_258_HF__hfset,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_259_type__definition_ORep,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( member_nat @ ( Rep @ X ) @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_260_type__definition__def,axiom,
% 0.30/1.56 ( type_d1794767497hf_nat
% 0.30/1.56 = ( ^ [Rep2: hF_Mirabelle_hf > nat,Abs2: nat > hF_Mirabelle_hf,A6: set_nat] :
% 0.30/1.56 ( ! [X2: hF_Mirabelle_hf] : ( member_nat @ ( Rep2 @ X2 ) @ A6 )
% 0.30/1.56 & ! [X2: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.30/1.56 = X2 )
% 0.30/1.56 & ! [Y: nat] :
% 0.30/1.56 ( ( member_nat @ Y @ A6 )
% 0.30/1.56 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.30/1.56 = Y ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_261_Int__UNIV,axiom,
% 0.30/1.56 ! [A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.30/1.56 = top_top_set_nat )
% 0.30/1.56 = ( ( A = top_top_set_nat )
% 0.30/1.56 & ( B4 = top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_262_UnCI,axiom,
% 0.30/1.56 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.30/1.56 ( ( ~ ( member_nat @ C @ B4 )
% 0.30/1.56 => ( member_nat @ C @ A ) )
% 0.30/1.56 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_263_UnCI,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( ~ ( member1367349282lle_hf @ C @ B4 )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ A ) )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_264_Un__iff,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.30/1.56 = ( ( member_nat @ C @ A )
% 0.30/1.56 | ( member_nat @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_265_Un__iff,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
% 0.30/1.56 = ( ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 | ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_266_UnE,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.30/1.56 => ( ~ ( member_nat @ C @ A )
% 0.30/1.56 => ( member_nat @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_267_UnE,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
% 0.30/1.56 => ( ~ ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_268_UnI1,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ A )
% 0.30/1.56 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_269_UnI1,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_270_UnI2,axiom,
% 0.30/1.56 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ B4 )
% 0.30/1.56 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_271_UnI2,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ B4 )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_272_Int__UNIV__left,axiom,
% 0.30/1.56 ! [B4: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
% 0.30/1.56 = B4 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_273_Int__UNIV__right,axiom,
% 0.30/1.56 ! [A: set_nat] :
% 0.30/1.56 ( ( inf_inf_set_nat @ A @ top_top_set_nat )
% 0.30/1.56 = A ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_274_type__definition_ORep__inverse,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( Abs @ ( Rep @ X ) )
% 0.30/1.56 = X ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_275_type__definition_OAbs__inverse,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( member_nat @ Y3 @ A )
% 0.30/1.56 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.30/1.56 = Y3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_276_type__definition_ORep__inject,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( ( Rep @ X )
% 0.30/1.56 = ( Rep @ Y3 ) )
% 0.30/1.56 = ( X = Y3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_277_type__definition_ORep__induct,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat,P: nat > $o] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( member_nat @ Y3 @ A )
% 0.30/1.56 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( Rep @ X3 ) )
% 0.30/1.56 => ( P @ Y3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_278_type__definition_OAbs__inject,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: nat,Y3: nat] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( member_nat @ X @ A )
% 0.30/1.56 => ( ( member_nat @ Y3 @ A )
% 0.30/1.56 => ( ( ( Abs @ X )
% 0.30/1.56 = ( Abs @ Y3 ) )
% 0.30/1.56 = ( X = Y3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_279_type__definition_OAbs__induct,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ! [Y5: nat] :
% 0.30/1.56 ( ( member_nat @ Y5 @ A )
% 0.30/1.56 => ( P @ ( Abs @ Y5 ) ) )
% 0.30/1.56 => ( P @ X ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_280_type__definition_ORep__cases,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ( ( member_nat @ Y3 @ A )
% 0.30/1.56 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.30/1.56 ( Y3
% 0.30/1.56 != ( Rep @ X3 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_281_type__definition_OAbs__cases,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
% 0.30/1.56 => ~ ! [Y5: nat] :
% 0.30/1.56 ( ( X
% 0.30/1.56 = ( Abs @ Y5 ) )
% 0.30/1.56 => ~ ( member_nat @ Y5 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_282_type__definition_Ointro,axiom,
% 0.30/1.56 ! [Rep: hF_Mirabelle_hf > nat,A: set_nat,Abs: nat > hF_Mirabelle_hf] :
% 0.30/1.56 ( ! [X3: hF_Mirabelle_hf] : ( member_nat @ ( Rep @ X3 ) @ A )
% 0.30/1.56 => ( ! [X3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( Abs @ ( Rep @ X3 ) )
% 0.30/1.56 = X3 )
% 0.30/1.56 => ( ! [Y5: nat] :
% 0.30/1.56 ( ( member_nat @ Y5 @ A )
% 0.30/1.56 => ( ( Rep @ ( Abs @ Y5 ) )
% 0.30/1.56 = Y5 ) )
% 0.30/1.56 => ( type_d1794767497hf_nat @ Rep @ Abs @ A ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_283_top__empty__eq,axiom,
% 0.30/1.56 ( top_to22270292e_hf_o
% 0.30/1.56 = ( ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ top_to489427057lle_hf ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_284_top__empty__eq,axiom,
% 0.30/1.56 ( top_top_nat_o
% 0.30/1.56 = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_285_hcard__0,axiom,
% 0.30/1.56 ( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
% 0.30/1.56 = zero_zero_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_286_hinsert__def,axiom,
% 0.30/1.56 ( hF_Mirabelle_hinsert
% 0.30/1.56 = ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_287_insert__absorb2,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ X @ A ) )
% 0.30/1.56 = ( insert9649339lle_hf @ X @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_288_insert__iff,axiom,
% 0.30/1.56 ! [A2: nat,B: nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.30/1.56 = ( ( A2 = B )
% 0.30/1.56 | ( member_nat @ A2 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_289_insert__iff,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
% 0.30/1.56 = ( ( A2 = B )
% 0.30/1.56 | ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_290_insertCI,axiom,
% 0.30/1.56 ! [A2: nat,B4: set_nat,B: nat] :
% 0.30/1.56 ( ( ~ ( member_nat @ A2 @ B4 )
% 0.30/1.56 => ( A2 = B ) )
% 0.30/1.56 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_291_insertCI,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
% 0.30/1.56 => ( A2 = B ) )
% 0.30/1.56 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_292_IntI,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ A )
% 0.30/1.56 => ( ( member_nat @ C @ B4 )
% 0.30/1.56 => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_293_IntI,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 => ( ( member1367349282lle_hf @ C @ B4 )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_294_Int__iff,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.30/1.56 = ( ( member_nat @ C @ A )
% 0.30/1.56 & ( member_nat @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_295_Int__iff,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.30/1.56 = ( ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 & ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_296_Int__insert__left__if0,axiom,
% 0.30/1.56 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.30/1.56 ( ~ ( member_nat @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_297_Int__insert__left__if0,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ~ ( member1367349282lle_hf @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_298_Int__insert__left__if1,axiom,
% 0.30/1.56 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_299_Int__insert__left__if1,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_300_insert__inter__insert,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ A ) @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_301_Int__insert__right__if0,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ~ ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.30/1.56 = ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_302_Int__insert__right__if0,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_303_Int__insert__right__if1,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.30/1.56 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_304_Int__insert__right__if1,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_305_Un__insert__left,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B4 @ C2 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_306_Un__insert__right,axiom,
% 0.30/1.56 ! [A: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( sup_su1790843629lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_307_insert__UNIV,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( insert9649339lle_hf @ X @ top_to489427057lle_hf )
% 0.30/1.56 = top_to489427057lle_hf ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_308_insert__UNIV,axiom,
% 0.30/1.56 ! [X: nat] :
% 0.30/1.56 ( ( insert_nat @ X @ top_top_set_nat )
% 0.30/1.56 = top_top_set_nat ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_309_IntE,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.30/1.56 => ~ ( ( member_nat @ C @ A )
% 0.30/1.56 => ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_310_IntE,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.30/1.56 => ~ ( ( member1367349282lle_hf @ C @ A )
% 0.30/1.56 => ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_311_IntD1,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.30/1.56 => ( member_nat @ C @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_312_IntD1,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_313_IntD2,axiom,
% 0.30/1.56 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.30/1.56 => ( member_nat @ C @ B4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_314_IntD2,axiom,
% 0.30/1.56 ! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
% 0.30/1.56 => ( member1367349282lle_hf @ C @ B4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_315_Int__insert__left,axiom,
% 0.30/1.56 ! [A2: nat,C2: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( ( member_nat @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
% 0.30/1.56 & ( ~ ( member_nat @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_316_Int__insert__left,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( member1367349282lle_hf @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) )
% 0.30/1.56 & ( ~ ( member1367349282lle_hf @ A2 @ C2 )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
% 0.30/1.56 = ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_317_Int__insert__right,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ( ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.30/1.56 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
% 0.30/1.56 & ( ~ ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.30/1.56 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_318_Int__insert__right,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) )
% 0.30/1.56 & ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
% 0.30/1.56 = ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_319_mk__disjoint__insert,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ A )
% 0.30/1.56 => ? [B6: set_nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert_nat @ A2 @ B6 ) )
% 0.30/1.56 & ~ ( member_nat @ A2 @ B6 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_320_mk__disjoint__insert,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ? [B6: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ B6 ) )
% 0.30/1.56 & ~ ( member1367349282lle_hf @ A2 @ B6 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_321_insert__commute,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ Y3 @ A ) )
% 0.30/1.56 = ( insert9649339lle_hf @ Y3 @ ( insert9649339lle_hf @ X @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_322_insert__eq__iff,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
% 0.30/1.56 ( ~ ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ~ ( member_nat @ B @ B4 )
% 0.30/1.56 => ( ( ( insert_nat @ A2 @ A )
% 0.30/1.56 = ( insert_nat @ B @ B4 ) )
% 0.30/1.56 = ( ( ( A2 = B )
% 0.30/1.56 => ( A = B4 ) )
% 0.30/1.56 & ( ( A2 != B )
% 0.30/1.56 => ? [C3: set_nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert_nat @ B @ C3 ) )
% 0.30/1.56 & ~ ( member_nat @ B @ C3 )
% 0.30/1.56 & ( B4
% 0.30/1.56 = ( insert_nat @ A2 @ C3 ) )
% 0.30/1.56 & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_323_insert__eq__iff,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ~ ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ~ ( member1367349282lle_hf @ B @ B4 )
% 0.30/1.56 => ( ( ( insert9649339lle_hf @ A2 @ A )
% 0.30/1.56 = ( insert9649339lle_hf @ B @ B4 ) )
% 0.30/1.56 = ( ( ( A2 = B )
% 0.30/1.56 => ( A = B4 ) )
% 0.30/1.56 & ( ( A2 != B )
% 0.30/1.56 => ? [C3: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert9649339lle_hf @ B @ C3 ) )
% 0.30/1.56 & ~ ( member1367349282lle_hf @ B @ C3 )
% 0.30/1.56 & ( B4
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ C3 ) )
% 0.30/1.56 & ~ ( member1367349282lle_hf @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_324_insert__absorb,axiom,
% 0.30/1.56 ! [A2: nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ A )
% 0.30/1.56 => ( ( insert_nat @ A2 @ A )
% 0.30/1.56 = A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_325_insert__absorb,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ A )
% 0.30/1.56 => ( ( insert9649339lle_hf @ A2 @ A )
% 0.30/1.56 = A ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_326_insert__ident,axiom,
% 0.30/1.56 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.30/1.56 ( ~ ( member_nat @ X @ A )
% 0.30/1.56 => ( ~ ( member_nat @ X @ B4 )
% 0.30/1.56 => ( ( ( insert_nat @ X @ A )
% 0.30/1.56 = ( insert_nat @ X @ B4 ) )
% 0.30/1.56 = ( A = B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_327_insert__ident,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ~ ( member1367349282lle_hf @ X @ A )
% 0.30/1.56 => ( ~ ( member1367349282lle_hf @ X @ B4 )
% 0.30/1.56 => ( ( ( insert9649339lle_hf @ X @ A )
% 0.30/1.56 = ( insert9649339lle_hf @ X @ B4 ) )
% 0.30/1.56 = ( A = B4 ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_328_Set_Oset__insert,axiom,
% 0.30/1.56 ! [X: nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ X @ A )
% 0.30/1.56 => ~ ! [B6: set_nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert_nat @ X @ B6 ) )
% 0.30/1.56 => ( member_nat @ X @ B6 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_329_Set_Oset__insert,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ X @ A )
% 0.30/1.56 => ~ ! [B6: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( insert9649339lle_hf @ X @ B6 ) )
% 0.30/1.56 => ( member1367349282lle_hf @ X @ B6 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_330_insertI2,axiom,
% 0.30/1.56 ! [A2: nat,B4: set_nat,B: nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ B4 )
% 0.30/1.56 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_331_insertI2,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ B4 )
% 0.30/1.56 => ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_332_insertI1,axiom,
% 0.30/1.56 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_333_insertI1,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_334_insertE,axiom,
% 0.30/1.56 ! [A2: nat,B: nat,A: set_nat] :
% 0.30/1.56 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.30/1.56 => ( ( A2 != B )
% 0.30/1.56 => ( member_nat @ A2 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_335_insertE,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.30/1.56 ( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
% 0.30/1.56 => ( ( A2 != B )
% 0.30/1.56 => ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_336_hfset__hinsert,axiom,
% 0.30/1.56 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ A2 @ B ) )
% 0.30/1.56 = ( insert9649339lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_337_hcard__union__inter,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( plus_plus_nat @ ( hF_Mirabelle_hcard @ ( sup_su638957495lle_hf @ X @ Y3 ) ) @ ( hF_Mirabelle_hcard @ ( inf_in956532509lle_hf @ X @ Y3 ) ) )
% 0.30/1.56 = ( plus_plus_nat @ ( hF_Mirabelle_hcard @ X ) @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_338_hcard__hinsert__if,axiom,
% 0.30/1.56 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.30/1.56 ( ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.30/1.56 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.30/1.56 = ( hF_Mirabelle_hcard @ Y3 ) ) )
% 0.30/1.56 & ( ~ ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.30/1.56 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.30/1.56 = ( suc @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_339_add__left__cancel,axiom,
% 0.30/1.56 ! [A2: nat,B: nat,C: nat] :
% 0.30/1.56 ( ( ( plus_plus_nat @ A2 @ B )
% 0.30/1.56 = ( plus_plus_nat @ A2 @ C ) )
% 0.30/1.56 = ( B = C ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_340_add__right__cancel,axiom,
% 0.30/1.56 ! [B: nat,A2: nat,C: nat] :
% 0.30/1.56 ( ( ( plus_plus_nat @ B @ A2 )
% 0.30/1.56 = ( plus_plus_nat @ C @ A2 ) )
% 0.30/1.56 = ( B = C ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_341_zero__eq__add__iff__both__eq__0,axiom,
% 0.30/1.56 ! [X: nat,Y3: nat] :
% 0.30/1.56 ( ( zero_zero_nat
% 0.30/1.56 = ( plus_plus_nat @ X @ Y3 ) )
% 0.30/1.56 = ( ( X = zero_zero_nat )
% 0.30/1.56 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_342_add__eq__0__iff__both__eq__0,axiom,
% 0.30/1.56 ! [X: nat,Y3: nat] :
% 0.30/1.56 ( ( ( plus_plus_nat @ X @ Y3 )
% 0.30/1.56 = zero_zero_nat )
% 0.30/1.56 = ( ( X = zero_zero_nat )
% 0.30/1.56 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_343_add__cancel__right__right,axiom,
% 0.30/1.56 ! [A2: nat,B: nat] :
% 0.30/1.56 ( ( A2
% 0.30/1.56 = ( plus_plus_nat @ A2 @ B ) )
% 0.30/1.56 = ( B = zero_zero_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_344_add__cancel__right__left,axiom,
% 0.30/1.56 ! [A2: nat,B: nat] :
% 0.30/1.56 ( ( A2
% 0.30/1.56 = ( plus_plus_nat @ B @ A2 ) )
% 0.30/1.56 = ( B = zero_zero_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_345_add__cancel__left__right,axiom,
% 0.30/1.56 ! [A2: nat,B: nat] :
% 0.30/1.56 ( ( ( plus_plus_nat @ A2 @ B )
% 0.30/1.56 = A2 )
% 0.30/1.56 = ( B = zero_zero_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_346_add__cancel__left__left,axiom,
% 0.30/1.56 ! [B: nat,A2: nat] :
% 0.30/1.56 ( ( ( plus_plus_nat @ B @ A2 )
% 0.30/1.56 = A2 )
% 0.30/1.56 = ( B = zero_zero_nat ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_347_add_Oright__neutral,axiom,
% 0.30/1.56 ! [A2: nat] :
% 0.30/1.56 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_348_add_Oleft__neutral,axiom,
% 0.30/1.56 ! [A2: nat] :
% 0.30/1.56 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.30/1.56 = A2 ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/1.56 ! [A2: nat,B: nat,C: nat] :
% 0.30/1.56 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.30/1.56 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_350_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/1.56 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.56 ( ( ( I = J )
% 0.30/1.56 & ( K = L ) )
% 0.30/1.56 => ( ( plus_plus_nat @ I @ K )
% 0.30/1.56 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_351_group__cancel_Oadd1,axiom,
% 0.30/1.56 ! [A: nat,K: nat,A2: nat,B: nat] :
% 0.30/1.56 ( ( A
% 0.30/1.56 = ( plus_plus_nat @ K @ A2 ) )
% 0.30/1.56 => ( ( plus_plus_nat @ A @ B )
% 0.30/1.56 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_352_group__cancel_Oadd2,axiom,
% 0.30/1.56 ! [B4: nat,K: nat,B: nat,A2: nat] :
% 0.30/1.56 ( ( B4
% 0.30/1.56 = ( plus_plus_nat @ K @ B ) )
% 0.30/1.56 => ( ( plus_plus_nat @ A2 @ B4 )
% 0.30/1.56 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(fact_353_add_Oassoc,axiom,
% 0.30/1.56 ! [A2: nat,B: nat,C: nat] :
% 0.30/1.56 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.30/1.56 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.56
% 0.30/1.56 thf(conj_0,hypothesis,
% 0.30/1.56 hF_Mir199975595nction @ r ).
% 0.30/1.56
% 0.30/1.56 thf(conj_1,conjecture,
% 0.30/1.56 hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ).
% 0.30/1.56 ------------------------
% 0.30/1.59 % (3891)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_3790 for (2999ds/27Mi)
% 0.30/1.59 % (3890)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_3790 for (2999ds/4Mi)
% 0.30/1.59 % (3890)Instruction limit reached!
% 0.30/1.59 % (3890)------------------------------
% 0.30/1.59 % (3890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.59 % (3890)Termination reason: Unknown
% 0.30/1.59 % (3890)Termination phase: shuffling
% 0.30/1.59
% 0.30/1.59 % (3890)Memory used [KB]: 1407
% 0.30/1.59 % (3890)Time elapsed: 0.003 s
% 0.30/1.59 % (3890)Instructions burned: 6 (million)
% 0.30/1.59 % (3890)------------------------------
% 0.30/1.59 % (3890)------------------------------
% 0.30/1.59 % (3894)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_3790 for (2999ds/275Mi)
% 0.30/1.60 % (3891)Instruction limit reached!
% 0.30/1.60 % (3891)------------------------------
% 0.30/1.60 % (3891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.60 % (3889)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/183Mi)
% 0.30/1.60 % (3891)Termination reason: Unknown
% 0.30/1.60 % (3891)Termination phase: Property scanning
% 0.30/1.60
% 0.30/1.60 % (3891)Memory used [KB]: 1535
% 0.30/1.60 % (3891)Time elapsed: 0.009 s
% 0.30/1.60 % (3891)Instructions burned: 27 (million)
% 0.30/1.60 % (3891)------------------------------
% 0.30/1.60 % (3891)------------------------------
% 2.70/1.60 % (3892)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/2Mi)
% 2.70/1.60 % (3895)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/18Mi)
% 2.70/1.60 % (3893)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_3790 for (2999ds/2Mi)
% 2.70/1.60 % (3892)Instruction limit reached!
% 2.70/1.60 % (3892)------------------------------
% 2.70/1.60 % (3892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (3892)Termination reason: Unknown
% 2.70/1.60 % (3892)Termination phase: shuffling
% 2.70/1.60 % (3893)Instruction limit reached!
% 2.70/1.60 % (3893)------------------------------
% 2.70/1.60 % (3893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (3893)Termination reason: Unknown
% 2.70/1.60 % (3893)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (3893)Memory used [KB]: 1407
% 2.70/1.60 % (3893)Time elapsed: 0.004 s
% 2.70/1.60 % (3893)Instructions burned: 3 (million)
% 2.70/1.60 % (3893)------------------------------
% 2.70/1.60 % (3893)------------------------------
% 2.70/1.60
% 2.70/1.60 % (3892)Memory used [KB]: 1407
% 2.70/1.60 % (3892)Time elapsed: 0.004 s
% 2.70/1.60 % (3892)Instructions burned: 3 (million)
% 2.70/1.60 % (3892)------------------------------
% 2.70/1.60 % (3892)------------------------------
% 2.70/1.60 % (3896)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/3Mi)
% 2.70/1.60 % (3896)Instruction limit reached!
% 2.70/1.60 % (3896)------------------------------
% 2.70/1.60 % (3896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (3896)Termination reason: Unknown
% 2.70/1.60 % (3896)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (3896)Memory used [KB]: 1407
% 2.70/1.60 % (3896)Time elapsed: 0.002 s
% 2.70/1.60 % (3896)Instructions burned: 4 (million)
% 2.70/1.60 % (3896)------------------------------
% 2.70/1.60 % (3896)------------------------------
% 2.70/1.61 % (3897)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_3790 for (2999ds/37Mi)
% 2.70/1.61 % (3894)Refutation not found, incomplete strategy
% 2.70/1.61 % (3894)------------------------------
% 2.70/1.61 % (3894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (3894)Termination reason: Refutation not found, incomplete strategy
% 2.70/1.61
% 2.70/1.61
% 2.70/1.61 % (3894)Memory used [KB]: 6268
% 2.70/1.61 % (3894)Time elapsed: 0.014 s
% 2.70/1.61 % (3894)Instructions burned: 42 (million)
% 2.70/1.61 % (3894)------------------------------
% 2.70/1.61 % (3894)------------------------------
% 2.70/1.61 % (3895)Instruction limit reached!
% 2.70/1.61 % (3895)------------------------------
% 2.70/1.61 % (3895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.61 % (3895)Termination reason: Unknown
% 2.70/1.61 % (3895)Termination phase: Property scanning
% 2.70/1.61
% 2.70/1.61 % (3895)Memory used [KB]: 1535
% 2.70/1.61 % (3895)Time elapsed: 0.013 s
% 2.70/1.61 % (3895)Instructions burned: 19 (million)
% 2.70/1.61 % (3895)------------------------------
% 2.70/1.61 % (3895)------------------------------
% 2.70/1.61 % (3900)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_3790 for (2999ds/1041Mi)
% 2.70/1.62 % (3901)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/7Mi)
% 2.70/1.62 % (3897)Instruction limit reached!
% 2.70/1.62 % (3897)------------------------------
% 2.70/1.62 % (3897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (3897)Termination reason: Unknown
% 2.70/1.62 % (3897)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (3897)Memory used [KB]: 2046
% 2.70/1.62 % (3897)Time elapsed: 0.011 s
% 2.70/1.62 % (3897)Instructions burned: 38 (million)
% 2.70/1.62 % (3897)------------------------------
% 2.70/1.62 % (3897)------------------------------
% 2.70/1.62 % (3898)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_3790 for (2999ds/15Mi)
% 2.70/1.62 % (3899)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_3790 for (2999ds/3Mi)
% 2.70/1.62 % (3899)Instruction limit reached!
% 2.70/1.62 % (3899)------------------------------
% 2.70/1.62 % (3899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (3899)Termination reason: Unknown
% 2.70/1.62 % (3899)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (3899)Memory used [KB]: 1407
% 2.70/1.62 % (3899)Time elapsed: 0.004 s
% 2.70/1.62 % (3899)Instructions burned: 3 (million)
% 2.70/1.62 % (3899)------------------------------
% 2.70/1.62 % (3899)------------------------------
% 2.70/1.62 % (3901)Instruction limit reached!
% 2.70/1.62 % (3901)------------------------------
% 2.70/1.62 % (3901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (3901)Termination reason: Unknown
% 2.70/1.62 % (3901)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (3901)Memory used [KB]: 1535
% 2.70/1.62 % (3901)Time elapsed: 0.004 s
% 2.70/1.62 % (3901)Instructions burned: 10 (million)
% 2.70/1.62 % (3901)------------------------------
% 2.70/1.62 % (3901)------------------------------
% 2.70/1.62 % (3898)Instruction limit reached!
% 2.70/1.62 % (3898)------------------------------
% 2.70/1.62 % (3898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (3903)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_3790 for (2999ds/3Mi)
% 2.70/1.62 % (3898)Termination reason: Unknown
% 2.70/1.62 % (3898)Termination phase: Property scanning
% 2.70/1.62
% 2.70/1.62 % (3898)Memory used [KB]: 1535
% 2.70/1.62 % (3898)Time elapsed: 0.011 s
% 2.70/1.62 % (3898)Instructions burned: 15 (million)
% 2.70/1.63 % (3898)------------------------------
% 2.70/1.63 % (3898)------------------------------
% 2.70/1.63 % (3902)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/16Mi)
% 2.70/1.63 % (3903)Instruction limit reached!
% 2.70/1.63 % (3903)------------------------------
% 2.70/1.63 % (3903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (3903)Termination reason: Unknown
% 2.70/1.63 % (3903)Termination phase: shuffling
% 2.70/1.63
% 2.70/1.63 % (3903)Memory used [KB]: 1407
% 2.70/1.63 % (3903)Time elapsed: 0.002 s
% 2.70/1.63 % (3903)Instructions burned: 4 (million)
% 2.70/1.63 % (3903)------------------------------
% 2.70/1.63 % (3903)------------------------------
% 2.70/1.63 % (3905)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/7Mi)
% 2.70/1.63 % (3905)Instruction limit reached!
% 2.70/1.63 % (3905)------------------------------
% 2.70/1.63 % (3905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (3905)Termination reason: Unknown
% 2.70/1.63 % (3905)Termination phase: shuffling
% 2.70/1.63
% 2.70/1.63 % (3905)Memory used [KB]: 1407
% 2.70/1.63 % (3905)Time elapsed: 0.003 s
% 2.70/1.63 % (3905)Instructions burned: 7 (million)
% 2.70/1.63 % (3905)------------------------------
% 2.70/1.63 % (3905)------------------------------
% 2.70/1.63 % (3902)Instruction limit reached!
% 2.70/1.63 % (3902)------------------------------
% 2.70/1.63 % (3902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (3902)Termination reason: Unknown
% 2.70/1.63 % (3902)Termination phase: Property scanning
% 2.70/1.63
% 2.70/1.63 % (3902)Memory used [KB]: 1535
% 2.70/1.63 % (3902)Time elapsed: 0.010 s
% 2.70/1.63 % (3902)Instructions burned: 16 (million)
% 2.70/1.63 % (3902)------------------------------
% 2.70/1.63 % (3902)------------------------------
% 2.70/1.63 % (3904)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_3790 for (2999ds/3Mi)
% 2.70/1.64 % (3907)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/4Mi)
% 2.70/1.64 % (3904)Instruction limit reached!
% 2.70/1.64 % (3904)------------------------------
% 2.70/1.64 % (3904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (3904)Termination reason: Unknown
% 2.70/1.64 % (3904)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (3904)Memory used [KB]: 1407
% 2.70/1.64 % (3904)Time elapsed: 0.003 s
% 2.70/1.64 % (3904)Instructions burned: 3 (million)
% 2.70/1.64 % (3904)------------------------------
% 2.70/1.64 % (3904)------------------------------
% 2.70/1.64 % (3907)Instruction limit reached!
% 2.70/1.64 % (3907)------------------------------
% 2.70/1.64 % (3907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (3907)Termination reason: Unknown
% 2.70/1.64 % (3907)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (3907)Memory used [KB]: 1407
% 2.70/1.64 % (3907)Time elapsed: 0.002 s
% 2.70/1.64 % (3907)Instructions burned: 4 (million)
% 2.70/1.64 % (3907)------------------------------
% 2.70/1.64 % (3907)------------------------------
% 2.70/1.64 % (3908)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/18Mi)
% 2.70/1.64 % (3906)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/3Mi)
% 2.70/1.64 % (3906)Instruction limit reached!
% 2.70/1.64 % (3906)------------------------------
% 2.70/1.64 % (3906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (3906)Termination reason: Unknown
% 2.70/1.64 % (3906)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (3906)Memory used [KB]: 1407
% 2.70/1.64 % (3906)Time elapsed: 0.004 s
% 2.70/1.64 % (3906)Instructions burned: 4 (million)
% 2.70/1.64 % (3906)------------------------------
% 2.70/1.64 % (3906)------------------------------
% 2.70/1.65 % (3908)Instruction limit reached!
% 2.70/1.65 % (3908)------------------------------
% 2.70/1.65 % (3908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (3908)Termination reason: Unknown
% 2.70/1.65 % (3908)Termination phase: Property scanning
% 2.70/1.65
% 2.70/1.65 % (3908)Memory used [KB]: 1535
% 2.70/1.65 % (3908)Time elapsed: 0.007 s
% 2.70/1.65 % (3908)Instructions burned: 21 (million)
% 2.70/1.65 % (3908)------------------------------
% 2.70/1.65 % (3908)------------------------------
% 2.70/1.65 % (3909)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/710Mi)
% 2.70/1.65 % (3911)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_3790 for (2999ds/902Mi)
% 2.70/1.65 % (3910)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/6Mi)
% 2.70/1.65 % (3910)Instruction limit reached!
% 2.70/1.65 % (3910)------------------------------
% 2.70/1.65 % (3910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.65 % (3910)Termination reason: Unknown
% 2.70/1.65 % (3910)Termination phase: shuffling
% 2.70/1.65
% 2.70/1.65 % (3910)Memory used [KB]: 1407
% 2.70/1.65 % (3910)Time elapsed: 0.004 s
% 2.70/1.65 % (3910)Instructions burned: 6 (million)
% 2.70/1.65 % (3910)------------------------------
% 2.70/1.65 % (3910)------------------------------
% 2.70/1.65 % (3913)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3790 for (2999ds/5Mi)
% 2.70/1.66 % (3913)Instruction limit reached!
% 2.70/1.66 % (3913)------------------------------
% 2.70/1.66 % (3913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.66 % (3913)Termination reason: Unknown
% 2.70/1.66 % (3913)Termination phase: shuffling
% 2.70/1.66
% 2.70/1.66 % (3913)Memory used [KB]: 1407
% 2.70/1.66 % (3913)Time elapsed: 0.003 s
% 2.70/1.66 % (3913)Instructions burned: 8 (million)
% 2.70/1.66 % (3913)------------------------------
% 2.70/1.66 % (3913)------------------------------
% 2.70/1.66 % (3909)First to succeed.
% 2.70/1.66 % (3914)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_3790 for (2999ds/6Mi)
% 2.70/1.66 % (3909)Refutation found. Thanks to Tanya!
% 2.70/1.66 % SZS status Theorem for DTF2THF_3790
% 2.70/1.66 % SZS output start Proof for DTF2THF_3790
% 2.70/1.66 thf(type_def_5, type, nat: $tType).
% 2.70/1.66 thf(type_def_7, type, hF_Mirabelle_hf: $tType).
% 2.70/1.66 thf(type_def_8, type, set_HF_Mirabelle_hf: $tType).
% 2.70/1.66 thf(type_def_9, type, set_nat: $tType).
% 2.70/1.66 thf(func_def_0, type, set_HF_Mirabelle_hf: $tType).
% 2.70/1.66 thf(func_def_1, type, hF_Mirabelle_hf: $tType).
% 2.70/1.66 thf(func_def_2, type, set_nat: $tType).
% 2.70/1.66 thf(func_def_3, type, nat: $tType).
% 2.70/1.66 thf(func_def_4, type, plus_plus_nat: nat > nat > nat).
% 2.70/1.66 thf(func_def_5, type, zero_z189798548lle_hf: hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_6, type, zero_zero_nat: nat).
% 2.70/1.66 thf(func_def_7, type, hF_Mir818139703ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_8, type, hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_9, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_10, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_11, type, hF_Mir1248913145eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_12, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_13, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_14, type, hF_Mirabelle_hcard: hF_Mirabelle_hf > nat).
% 2.70/1.66 thf(func_def_15, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_16, type, hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat).
% 2.70/1.66 thf(func_def_17, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_18, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_19, type, hF_Mir199975595nction: hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_20, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_21, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_22, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_23, type, hF_Mir434065167lation: hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_24, type, hF_Mir1653039215strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_25, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_26, type, hF_Mir137170979_hpair: hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_27, type, inf_in307783154e_hf_o: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_28, type, inf_inf_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 2.70/1.66 thf(func_def_29, type, inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_30, type, inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_31, type, inf_inf_set_nat: set_nat > set_nat > set_nat).
% 2.70/1.66 thf(func_def_32, type, sup_su1199008216e_hf_o: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_33, type, sup_sup_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 2.70/1.66 thf(func_def_34, type, sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_35, type, sup_su1790843629lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_36, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 2.70/1.66 thf(func_def_37, type, suc: nat > nat).
% 2.70/1.66 thf(func_def_38, type, top_to22270292e_hf_o: hF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_39, type, top_top_nat_o: nat > $o).
% 2.70/1.66 thf(func_def_41, type, top_to489427057lle_hf: set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_42, type, top_top_set_nat: set_nat).
% 2.70/1.66 thf(func_def_43, type, collec2046588256lle_hf: (hF_Mirabelle_hf > $o) > set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_44, type, collect_nat: (nat > $o) > set_nat).
% 2.70/1.66 thf(func_def_45, type, insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_46, type, insert_nat: nat > set_nat > set_nat).
% 2.70/1.66 thf(func_def_47, type, type_d1794767497hf_nat: (hF_Mirabelle_hf > nat) > (nat > hF_Mirabelle_hf) > set_nat > $o).
% 2.70/1.66 thf(func_def_48, type, member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 2.70/1.66 thf(func_def_49, type, member_nat: nat > set_nat > $o).
% 2.70/1.66 thf(func_def_50, type, r: hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_51, type, x: hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_65, type, sK0: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_66, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_68, type, sK3: hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_69, type, sK4: hF_Mirabelle_hf).
% 2.70/1.66 thf(func_def_70, type, sK5: hF_Mirabelle_hf).
% 2.70/1.66 thf(f800,plain,(
% 2.70/1.66 $false),
% 2.70/1.66 inference(subsumption_resolution,[status(thm)],[f799,f753])).
% 2.70/1.66 thf(f753,plain,(
% 2.70/1.66 (sK4 != sK5)),
% 2.70/1.66 inference(equality_proxy_clausification,[status(thm)],[f751])).
% 2.70/1.66 thf(f751,plain,(
% 2.70/1.66 ((sK4 = sK5) = $false)),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f750])).
% 2.70/1.66 thf(f750,plain,(
% 2.70/1.66 (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK5) @ (hF_Mir1653039215strict @ r @ x)) => (sK4 = sK5)) = $false)),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f749])).
% 2.70/1.66 thf(f749,plain,(
% 2.70/1.66 (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y0) @ (hF_Mir1653039215strict @ r @ x)) => (sK4 = Y0))) @ sK5) = $false)),
% 2.70/1.66 inference(sigma_clausification,[status(thm)],[f747])).
% 2.70/1.66 thf(f747,plain,(
% 2.70/1.66 ($false = (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y0) @ (hF_Mir1653039215strict @ r @ x)) => (sK4 = Y0)))))),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f746])).
% 2.70/1.66 thf(f746,plain,(
% 2.70/1.66 (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK4) @ (hF_Mir1653039215strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y0) @ (hF_Mir1653039215strict @ r @ x)) => (sK4 = Y0))))) = $false)),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f745])).
% 2.70/1.66 thf(f745,plain,(
% 2.70/1.66 (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y0) @ (hF_Mir1653039215strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y1) @ (hF_Mir1653039215strict @ r @ x)) => (Y0 = Y1)))))) @ sK4) = $false)),
% 2.70/1.66 inference(sigma_clausification,[status(thm)],[f744])).
% 2.70/1.66 thf(f744,plain,(
% 2.70/1.66 ((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y0) @ (hF_Mir1653039215strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ Y1) @ (hF_Mir1653039215strict @ r @ x)) => (Y0 = Y1))))))) = $false)),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f743])).
% 2.70/1.66 thf(f743,plain,(
% 2.70/1.66 (((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ (hF_Mir1653039215strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y2) @ (hF_Mir1653039215strict @ r @ x)) => (Y1 = Y2)))))))) @ sK3) = $false)),
% 2.70/1.66 inference(sigma_clausification,[status(thm)],[f742])).
% 2.70/1.66 thf(f742,plain,(
% 2.70/1.66 ($true != (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ (hF_Mir1653039215strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y2) @ (hF_Mir1653039215strict @ r @ x)) => (Y1 = Y2))))))))))),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f739])).
% 2.70/1.66 thf(f739,plain,(
% 2.70/1.66 (((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y3) @ Y0) => (Y2 = Y3)))))))))) @ (hF_Mir1653039215strict @ r @ x)) != $true)),
% 2.70/1.66 inference(definition_unfolding,[status(thm)],[f731,f737])).
% 2.70/1.66 thf(f737,plain,(
% 2.70/1.66 (hF_Mir199975595nction = (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y3) @ Y0) => (Y2 = Y3)))))))))))),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f558])).
% 2.70/1.66 thf(f558,plain,(
% 2.70/1.66 (hF_Mir199975595nction = (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y3) @ Y0) => (Y2 = Y3)))))))))))),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f557])).
% 2.70/1.66 thf(f557,plain,(
% 2.70/1.66 ((^[X0 : hF_Mirabelle_hf] : (! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ X0) => ! [X3 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X3) @ X0) => (X1 = X3))))) = hF_Mir199975595nction)),
% 2.70/1.66 inference(rectify,[status(thm)],[f2])).
% 2.70/1.66 thf(f2,axiom,(
% 2.70/1.66 ((^[X2 : hF_Mirabelle_hf] : (! [X4 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X3 @ X4) @ X2) => ! [X5 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X3 @ X5) @ X2) => (X4 = X5))))) = hF_Mir199975595nction)),
% 2.70/1.66 file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',fact_1_hfunction__def)).
% 2.70/1.66 thf(f731,plain,(
% 2.70/1.66 ((hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x)) != $true)),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f725])).
% 2.70/1.66 thf(f725,plain,(
% 2.70/1.66 ((hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x)) != $true)),
% 2.70/1.66 inference(flattening,[status(thm)],[f580])).
% 2.70/1.66 thf(f580,plain,(
% 2.70/1.66 ~((hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x)) = $true)),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f579])).
% 2.70/1.66 thf(f579,plain,(
% 2.70/1.66 ~(hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x))),
% 2.70/1.66 inference(rectify,[status(thm)],[f357])).
% 2.70/1.66 thf(f357,negated_conjecture,(
% 2.70/1.66 ~(hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x))),
% 2.70/1.66 inference(negated_conjecture,[status(cth)],[f356])).
% 2.70/1.66 thf(f356,conjecture,(
% 2.70/1.66 (hF_Mir199975595nction @ (hF_Mir1653039215strict @ r @ x))),
% 2.70/1.66 file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',conj_1)).
% 2.70/1.66 thf(f799,plain,(
% 2.70/1.66 (sK4 = sK5)),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f795])).
% 2.70/1.66 thf(f795,plain,(
% 2.70/1.66 ($false = $true) | (sK4 = sK5)),
% 2.70/1.66 inference(superposition,[status(thm)],[f781,f771])).
% 2.70/1.66 thf(f771,plain,(
% 2.70/1.66 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK5) @ r) = $true)),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f769])).
% 2.70/1.66 thf(f769,plain,(
% 2.70/1.66 ($true != $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK5) @ r) = $true)),
% 2.70/1.66 inference(superposition,[status(thm)],[f733,f752])).
% 2.70/1.66 thf(f752,plain,(
% 2.70/1.66 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK5) @ (hF_Mir1653039215strict @ r @ x)) = $true)),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f750])).
% 2.70/1.66 thf(f733,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) != $true) | ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) )),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f730])).
% 2.70/1.66 thf(f730,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((((((hF_Mirabelle_hmem @ (sK1 @ X2 @ X1) @ X2) = $true) & ((hF_Mirabelle_hpair @ (sK1 @ X2 @ X1) @ (sK0 @ X2 @ X1)) = X1)) & ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) | ((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) != $true)) & (((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) = $true) | ! [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X6 @ X2) != $true) | ((hF_Mirabelle_hpair @ X6 @ X5) != X1)) | ((hF_Mirabelle_hmem @ X1 @ X0) != $true)))),
% 2.70/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f728,f729])).
% 2.70/1.66 thf(f729,plain,(
% 2.70/1.66 ! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X4 @ X2)) & ((hF_Mirabelle_hpair @ X4 @ X3) = X1)) => (((hF_Mirabelle_hmem @ (sK1 @ X2 @ X1) @ X2) = $true) & ((hF_Mirabelle_hpair @ (sK1 @ X2 @ X1) @ (sK0 @ X2 @ X1)) = X1)))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f728,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X4 @ X2)) & ((hF_Mirabelle_hpair @ X4 @ X3) = X1)) & ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) | ((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) != $true)) & (((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) = $true) | ! [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X6 @ X2) != $true) | ((hF_Mirabelle_hpair @ X6 @ X5) != X1)) | ((hF_Mirabelle_hmem @ X1 @ X0) != $true)))),
% 2.70/1.66 inference(rectify,[status(thm)],[f727])).
% 2.70/1.66 thf(f727,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X4 @ X2)) & ((hF_Mirabelle_hpair @ X4 @ X3) = X1)) & ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) | ((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) != $true)) & (((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) = $true) | ! [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X4 @ X2)) | ((hF_Mirabelle_hpair @ X4 @ X3) != X1)) | ((hF_Mirabelle_hmem @ X1 @ X0) != $true)))),
% 2.70/1.66 inference(flattening,[status(thm)],[f726])).
% 2.70/1.66 thf(f726,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X4 @ X2)) & ((hF_Mirabelle_hpair @ X4 @ X3) = X1)) & ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) | ((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) != $true)) & (((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) = $true) | (! [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X4 @ X2)) | ((hF_Mirabelle_hpair @ X4 @ X3) != X1)) | ((hF_Mirabelle_hmem @ X1 @ X0) != $true))))),
% 2.70/1.66 inference(nnf_transformation,[status(thm)],[f474])).
% 2.70/1.66 thf(f474,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X4 @ X2)) & ((hF_Mirabelle_hpair @ X4 @ X3) = X1)) & ((hF_Mirabelle_hmem @ X1 @ X0) = $true)) <=> ((hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)) = $true))),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f473])).
% 2.70/1.66 thf(f473,plain,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ X0) & ? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (((hF_Mirabelle_hpair @ X4 @ X3) = X1) & (hF_Mirabelle_hmem @ X4 @ X2)) = (hF_Mirabelle_hmem @ X1 @ (hF_Mir1653039215strict @ X0 @ X2)))),
% 2.70/1.66 inference(rectify,[status(thm)],[f3])).
% 2.70/1.66 thf(f3,axiom,(
% 2.70/1.66 ! [X0 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf,X7 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X6 @ X0) & ? [X4 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf] : (((hF_Mirabelle_hpair @ X3 @ X4) = X6) & (hF_Mirabelle_hmem @ X3 @ X7)) = (hF_Mirabelle_hmem @ X6 @ (hF_Mir1653039215strict @ X0 @ X7)))),
% 2.70/1.66 file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',fact_2_hrestrict__iff)).
% 2.70/1.66 thf(f781,plain,(
% 2.70/1.66 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ X0) @ r) = $false) | (sK4 = X0)) )),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f772])).
% 2.70/1.66 thf(f772,plain,(
% 2.70/1.66 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ X0) @ r) = $false) | (sK4 = X0) | ($false = $true)) )),
% 2.70/1.66 inference(superposition,[status(thm)],[f763,f765])).
% 2.70/1.66 thf(f765,plain,(
% 2.70/1.66 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK4) @ r) = $true)),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f764])).
% 2.70/1.66 thf(f764,plain,(
% 2.70/1.66 ($true != $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK4) @ r) = $true)),
% 2.70/1.66 inference(superposition,[status(thm)],[f733,f748])).
% 2.70/1.66 thf(f748,plain,(
% 2.70/1.66 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK3 @ sK4) @ (hF_Mir1653039215strict @ r @ x)) = $true)),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f746])).
% 2.70/1.66 thf(f763,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((X2 = X3) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X3) @ r) = $false) | ($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r))) )),
% 2.70/1.66 inference(equality_proxy_clausification,[status(thm)],[f762])).
% 2.70/1.66 thf(f762,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X3) @ r) = $false) | ((X2 = X3) = $true) | ($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r))) )),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f761])).
% 2.70/1.66 thf(f761,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r)) | (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X3) @ r) => (X2 = X3)) = $true)) )),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f760])).
% 2.70/1.66 thf(f760,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r)) | (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y0) @ r) => (X2 = Y0))) @ X3) = $true)) )),
% 2.70/1.66 inference(pi_clausification,[status(thm)],[f759])).
% 2.70/1.66 thf(f759,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r)) | ($true = (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y0) @ r) => (X2 = Y0)))))) )),
% 2.70/1.66 inference(binary_proxy_clausification,[status(thm)],[f758])).
% 2.70/1.66 thf(f758,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y0) @ r) => (X2 = Y0))))) = $true)) )),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f757])).
% 2.70/1.66 thf(f757,plain,(
% 2.70/1.66 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y0) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y1) @ r) => (Y0 = Y1)))))) @ X2) = $true)) )),
% 2.70/1.66 inference(pi_clausification,[status(thm)],[f756])).
% 2.70/1.66 thf(f756,plain,(
% 2.70/1.66 ( ! [X1 : hF_Mirabelle_hf] : (((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y0) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ Y1) @ r) => (Y0 = Y1))))))) = $true)) )),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f755])).
% 2.70/1.66 thf(f755,plain,(
% 2.70/1.66 ( ! [X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y2) @ r) => (Y1 = Y2)))))))) @ X1) = $true)) )),
% 2.70/1.66 inference(pi_clausification,[status(thm)],[f754])).
% 2.70/1.66 thf(f754,plain,(
% 2.70/1.66 ($true = (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y2) @ r) => (Y1 = Y2))))))))))),
% 2.70/1.66 inference(beta_eta_normalization,[status(thm)],[f740])).
% 2.70/1.66 thf(f740,plain,(
% 2.70/1.66 (((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y3) @ Y0) => (Y2 = Y3)))))))))) @ r) = $true)),
% 2.70/1.66 inference(definition_unfolding,[status(thm)],[f736,f737])).
% 2.70/1.66 thf(f736,plain,(
% 2.70/1.66 ((hF_Mir199975595nction @ r) = $true)),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f588])).
% 2.70/1.66 thf(f588,plain,(
% 2.70/1.66 ((hF_Mir199975595nction @ r) = $true)),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f587])).
% 2.70/1.66 thf(f587,plain,(
% 2.70/1.66 (hF_Mir199975595nction @ r)),
% 2.70/1.66 inference(rectify,[status(thm)],[f355])).
% 2.70/1.66 thf(f355,axiom,(
% 2.70/1.66 (hF_Mir199975595nction @ r)),
% 2.70/1.66 file('/export/starexec/sandbox2/tmp/tmp.QgTZs0ghIr/DTF2THF_3790.p',conj_0)).
% 2.70/1.66 % SZS output end Proof for DTF2THF_3790
% 2.70/1.66 % (3909)------------------------------
% 2.70/1.66 % (3909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.66 % (3909)Termination reason: Refutation
% 2.70/1.66
% 2.70/1.66 % (3909)Memory used [KB]: 6268
% 2.70/1.66 % (3909)Time elapsed: 0.017 s
% 2.70/1.66 % (3909)Instructions burned: 39 (million)
% 2.70/1.66 % (3909)------------------------------
% 2.70/1.66 % (3909)------------------------------
% 2.70/1.66 % (3888)Success in time 0.104 s
%------------------------------------------------------------------------------