%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP074^1 : 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 : n019.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 08:55:20 AM UTC 2026
% Result : Theorem 2.74s 1.77s
% Output : Refutation 2.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 4
% Syntax : Number of formulae : 19 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 19 ( 18 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 112 ( 8 ~; 0 |; 0 &; 104 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 38 usr; 5 con; 0-2 aty)
% Number of variables : 20 ( 12 ^; 8 !; 0 ?; 20 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
hF_Mirabelle_hf: $tType ).
thf(type_def_7,type,
nat: $tType ).
thf(type_def_8,type,
set_HF_Mirabelle_hf: $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,
nat: $tType ).
thf(func_def_3,type,
minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_4,type,
minus_minus_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_Abs_hf: nat > hF_Mirabelle_hf ).
thf(func_def_15,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_16,type,
hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_17,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_18,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_19,type,
hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_20,type,
hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_21,type,
inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_22,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_23,type,
sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_24,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_25,type,
semiri1382578993at_nat: nat > nat ).
thf(func_def_26,type,
ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_27,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_28,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(func_def_29,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_30,type,
x: hF_Mirabelle_hf ).
thf(func_def_31,type,
y: hF_Mirabelle_hf ).
thf(func_def_46,type,
sK0: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_47,type,
sK1: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_48,type,
sK2: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_49,type,
sK3: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_50,type,
sK4: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_51,type,
sK5: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_52,type,
sK6: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_54,type,
sK8: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_55,type,
sK9: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(f1002,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1001]) ).
thf(f1001,plain,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
!= ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ),
inference(superposition,[status(thm)],[f921,f967]) ).
thf(f967,plain,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ x @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ x @ ( hF_Mirabelle_hinsert @ y @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) ) @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f905]) ).
thf(f905,plain,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_hinsert @ Y1 @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) ) @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) )
@ x
@ y ) ),
inference(definition_unfolding,[status(thm)],[f860,f851,f880]) ).
thf(f880,plain,
( hF_Mirabelle_hpair
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_hinsert @ Y1 @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) ) @ ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f834,f851,f851,f851]) ).
thf(f834,plain,
( hF_Mirabelle_hpair
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_hinsert @ Y1 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ),
inference(cnf_transformation,[status(thm)],[f364]) ).
thf(f364,plain,
( hF_Mirabelle_hpair
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ Y0 @ ( hF_Mirabelle_hinsert @ Y1 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ),
inference(fool_elimination,[status(thm)],[f363]) ).
thf(f363,plain,
( hF_Mirabelle_hpair
= ( ^ [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ X0 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ X0 @ ( hF_Mirabelle_hinsert @ X1 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
( hF_Mirabelle_hpair
= ( ^ [X8: hF_Mirabelle_hf,X9: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ X8 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ X8 @ ( hF_Mirabelle_hinsert @ X9 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_13_hpair__def) ).
thf(f851,plain,
( zero_z189798548lle_hf
= ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ),
inference(cnf_transformation,[status(thm)],[f23]) ).
thf(f23,axiom,
( zero_z189798548lle_hf
= ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ),
file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_22_Abs__hf__0) ).
thf(f860,plain,
( zero_z189798548lle_hf
= ( hF_Mirabelle_hpair @ x @ y ) ),
inference(cnf_transformation,[status(thm)],[f796]) ).
thf(f796,plain,
( zero_z189798548lle_hf
= ( hF_Mirabelle_hpair @ x @ y ) ),
inference(flattening,[status(thm)],[f345]) ).
thf(f345,negated_conjecture,
~ ( ( zero_z189798548lle_hf
!= ( hF_Mirabelle_hpair @ x @ y ) ) ),
inference(negated_conjecture,[status(cth)],[f344]) ).
thf(f344,conjecture,
( zero_z189798548lle_hf
!= ( hF_Mirabelle_hpair @ x @ y ) ),
file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',conj_0) ).
thf(f921,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ X0 @ X1 )
!= ( hF_Mirabelle_Abs_hf @ zero_zero_nat ) ),
inference(definition_unfolding,[status(thm)],[f876,f851]) ).
thf(f876,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( zero_z189798548lle_hf
!= ( hF_Mirabelle_hinsert @ X0 @ X1 ) ),
inference(cnf_transformation,[status(thm)],[f797]) ).
thf(f797,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( zero_z189798548lle_hf
!= ( hF_Mirabelle_hinsert @ X0 @ X1 ) ),
inference(rectify,[status(thm)],[f19]) ).
thf(f19,axiom,
! [X0: hF_Mirabelle_hf,X12: hF_Mirabelle_hf] :
( zero_z189798548lle_hf
!= ( hF_Mirabelle_hinsert @ X0 @ X12 ) ),
file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_18_hinsert__nonempty) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP074^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n019.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 08:25:09 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.47 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : ITP074^1 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.47 % Domain : Interactive Theorem Proving
% 0.26/0.47 % Problem : Sledgehammer HF problem prob_612__5334652_1
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English :
% 0.26/0.47
% 0.26/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.47 % Source : [Des21]
% 0.26/0.47 % Names : HF/prob_612__5334652_1 [Des21]
% 0.26/0.47
% 0.26/0.47 % Status : Theorem
% 0.26/0.47 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.50 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0
% 0.26/0.47 % Syntax : Number of formulae : 376 ( 220 unt; 32 typ; 0 def)
% 0.26/0.47 % Number of atoms : 708 ( 335 equ; 0 cnn)
% 0.26/0.47 % Maximal formula atoms : 6 ( 2 avg)
% 0.26/0.47 % Number of connectives : 2286 ( 42 ~; 7 |; 34 &;1996 @)
% 0.26/0.47 % ( 0 <=>; 207 =>; 0 <=; 0 <~>)
% 0.26/0.47 % Maximal formula depth : 16 ( 5 avg)
% 0.26/0.47 % Number of types : 4 ( 3 usr)
% 0.26/0.47 % Number of type conns : 86 ( 86 >; 0 *; 0 +; 0 <<)
% 0.26/0.47 % Number of symbols : 30 ( 29 usr; 4 con; 0-2 aty)
% 0.26/0.47 % Number of variables : 876 ( 79 ^; 782 !; 15 ?; 876 :)
% 0.26/0.47 % SPC : TH0_THM_EQU_NAR
% 0.26/0.47
% 0.26/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:37:44.587
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % Could-be-implicit typings (3)
% 0.26/0.47 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.26/0.47 set_HF_Mirabelle_hf: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 hF_Mirabelle_hf: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.26/0.47 nat: $tType ).
% 0.26/0.47
% 0.26/0.47 % Explicit typings (29)
% 0.26/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.26/0.47 minus_minus_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.26/0.47 zero_zero_nat: nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.26/0.47 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.26/0.47 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 0.26/0.47 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.26/0.47 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.26/0.47 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.26/0.47 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.26/0.47 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.26/0.47 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.26/0.47 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 0.26/0.47 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.26/0.47 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.26/0.47 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 0.26/0.47 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 0.26/0.47 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.26/0.47 inf_inf_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.26/0.47 sup_sup_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.26/0.47 semiri1382578993at_nat: nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.26/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.26/0.47 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_x,type,
% 0.26/0.47 x: hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_y,type,
% 0.26/0.47 y: hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 % Relevant facts (343)
% 0.26/0.47 thf(fact_0_hpair__iff,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.26/0.47 = ( hF_Mirabelle_hpair @ A2 @ B2 ) )
% 0.26/0.47 = ( ( A = A2 )
% 0.26/0.47 & ( B = B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_iff
% 0.26/0.47 thf(fact_1_hpair__inject,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.26/0.47 = ( hF_Mirabelle_hpair @ A2 @ B2 ) )
% 0.26/0.47 => ~ ( ( A = A2 )
% 0.26/0.47 => ( B != B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_inject
% 0.26/0.47 thf(fact_2_hpair__neq__fst,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.26/0.47 != A ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_neq_fst
% 0.26/0.47 thf(fact_3_hpair__neq__snd,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.26/0.47 != B ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_neq_snd
% 0.26/0.47 thf(fact_4_hfst__conv,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.26/0.47 = A ) ).
% 0.26/0.47
% 0.26/0.47 % hfst_conv
% 0.26/0.47 thf(fact_5_hsnd__conv,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.26/0.47 = B ) ).
% 0.26/0.47
% 0.26/0.47 % hsnd_conv
% 0.26/0.47 thf(fact_6_HInter__hempty,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 0.26/0.47 = zero_z189798548lle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % HInter_hempty
% 0.26/0.47 thf(fact_7_HCollect__hempty,axiom,
% 0.26/0.47 ! [P: hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 0.26/0.47 = zero_z189798548lle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % HCollect_hempty
% 0.26/0.47 thf(fact_8_Replace__0,axiom,
% 0.26/0.47 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 0.26/0.47 = zero_z189798548lle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % Replace_0
% 0.26/0.47 thf(fact_9_HUnion__hempty,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.26/0.47 = zero_z189798548lle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % HUnion_hempty
% 0.26/0.47 thf(fact_10_zero__reorient,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( zero_z189798548lle_hf = X )
% 0.26/0.47 = ( X = zero_z189798548lle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_reorient
% 0.26/0.47 thf(fact_11_zero__reorient,axiom,
% 0.26/0.47 ! [X: nat] :
% 0.26/0.47 ( ( zero_zero_nat = X )
% 0.26/0.47 = ( X = zero_zero_nat ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_reorient
% 0.26/0.47 thf(fact_12_RepFun__0,axiom,
% 0.26/0.47 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.26/0.47 = zero_z189798548lle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_0
% 0.26/0.47 thf(fact_13_hpair__def,axiom,
% 0.26/0.47 ( hF_Mirabelle_hpair
% 0.26/0.47 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_def
% 0.26/0.47 thf(fact_14_hpair__def_H,axiom,
% 0.26/0.47 ( hF_Mirabelle_hpair
% 0.26/0.47 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_def'
% 0.26/0.47 thf(fact_15_singleton__eq__iff,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.26/0.47 = ( A = B ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_eq_iff
% 0.26/0.47 thf(fact_16_RepFun__hinsert,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A @ B ) @ F )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ ( F @ A ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_hinsert
% 0.26/0.47 thf(fact_17_hinsert__commute,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y @ Z ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ Y @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_commute
% 0.26/0.47 thf(fact_18_hinsert__nonempty,axiom,
% 0.26/0.47 ! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hinsert @ A @ A4 )
% 0.26/0.48 != zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % hinsert_nonempty
% 0.26/0.48 thf(fact_19_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ( hF_Mirabelle_hinsert @ A @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 0.26/0.48 = ( ( ( A = C )
% 0.26/0.48 & ( B = D ) )
% 0.26/0.48 | ( ( A = D )
% 0.26/0.48 & ( B = C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HF_Mirabelle_glliljednj.doubleton_eq_iff
% 0.26/0.48 thf(fact_20_hf__induct__ax,axiom,
% 0.26/0.48 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ zero_z189798548lle_hf )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ X2 )
% 0.26/0.48 => ! [Y2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ Y2 )
% 0.26/0.48 => ( P @ ( hF_Mirabelle_hinsert @ Y2 @ X2 ) ) ) )
% 0.26/0.48 => ( P @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_induct_ax
% 0.26/0.48 thf(fact_21_HInter__hinsert,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4 != zero_z189798548lle_hf )
% 0.26/0.48 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HInter_hinsert
% 0.26/0.48 thf(fact_22_Abs__hf__0,axiom,
% 0.26/0.48 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.26/0.48 = zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % Abs_hf_0
% 0.26/0.48 thf(fact_23_HInter__iff,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4 != zero_z189798548lle_hf )
% 0.26/0.48 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A4 ) )
% 0.26/0.48 = ( ! [Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HInter_iff
% 0.26/0.48 thf(fact_24_HUnion__hinsert,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ ( hF_Mirabelle_HUnion @ A4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HUnion_hinsert
% 0.26/0.48 thf(fact_25_hdiff__insert,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( minus_1232880740lle_hf @ A4 @ ( hF_Mirabelle_hinsert @ A @ B4 ) )
% 0.26/0.48 = ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A4 @ B4 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hdiff_insert
% 0.26/0.48 thf(fact_26_hinsert__eq__sup,axiom,
% 0.26/0.48 ( hF_Mirabelle_hinsert
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinsert_eq_sup
% 0.26/0.48 thf(fact_27_hf__induct,axiom,
% 0.26/0.48 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ zero_z189798548lle_hf )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ X2 )
% 0.26/0.48 => ( ( P @ Y2 )
% 0.26/0.48 => ( ~ ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.26/0.48 => ( P @ ( hF_Mirabelle_hinsert @ X2 @ Y2 ) ) ) ) )
% 0.26/0.48 => ( P @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_induct
% 0.26/0.48 thf(fact_28_hf__cases,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( Y != zero_z189798548lle_hf )
% 0.26/0.48 => ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( Y
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_cases
% 0.26/0.48 thf(fact_29_Replace__hunion,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ R )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A4 @ R ) @ ( hF_Mirabelle_Replace @ B4 @ R ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Replace_hunion
% 0.26/0.48 thf(fact_30_HCollect__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A4 ) )
% 0.26/0.48 = ( ( P @ X )
% 0.26/0.48 & ( hF_Mirabelle_hmem @ X @ A4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HCollect_iff
% 0.26/0.48 thf(fact_31_hf__equalityI,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X2 @ A )
% 0.26/0.48 = ( hF_Mirabelle_hmem @ X2 @ B ) )
% 0.26/0.48 => ( A = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_equalityI
% 0.26/0.48 thf(fact_32_zero__diff,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.26/0.48 = zero_zero_nat ) ).
% 0.26/0.48
% 0.26/0.48 % zero_diff
% 0.26/0.48 thf(fact_33_diff__zero,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % diff_zero
% 0.26/0.48 thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ A @ A )
% 0.26/0.48 = zero_zero_nat ) ).
% 0.26/0.48
% 0.26/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.26/0.48 thf(fact_35_hmem__hinsert,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.26/0.48 = ( ( A = B )
% 0.26/0.48 | ( hF_Mirabelle_hmem @ A @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_hinsert
% 0.26/0.48 thf(fact_36_hunion__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A @ B ) )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ X @ A )
% 0.26/0.48 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hunion_iff
% 0.26/0.48 thf(fact_37_hinter__iff,axiom,
% 0.26/0.48 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U @ ( inf_in956532509lle_hf @ X @ Y ) )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ U @ X )
% 0.26/0.48 & ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinter_iff
% 0.26/0.48 thf(fact_38_RepFun__iff,axiom,
% 0.26/0.48 ! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
% 0.26/0.48 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.26/0.48 & ( V
% 0.26/0.48 = ( F @ U2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % RepFun_iff
% 0.26/0.48 thf(fact_39_hunion__hempty__left,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A4 )
% 0.26/0.48 = A4 ) ).
% 0.26/0.48
% 0.26/0.48 % hunion_hempty_left
% 0.26/0.48 thf(fact_40_hunion__hempty__right,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ A4 @ zero_z189798548lle_hf )
% 0.26/0.48 = A4 ) ).
% 0.26/0.48
% 0.26/0.48 % hunion_hempty_right
% 0.26/0.48 thf(fact_41_HUnion__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A4 ) )
% 0.26/0.48 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.26/0.48 & ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HUnion_iff
% 0.26/0.48 thf(fact_42_mem__Collect__eq,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 0.26/0.48 = ( P @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_Collect_eq
% 0.26/0.48 thf(fact_43_Collect__mem__eq,axiom,
% 0.26/0.48 ! [A4: set_HF_Mirabelle_hf] :
% 0.26/0.48 ( ( collec2046588256lle_hf
% 0.26/0.48 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 ) )
% 0.26/0.48 = A4 ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_mem_eq
% 0.26/0.48 thf(fact_44_Collect__cong,axiom,
% 0.26/0.48 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( P @ X2 )
% 0.26/0.48 = ( Q @ X2 ) )
% 0.26/0.48 => ( ( collec2046588256lle_hf @ P )
% 0.26/0.48 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_cong
% 0.26/0.48 thf(fact_45_hdiff__iff,axiom,
% 0.26/0.48 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U @ ( minus_1232880740lle_hf @ X @ Y ) )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ U @ X )
% 0.26/0.48 & ~ ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hdiff_iff
% 0.26/0.48 thf(fact_46_hinter__hempty__left,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A4 )
% 0.26/0.48 = zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % hinter_hempty_left
% 0.26/0.48 thf(fact_47_hinter__hempty__right,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ A4 @ zero_z189798548lle_hf )
% 0.26/0.48 = zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % hinter_hempty_right
% 0.26/0.48 thf(fact_48_Replace__iff,axiom,
% 0.26/0.48 ! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_Replace @ A4 @ R ) )
% 0.26/0.48 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.26/0.48 & ( R @ U2 @ V )
% 0.26/0.48 & ! [Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( R @ U2 @ Y3 )
% 0.26/0.48 => ( Y3 = V ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Replace_iff
% 0.26/0.48 thf(fact_49_hdiff__zero,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( minus_1232880740lle_hf @ X @ zero_z189798548lle_hf )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % hdiff_zero
% 0.26/0.48 thf(fact_50_zero__hdiff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X )
% 0.26/0.48 = zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % zero_hdiff
% 0.26/0.48 thf(fact_51_RepFun__hunion,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ F )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A4 @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % RepFun_hunion
% 0.26/0.48 thf(fact_52_HUnion__hunion,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A4 @ B4 ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A4 ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % HUnion_hunion
% 0.26/0.48 thf(fact_53_diff__right__commute,axiom,
% 0.26/0.48 ! [A: nat,C: nat,B: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.26/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_right_commute
% 0.26/0.48 thf(fact_54_replacement__fun,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.48 ? [Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ! [V2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 0.26/0.48 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.48 & ( V2
% 0.26/0.48 = ( F @ U2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % replacement_fun
% 0.26/0.48 thf(fact_55_hmem__not__refl,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_not_refl
% 0.26/0.48 thf(fact_56_comprehension,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.26/0.48 ? [Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.26/0.48 & ( P @ U3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % comprehension
% 0.26/0.48 thf(fact_57_union__of__set,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ? [Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.26/0.48 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ Y3 @ X )
% 0.26/0.48 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % union_of_set
% 0.26/0.48 thf(fact_58_hmem__not__sym,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ~ ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.26/0.48 & ( hF_Mirabelle_hmem @ Y @ X ) ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_not_sym
% 0.26/0.48 thf(fact_59_binary__union,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ? [Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.26/0.48 | ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % binary_union
% 0.26/0.48 thf(fact_60_replacement,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.26/0.48 => ( ( R @ U4 @ V3 )
% 0.26/0.48 => ( ( R @ U4 @ V4 )
% 0.26/0.48 => ( V4 = V3 ) ) ) )
% 0.26/0.48 => ? [Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ! [V2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 0.26/0.48 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.48 & ( R @ U2 @ V2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % replacement
% 0.26/0.48 thf(fact_61_hmem__ne,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.26/0.48 => ( X != Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_ne
% 0.26/0.48 thf(fact_62_hf__ext,axiom,
% 0.26/0.48 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.48 ! [X3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.26/0.48 = ( hF_Mirabelle_hmem @ X3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_ext
% 0.26/0.48 thf(fact_63_hinter__hinsert__right,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) )
% 0.26/0.48 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinter_hinsert_right
% 0.26/0.48 thf(fact_64_hinter__hinsert__left,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) )
% 0.26/0.48 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinter_hinsert_left
% 0.26/0.48 thf(fact_65_foundation,axiom,
% 0.26/0.48 ! [Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( Z != zero_z189798548lle_hf )
% 0.26/0.48 => ? [W: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.26/0.48 & ( ( inf_in956532509lle_hf @ W @ Z )
% 0.26/0.48 = zero_z189798548lle_hf ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % foundation
% 0.26/0.48 thf(fact_66_hinsert__hdiff__if,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 0.26/0.48 = ( minus_1232880740lle_hf @ A4 @ B4 ) ) )
% 0.26/0.48 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 0.26/0.48 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ X @ ( minus_1232880740lle_hf @ A4 @ B4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinsert_hdiff_if
% 0.26/0.48 thf(fact_67_hinsert__iff,axiom,
% 0.26/0.48 ! [Z: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( Z
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ Y @ X ) )
% 0.26/0.48 = ( ! [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.48 | ( U2 = Y ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hinsert_iff
% 0.26/0.48 thf(fact_68_hemptyE,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf] :
% 0.26/0.48 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % hemptyE
% 0.26/0.48 thf(fact_69_hempty__iff,axiom,
% 0.26/0.48 ! [Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( Z = zero_z189798548lle_hf )
% 0.26/0.48 = ( ! [X3: hF_Mirabelle_hf] :
% 0.26/0.48 ~ ( hF_Mirabelle_hmem @ X3 @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hempty_iff
% 0.26/0.48 thf(fact_70_hmem__hempty,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf] :
% 0.26/0.48 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_hempty
% 0.26/0.48 thf(fact_71_hunion__hinsert__right,axiom,
% 0.26/0.48 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hunion_hinsert_right
% 0.26/0.48 thf(fact_72_hunion__hinsert__left,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 0.26/0.48 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A4 @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hunion_hinsert_left
% 0.26/0.48 thf(fact_73_RepFun__cong,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4 = B4 )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 0.26/0.48 => ( ( F @ X2 )
% 0.26/0.48 = ( G @ X2 ) ) )
% 0.26/0.48 => ( ( hF_Mirabelle_RepFun @ A4 @ F )
% 0.26/0.48 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % RepFun_cong
% 0.26/0.48 thf(fact_74_hmem__Sup__ne,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X @ Y )
% 0.26/0.48 => ( ( hF_Mirabelle_HUnion @ X )
% 0.26/0.48 != Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_Sup_ne
% 0.26/0.48 thf(fact_75_Replace__cong,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.48 ( ( A4 = B4 )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 0.26/0.48 => ( ( P @ X2 @ Y2 )
% 0.26/0.48 = ( Q @ X2 @ Y2 ) ) )
% 0.26/0.48 => ( ( hF_Mirabelle_Replace @ A4 @ P )
% 0.26/0.48 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Replace_cong
% 0.26/0.48 thf(fact_76_sup__inf__absorb,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_absorb
% 0.26/0.48 thf(fact_77_sup__inf__absorb,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_absorb
% 0.26/0.48 thf(fact_78_inf__sup__absorb,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_absorb
% 0.26/0.48 thf(fact_79_inf__sup__absorb,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_absorb
% 0.26/0.48 thf(fact_80_sup_Oright__idem,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
% 0.26/0.48 = ( sup_sup_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.right_idem
% 0.26/0.48 thf(fact_81_sup_Oright__idem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ B )
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.right_idem
% 0.26/0.48 thf(fact_82_sup__left__idem,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 0.26/0.48 = ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_left_idem
% 0.26/0.48 thf(fact_83_sup__left__idem,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_left_idem
% 0.26/0.48 thf(fact_84_sup_Oleft__idem,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
% 0.26/0.48 = ( sup_sup_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.left_idem
% 0.26/0.48 thf(fact_85_sup_Oleft__idem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.left_idem
% 0.26/0.48 thf(fact_86_sup__idem,axiom,
% 0.26/0.48 ! [X: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ X )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % sup_idem
% 0.26/0.48 thf(fact_87_sup__idem,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ X )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % sup_idem
% 0.26/0.48 thf(fact_88_sup_Oidem,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ A @ A )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % sup.idem
% 0.26/0.48 thf(fact_89_sup_Oidem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ A @ A )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % sup.idem
% 0.26/0.48 thf(fact_90_inf__right__idem,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
% 0.26/0.48 = ( inf_inf_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_right_idem
% 0.26/0.48 thf(fact_91_inf__right__idem,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y )
% 0.26/0.48 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_right_idem
% 0.26/0.48 thf(fact_92_inf_Oidem,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ A @ A )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % inf.idem
% 0.26/0.48 thf(fact_93_inf_Oidem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ A @ A )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % inf.idem
% 0.26/0.48 thf(fact_94_inf__idem,axiom,
% 0.26/0.48 ! [X: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ X )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_idem
% 0.26/0.48 thf(fact_95_inf__idem,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ X )
% 0.26/0.48 = X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_idem
% 0.26/0.48 thf(fact_96_inf_Oleft__idem,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
% 0.26/0.48 = ( inf_inf_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.left_idem
% 0.26/0.48 thf(fact_97_inf_Oleft__idem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ A @ B ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.left_idem
% 0.26/0.48 thf(fact_98_inf__left__idem,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 0.26/0.48 = ( inf_inf_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_left_idem
% 0.26/0.48 thf(fact_99_inf__left__idem,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_left_idem
% 0.26/0.48 thf(fact_100_inf_Oright__idem,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
% 0.26/0.48 = ( inf_inf_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.right_idem
% 0.26/0.48 thf(fact_101_inf_Oright__idem,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.right_idem
% 0.26/0.48 thf(fact_102_inf__sup__aci_I4_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 0.26/0.48 = ( inf_inf_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(4)
% 0.26/0.48 thf(fact_103_inf__sup__aci_I4_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(4)
% 0.26/0.48 thf(fact_104_inf__sup__aci_I3_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 0.26/0.48 = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(3)
% 0.26/0.48 thf(fact_105_inf__sup__aci_I3_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(3)
% 0.26/0.48 thf(fact_106_inf__sup__aci_I2_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
% 0.26/0.48 = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(2)
% 0.26/0.48 thf(fact_107_inf__sup__aci_I2_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
% 0.26/0.48 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(2)
% 0.26/0.48 thf(fact_108_inf__sup__aci_I1_J,axiom,
% 0.26/0.48 ( inf_inf_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(1)
% 0.26/0.48 thf(fact_109_inf__sup__aci_I1_J,axiom,
% 0.26/0.48 ( inf_in956532509lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(1)
% 0.26/0.48 thf(fact_110_boolean__algebra__cancel_Oinf1,axiom,
% 0.26/0.48 ! [A4: nat,K: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( A4
% 0.26/0.48 = ( inf_inf_nat @ K @ A ) )
% 0.26/0.48 => ( ( inf_inf_nat @ A4 @ B )
% 0.26/0.48 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.inf1
% 0.26/0.48 thf(fact_111_boolean__algebra__cancel_Oinf1,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4
% 0.26/0.48 = ( inf_in956532509lle_hf @ K @ A ) )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ A4 @ B )
% 0.26/0.48 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.inf1
% 0.26/0.48 thf(fact_112_boolean__algebra__cancel_Oinf2,axiom,
% 0.26/0.48 ! [B4: nat,K: nat,B: nat,A: nat] :
% 0.26/0.48 ( ( B4
% 0.26/0.48 = ( inf_inf_nat @ K @ B ) )
% 0.26/0.48 => ( ( inf_inf_nat @ A @ B4 )
% 0.26/0.48 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.inf2
% 0.26/0.48 thf(fact_113_boolean__algebra__cancel_Oinf2,axiom,
% 0.26/0.48 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( B4
% 0.26/0.48 = ( inf_in956532509lle_hf @ K @ B ) )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ A @ B4 )
% 0.26/0.48 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.inf2
% 0.26/0.48 thf(fact_114_inf_Oassoc,axiom,
% 0.26/0.48 ! [A: nat,B: nat,C: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
% 0.26/0.48 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.assoc
% 0.26/0.48 thf(fact_115_inf_Oassoc,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.assoc
% 0.26/0.48 thf(fact_116_inf__assoc,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
% 0.26/0.48 = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_assoc
% 0.26/0.48 thf(fact_117_inf__assoc,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
% 0.26/0.48 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_assoc
% 0.26/0.48 thf(fact_118_inf_Ocommute,axiom,
% 0.26/0.48 ( inf_inf_nat
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] : ( inf_inf_nat @ B3 @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.commute
% 0.26/0.48 thf(fact_119_inf_Ocommute,axiom,
% 0.26/0.48 ( inf_in956532509lle_hf
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.commute
% 0.26/0.48 thf(fact_120_inf__commute,axiom,
% 0.26/0.48 ( inf_inf_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_commute
% 0.26/0.48 thf(fact_121_inf__commute,axiom,
% 0.26/0.48 ( inf_in956532509lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_commute
% 0.26/0.48 thf(fact_122_inf_Oleft__commute,axiom,
% 0.26/0.48 ! [B: nat,A: nat,C: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
% 0.26/0.48 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.left_commute
% 0.26/0.48 thf(fact_123_inf_Oleft__commute,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A @ C ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.left_commute
% 0.26/0.48 thf(fact_124_inf__left__commute,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 0.26/0.48 = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_left_commute
% 0.26/0.48 thf(fact_125_inf__left__commute,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_left_commute
% 0.26/0.48 thf(fact_126_inf__sup__aci_I8_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 0.26/0.48 = ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(8)
% 0.26/0.48 thf(fact_127_inf__sup__aci_I8_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(8)
% 0.26/0.48 thf(fact_128_inf__sup__aci_I7_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 0.26/0.48 = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(7)
% 0.26/0.48 thf(fact_129_inf__sup__aci_I7_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(7)
% 0.26/0.48 thf(fact_130_inf__sup__aci_I6_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 0.26/0.48 = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(6)
% 0.26/0.48 thf(fact_131_inf__sup__aci_I6_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 0.26/0.48 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(6)
% 0.26/0.48 thf(fact_132_inf__sup__aci_I5_J,axiom,
% 0.26/0.48 ( sup_sup_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(5)
% 0.26/0.48 thf(fact_133_inf__sup__aci_I5_J,axiom,
% 0.26/0.48 ( sup_su638957495lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_aci(5)
% 0.26/0.48 thf(fact_134_boolean__algebra__cancel_Osup1,axiom,
% 0.26/0.48 ! [A4: nat,K: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( A4
% 0.26/0.48 = ( sup_sup_nat @ K @ A ) )
% 0.26/0.48 => ( ( sup_sup_nat @ A4 @ B )
% 0.26/0.48 = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.sup1
% 0.26/0.48 thf(fact_135_boolean__algebra__cancel_Osup1,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4
% 0.26/0.48 = ( sup_su638957495lle_hf @ K @ A ) )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ A4 @ B )
% 0.26/0.48 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.sup1
% 0.26/0.48 thf(fact_136_boolean__algebra__cancel_Osup2,axiom,
% 0.26/0.48 ! [B4: nat,K: nat,B: nat,A: nat] :
% 0.26/0.48 ( ( B4
% 0.26/0.48 = ( sup_sup_nat @ K @ B ) )
% 0.26/0.48 => ( ( sup_sup_nat @ A @ B4 )
% 0.26/0.48 = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.sup2
% 0.26/0.48 thf(fact_137_boolean__algebra__cancel_Osup2,axiom,
% 0.26/0.48 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( B4
% 0.26/0.48 = ( sup_su638957495lle_hf @ K @ B ) )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ A @ B4 )
% 0.26/0.48 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % boolean_algebra_cancel.sup2
% 0.26/0.48 thf(fact_138_sup_Oassoc,axiom,
% 0.26/0.48 ! [A: nat,B: nat,C: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
% 0.26/0.48 = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.assoc
% 0.26/0.48 thf(fact_139_sup_Oassoc,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ C )
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.assoc
% 0.26/0.48 thf(fact_140_sup__assoc,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 0.26/0.48 = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_assoc
% 0.26/0.48 thf(fact_141_sup__assoc,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 0.26/0.48 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_assoc
% 0.26/0.48 thf(fact_142_sup_Ocommute,axiom,
% 0.26/0.48 ( sup_sup_nat
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.commute
% 0.26/0.48 thf(fact_143_sup_Ocommute,axiom,
% 0.26/0.48 ( sup_su638957495lle_hf
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.commute
% 0.26/0.48 thf(fact_144_sup__commute,axiom,
% 0.26/0.48 ( sup_sup_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_commute
% 0.26/0.48 thf(fact_145_sup__commute,axiom,
% 0.26/0.48 ( sup_su638957495lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_commute
% 0.26/0.48 thf(fact_146_sup_Oleft__commute,axiom,
% 0.26/0.48 ! [B: nat,A: nat,C: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
% 0.26/0.48 = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.left_commute
% 0.26/0.48 thf(fact_147_sup_Oleft__commute,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A @ C ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.left_commute
% 0.26/0.48 thf(fact_148_sup__left__commute,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 0.26/0.48 = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_left_commute
% 0.26/0.48 thf(fact_149_sup__left__commute,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_left_commute
% 0.26/0.48 thf(fact_150_distrib__imp1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) )
% 0.26/0.48 = ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) )
% 0.26/0.48 => ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 0.26/0.48 = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_imp1
% 0.26/0.48 thf(fact_151_distrib__imp1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X2 @ ( sup_su638957495lle_hf @ Y2 @ Z2 ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ ( inf_in956532509lle_hf @ X2 @ Z2 ) ) )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_imp1
% 0.26/0.48 thf(fact_152_distrib__imp2,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
% 0.26/0.48 = ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) )
% 0.26/0.48 => ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 0.26/0.48 = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_imp2
% 0.26/0.48 thf(fact_153_distrib__imp2,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z2 ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X2 @ Y2 ) @ ( sup_su638957495lle_hf @ X2 @ Z2 ) ) )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_imp2
% 0.26/0.48 thf(fact_154_inf__sup__distrib1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 0.26/0.48 = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_distrib1
% 0.26/0.48 thf(fact_155_inf__sup__distrib1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_distrib1
% 0.26/0.48 thf(fact_156_inf__sup__distrib2,axiom,
% 0.26/0.48 ! [Y: nat,Z: nat,X: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
% 0.26/0.48 = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_distrib2
% 0.26/0.48 thf(fact_157_inf__sup__distrib2,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X )
% 0.26/0.48 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_distrib2
% 0.26/0.48 thf(fact_158_sup__inf__distrib1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 0.26/0.48 = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_distrib1
% 0.26/0.48 thf(fact_159_sup__inf__distrib1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_distrib1
% 0.26/0.48 thf(fact_160_sup__inf__distrib2,axiom,
% 0.26/0.48 ! [Y: nat,Z: nat,X: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
% 0.26/0.48 = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_distrib2
% 0.26/0.48 thf(fact_161_sup__inf__distrib2,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ Z ) @ X )
% 0.26/0.48 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_inf_distrib2
% 0.26/0.48 thf(fact_162_diff__self__eq__0,axiom,
% 0.26/0.48 ! [M: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ M @ M )
% 0.26/0.48 = zero_zero_nat ) ).
% 0.26/0.48
% 0.26/0.48 % diff_self_eq_0
% 0.26/0.48 thf(fact_163_diff__0__eq__0,axiom,
% 0.26/0.48 ! [N: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.26/0.48 = zero_zero_nat ) ).
% 0.26/0.48
% 0.26/0.48 % diff_0_eq_0
% 0.26/0.48 thf(fact_164_PrimReplace__iff,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U4 @ A4 )
% 0.26/0.48 => ( ( R @ U4 @ V3 )
% 0.26/0.48 => ( ( R @ U4 @ V4 )
% 0.26/0.48 => ( V4 = V3 ) ) ) )
% 0.26/0.48 => ( ( hF_Mirabelle_hmem @ V @ ( hF_Mir1248913145eplace @ A4 @ R ) )
% 0.26/0.48 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.26/0.48 & ( R @ U2 @ V ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % PrimReplace_iff
% 0.26/0.48 thf(fact_165_diffs0__imp__equal,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ( minus_minus_nat @ M @ N )
% 0.26/0.48 = zero_zero_nat )
% 0.26/0.48 => ( ( ( minus_minus_nat @ N @ M )
% 0.26/0.48 = zero_zero_nat )
% 0.26/0.48 => ( M = N ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % diffs0_imp_equal
% 0.26/0.48 thf(fact_166_minus__nat_Odiff__0,axiom,
% 0.26/0.48 ! [M: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.26/0.48 = M ) ).
% 0.26/0.48
% 0.26/0.48 % minus_nat.diff_0
% 0.26/0.48 thf(fact_167_zero__natural_Orsp,axiom,
% 0.26/0.48 zero_zero_nat = zero_zero_nat ).
% 0.26/0.48
% 0.26/0.48 % zero_natural.rsp
% 0.26/0.48 thf(fact_168_diff__commute,axiom,
% 0.26/0.48 ! [I: nat,J: nat,K: nat] :
% 0.26/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.26/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_commute
% 0.26/0.48 thf(fact_169_less__eq__insert1__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ ( hF_Mirabelle_hinsert @ X @ Y ) @ Z )
% 0.26/0.48 = ( ( hF_Mirabelle_hmem @ X @ Z )
% 0.26/0.48 & ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % less_eq_insert1_iff
% 0.26/0.48 thf(fact_170_hmem__def,axiom,
% 0.26/0.48 ( hF_Mirabelle_hmem
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hmem_def
% 0.26/0.48 thf(fact_171_diff__is__0__eq,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ( minus_minus_nat @ M @ N )
% 0.26/0.48 = zero_zero_nat )
% 0.26/0.48 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_is_0_eq
% 0.26/0.48 thf(fact_172_le__zero__eq,axiom,
% 0.26/0.48 ! [N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.26/0.48 = ( N = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_zero_eq
% 0.26/0.48 thf(fact_173_inf_Obounded__iff,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 0.26/0.48 = ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 & ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.bounded_iff
% 0.26/0.48 thf(fact_174_inf_Obounded__iff,axiom,
% 0.26/0.48 ! [A: nat,B: nat,C: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 0.26/0.48 = ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 & ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.bounded_iff
% 0.26/0.48 thf(fact_175_le__inf__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 0.26/0.48 = ( ( ord_le976219883lle_hf @ X @ Y )
% 0.26/0.48 & ( ord_le976219883lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_inf_iff
% 0.26/0.48 thf(fact_176_le__inf__iff,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 0.26/0.48 = ( ( ord_less_eq_nat @ X @ Y )
% 0.26/0.48 & ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_inf_iff
% 0.26/0.48 thf(fact_177_sup_Obounded__iff,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
% 0.26/0.48 = ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 & ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.bounded_iff
% 0.26/0.48 thf(fact_178_sup_Obounded__iff,axiom,
% 0.26/0.48 ! [B: nat,C: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
% 0.26/0.48 = ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.bounded_iff
% 0.26/0.48 thf(fact_179_le__sup__iff,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 0.26/0.48 = ( ( ord_le976219883lle_hf @ X @ Z )
% 0.26/0.48 & ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_sup_iff
% 0.26/0.48 thf(fact_180_le__sup__iff,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 0.26/0.48 = ( ( ord_less_eq_nat @ X @ Z )
% 0.26/0.48 & ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_sup_iff
% 0.26/0.48 thf(fact_181_bot__nat__0_Oextremum,axiom,
% 0.26/0.48 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.26/0.48
% 0.26/0.48 % bot_nat_0.extremum
% 0.26/0.48 thf(fact_182_le0,axiom,
% 0.26/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.26/0.48
% 0.26/0.48 % le0
% 0.26/0.48 thf(fact_183_diff__diff__cancel,axiom,
% 0.26/0.48 ! [I: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ I @ N )
% 0.26/0.48 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.26/0.48 = I ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_diff_cancel
% 0.26/0.48 thf(fact_184_hsubsetI,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X2 @ A4 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ X2 @ B4 ) )
% 0.26/0.48 => ( ord_le976219883lle_hf @ A4 @ B4 ) ) ).
% 0.26/0.48
% 0.26/0.48 % hsubsetI
% 0.26/0.48 thf(fact_185_less__eq__hempty,axiom,
% 0.26/0.48 ! [U: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ U @ zero_z189798548lle_hf )
% 0.26/0.48 = ( U = zero_z189798548lle_hf ) ) ).
% 0.26/0.48
% 0.26/0.48 % less_eq_hempty
% 0.26/0.48 thf(fact_186_diff__is__0__eq_H,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.26/0.48 => ( ( minus_minus_nat @ M @ N )
% 0.26/0.48 = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_is_0_eq'
% 0.26/0.48 thf(fact_187_diff__le__mono2,axiom,
% 0.26/0.48 ! [M: nat,N: nat,L: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.26/0.48 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_le_mono2
% 0.26/0.48 thf(fact_188_le__diff__iff_H,axiom,
% 0.26/0.48 ! [A: nat,C: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ C )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.26/0.48 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.26/0.48 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_diff_iff'
% 0.26/0.48 thf(fact_189_diff__le__self,axiom,
% 0.26/0.48 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.26/0.48
% 0.26/0.48 % diff_le_self
% 0.26/0.48 thf(fact_190_diff__le__mono,axiom,
% 0.26/0.48 ! [M: nat,N: nat,L: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.26/0.48 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % diff_le_mono
% 0.26/0.48 thf(fact_191_Nat_Odiff__diff__eq,axiom,
% 0.26/0.48 ! [K: nat,M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ K @ M )
% 0.26/0.48 => ( ( ord_less_eq_nat @ K @ N )
% 0.26/0.48 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.26/0.48 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Nat.diff_diff_eq
% 0.26/0.48 thf(fact_192_le__diff__iff,axiom,
% 0.26/0.48 ! [K: nat,M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ K @ M )
% 0.26/0.48 => ( ( ord_less_eq_nat @ K @ N )
% 0.26/0.48 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.26/0.48 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_diff_iff
% 0.26/0.48 thf(fact_193_eq__diff__iff,axiom,
% 0.26/0.48 ! [K: nat,M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ K @ M )
% 0.26/0.48 => ( ( ord_less_eq_nat @ K @ N )
% 0.26/0.48 => ( ( ( minus_minus_nat @ M @ K )
% 0.26/0.48 = ( minus_minus_nat @ N @ K ) )
% 0.26/0.48 = ( M = N ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_diff_iff
% 0.26/0.48 thf(fact_194_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.26/0.48 => ( A = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % bot_nat_0.extremum_uniqueI
% 0.26/0.48 thf(fact_195_bot__nat__0_Oextremum__unique,axiom,
% 0.26/0.48 ! [A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.26/0.48 = ( A = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % bot_nat_0.extremum_unique
% 0.26/0.48 thf(fact_196_le__0__eq,axiom,
% 0.26/0.48 ! [N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.26/0.48 = ( N = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_0_eq
% 0.26/0.48 thf(fact_197_less__eq__nat_Osimps_I1_J,axiom,
% 0.26/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.26/0.48
% 0.26/0.48 % less_eq_nat.simps(1)
% 0.26/0.48 thf(fact_198_Nat_Oex__has__greatest__nat,axiom,
% 0.26/0.48 ! [P: nat > $o,K: nat,B: nat] :
% 0.26/0.48 ( ( P @ K )
% 0.26/0.48 => ( ! [Y2: nat] :
% 0.26/0.48 ( ( P @ Y2 )
% 0.26/0.48 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.26/0.48 => ? [X2: nat] :
% 0.26/0.48 ( ( P @ X2 )
% 0.26/0.48 & ! [Y5: nat] :
% 0.26/0.48 ( ( P @ Y5 )
% 0.26/0.48 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Nat.ex_has_greatest_nat
% 0.26/0.48 thf(fact_199_nat__le__linear,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.26/0.48 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.26/0.48
% 0.26/0.48 % nat_le_linear
% 0.26/0.48 thf(fact_200_le__antisym,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.26/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.26/0.48 => ( M = N ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_antisym
% 0.26/0.48 thf(fact_201_eq__imp__le,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( M = N )
% 0.26/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_imp_le
% 0.26/0.48 thf(fact_202_le__trans,axiom,
% 0.26/0.48 ! [I: nat,J: nat,K: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ I @ J )
% 0.26/0.48 => ( ( ord_less_eq_nat @ J @ K )
% 0.26/0.48 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_trans
% 0.26/0.48 thf(fact_203_le__refl,axiom,
% 0.26/0.48 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.26/0.48
% 0.26/0.48 % le_refl
% 0.26/0.48 thf(fact_204_canonically__ordered__monoid__add__class_Ozero__le,axiom,
% 0.26/0.48 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.26/0.48
% 0.26/0.48 % canonically_ordered_monoid_add_class.zero_le
% 0.26/0.48 thf(fact_205_inf_OcoboundedI2,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ C )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.coboundedI2
% 0.26/0.48 thf(fact_206_inf_OcoboundedI2,axiom,
% 0.26/0.48 ! [B: nat,C: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ C )
% 0.26/0.48 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.coboundedI2
% 0.26/0.48 thf(fact_207_inf_OcoboundedI1,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ C )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.coboundedI1
% 0.26/0.48 thf(fact_208_inf_OcoboundedI1,axiom,
% 0.26/0.48 ! [A: nat,C: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ C )
% 0.26/0.48 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.coboundedI1
% 0.26/0.48 thf(fact_209_inf_Oabsorb__iff2,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ A3 @ B3 )
% 0.26/0.48 = B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb_iff2
% 0.26/0.48 thf(fact_210_inf_Oabsorb__iff2,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [B3: nat,A3: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ A3 @ B3 )
% 0.26/0.48 = B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb_iff2
% 0.26/0.48 thf(fact_211_inf_Oabsorb__iff1,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ A3 @ B3 )
% 0.26/0.48 = A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb_iff1
% 0.26/0.48 thf(fact_212_inf_Oabsorb__iff1,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ A3 @ B3 )
% 0.26/0.48 = A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb_iff1
% 0.26/0.48 thf(fact_213_inf_Ocobounded2,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B ) ).
% 0.26/0.48
% 0.26/0.48 % inf.cobounded2
% 0.26/0.48 thf(fact_214_inf_Ocobounded2,axiom,
% 0.26/0.48 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% 0.26/0.48
% 0.26/0.48 % inf.cobounded2
% 0.26/0.48 thf(fact_215_inf_Ocobounded1,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ A ) ).
% 0.26/0.48
% 0.26/0.48 % inf.cobounded1
% 0.26/0.48 thf(fact_216_inf_Ocobounded1,axiom,
% 0.26/0.48 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% 0.26/0.48
% 0.26/0.48 % inf.cobounded1
% 0.26/0.48 thf(fact_217_inf_Oorder__iff,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.48 ( A3
% 0.26/0.48 = ( inf_in956532509lle_hf @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.order_iff
% 0.26/0.48 thf(fact_218_inf_Oorder__iff,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] :
% 0.26/0.48 ( A3
% 0.26/0.48 = ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.order_iff
% 0.26/0.48 thf(fact_219_inf__greatest,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ Y )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ X @ Z )
% 0.26/0.48 => ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_greatest
% 0.26/0.48 thf(fact_220_inf__greatest,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X @ Z )
% 0.26/0.48 => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_greatest
% 0.26/0.48 thf(fact_221_inf_OboundedI,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ A @ C )
% 0.26/0.48 => ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.boundedI
% 0.26/0.48 thf(fact_222_inf_OboundedI,axiom,
% 0.26/0.48 ! [A: nat,B: nat,C: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ( ( ord_less_eq_nat @ A @ C )
% 0.26/0.48 => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.boundedI
% 0.26/0.48 thf(fact_223_inf_OboundedE,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 0.26/0.48 => ~ ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ~ ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.boundedE
% 0.26/0.48 thf(fact_224_inf_OboundedE,axiom,
% 0.26/0.48 ! [A: nat,B: nat,C: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.boundedE
% 0.26/0.48 thf(fact_225_inf__absorb2,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ Y @ X )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ X @ Y )
% 0.26/0.48 = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_absorb2
% 0.26/0.48 thf(fact_226_inf__absorb2,axiom,
% 0.26/0.48 ! [Y: nat,X: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y @ X )
% 0.26/0.48 => ( ( inf_inf_nat @ X @ Y )
% 0.26/0.48 = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_absorb2
% 0.26/0.48 thf(fact_227_inf__absorb1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ Y )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ X @ Y )
% 0.26/0.48 = X ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_absorb1
% 0.26/0.48 thf(fact_228_inf__absorb1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.26/0.48 => ( ( inf_inf_nat @ X @ Y )
% 0.26/0.48 = X ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_absorb1
% 0.26/0.48 thf(fact_229_inf_Oabsorb2,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ A @ B )
% 0.26/0.48 = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb2
% 0.26/0.48 thf(fact_230_inf_Oabsorb2,axiom,
% 0.26/0.48 ! [B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ( ( inf_inf_nat @ A @ B )
% 0.26/0.48 = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb2
% 0.26/0.48 thf(fact_231_inf_Oabsorb1,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ A @ B )
% 0.26/0.48 = A ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb1
% 0.26/0.48 thf(fact_232_inf_Oabsorb1,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ( ( inf_inf_nat @ A @ B )
% 0.26/0.48 = A ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.absorb1
% 0.26/0.48 thf(fact_233_le__iff__inf,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( inf_in956532509lle_hf @ X3 @ Y3 )
% 0.26/0.48 = X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_iff_inf
% 0.26/0.48 thf(fact_234_le__iff__inf,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] :
% 0.26/0.48 ( ( inf_inf_nat @ X3 @ Y3 )
% 0.26/0.48 = X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_iff_inf
% 0.26/0.48 thf(fact_235_inf__unique,axiom,
% 0.26/0.48 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ X2 )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ Y2 )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X2 @ Y2 )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ X2 @ Z2 )
% 0.26/0.48 => ( ord_le976219883lle_hf @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 0.26/0.48 => ( ( inf_in956532509lle_hf @ X @ Y )
% 0.26/0.48 = ( F @ X @ Y ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_unique
% 0.26/0.48 thf(fact_236_inf__unique,axiom,
% 0.26/0.48 ! [F: nat > nat > nat,X: nat,Y: nat] :
% 0.26/0.48 ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
% 0.26/0.48 => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
% 0.26/0.48 => ( ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X2 @ Z2 )
% 0.26/0.48 => ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 0.26/0.48 => ( ( inf_inf_nat @ X @ Y )
% 0.26/0.48 = ( F @ X @ Y ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_unique
% 0.26/0.48 thf(fact_237_inf_OorderI,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ B ) )
% 0.26/0.48 => ( ord_le976219883lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.orderI
% 0.26/0.48 thf(fact_238_inf_OorderI,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( A
% 0.26/0.48 = ( inf_inf_nat @ A @ B ) )
% 0.26/0.48 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.orderI
% 0.26/0.48 thf(fact_239_inf_OorderE,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ( A
% 0.26/0.48 = ( inf_in956532509lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.orderE
% 0.26/0.48 thf(fact_240_inf_OorderE,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ( A
% 0.26/0.48 = ( inf_inf_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf.orderE
% 0.26/0.48 thf(fact_241_le__infI2,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ X )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI2
% 0.26/0.48 thf(fact_242_le__infI2,axiom,
% 0.26/0.48 ! [B: nat,X: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ X )
% 0.26/0.48 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI2
% 0.26/0.48 thf(fact_243_le__infI1,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ X )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI1
% 0.26/0.48 thf(fact_244_le__infI1,axiom,
% 0.26/0.48 ! [A: nat,X: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ X )
% 0.26/0.48 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI1
% 0.26/0.48 thf(fact_245_inf__mono,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ C )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ B @ D )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ ( inf_in956532509lle_hf @ C @ D ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_mono
% 0.26/0.48 thf(fact_246_inf__mono,axiom,
% 0.26/0.48 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ C )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B @ D )
% 0.26/0.48 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_mono
% 0.26/0.48 thf(fact_247_le__infI,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ A )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ X @ B )
% 0.26/0.48 => ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI
% 0.26/0.48 thf(fact_248_le__infI,axiom,
% 0.26/0.48 ! [X: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ A )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X @ B )
% 0.26/0.48 => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infI
% 0.26/0.48 thf(fact_249_le__infE,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) )
% 0.26/0.48 => ~ ( ( ord_le976219883lle_hf @ X @ A )
% 0.26/0.48 => ~ ( ord_le976219883lle_hf @ X @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infE
% 0.26/0.48 thf(fact_250_le__infE,axiom,
% 0.26/0.48 ! [X: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ X @ A )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_infE
% 0.26/0.48 thf(fact_251_inf__le2,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% 0.26/0.48
% 0.26/0.48 % inf_le2
% 0.26/0.48 thf(fact_252_inf__le2,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% 0.26/0.48
% 0.26/0.48 % inf_le2
% 0.26/0.48 thf(fact_253_inf__le1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_le1
% 0.26/0.48 thf(fact_254_inf__le1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_le1
% 0.26/0.48 thf(fact_255_inf__sup__ord_I1_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(1)
% 0.26/0.48 thf(fact_256_inf__sup__ord_I1_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(1)
% 0.26/0.48 thf(fact_257_inf__sup__ord_I2_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(2)
% 0.26/0.48 thf(fact_258_inf__sup__ord_I2_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(2)
% 0.26/0.48 thf(fact_259_sup_OcoboundedI2,axiom,
% 0.26/0.48 ! [C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ C @ B )
% 0.26/0.48 => ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.coboundedI2
% 0.26/0.48 thf(fact_260_sup_OcoboundedI2,axiom,
% 0.26/0.48 ! [C: nat,B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ C @ B )
% 0.26/0.48 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.coboundedI2
% 0.26/0.48 thf(fact_261_sup_OcoboundedI1,axiom,
% 0.26/0.48 ! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ C @ A )
% 0.26/0.48 => ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.coboundedI1
% 0.26/0.48 thf(fact_262_sup_OcoboundedI1,axiom,
% 0.26/0.48 ! [C: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ C @ A )
% 0.26/0.48 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.coboundedI1
% 0.26/0.48 thf(fact_263_sup_Oabsorb__iff2,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ A3 @ B3 )
% 0.26/0.48 = B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb_iff2
% 0.26/0.48 thf(fact_264_sup_Oabsorb__iff2,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ A3 @ B3 )
% 0.26/0.48 = B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb_iff2
% 0.26/0.48 thf(fact_265_sup_Oabsorb__iff1,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ A3 @ B3 )
% 0.26/0.48 = A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb_iff1
% 0.26/0.48 thf(fact_266_sup_Oabsorb__iff1,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [B3: nat,A3: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ A3 @ B3 )
% 0.26/0.48 = A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb_iff1
% 0.26/0.48 thf(fact_267_sup_Ocobounded2,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ B @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.cobounded2
% 0.26/0.48 thf(fact_268_sup_Ocobounded2,axiom,
% 0.26/0.48 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.cobounded2
% 0.26/0.48 thf(fact_269_sup_Ocobounded1,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.cobounded1
% 0.26/0.48 thf(fact_270_sup_Ocobounded1,axiom,
% 0.26/0.48 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.cobounded1
% 0.26/0.48 thf(fact_271_sup_Oorder__iff,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.26/0.48 ( A3
% 0.26/0.48 = ( sup_su638957495lle_hf @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.order_iff
% 0.26/0.48 thf(fact_272_sup_Oorder__iff,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [B3: nat,A3: nat] :
% 0.26/0.48 ( A3
% 0.26/0.48 = ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.order_iff
% 0.26/0.48 thf(fact_273_sup_OboundedI,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ C @ A )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.boundedI
% 0.26/0.48 thf(fact_274_sup_OboundedI,axiom,
% 0.26/0.48 ! [B: nat,A: nat,C: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ( ( ord_less_eq_nat @ C @ A )
% 0.26/0.48 => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.boundedI
% 0.26/0.48 thf(fact_275_sup_OboundedE,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
% 0.26/0.48 => ~ ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ~ ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.boundedE
% 0.26/0.48 thf(fact_276_sup_OboundedE,axiom,
% 0.26/0.48 ! [B: nat,C: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.boundedE
% 0.26/0.48 thf(fact_277_sup__absorb2,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ Y )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ X @ Y )
% 0.26/0.48 = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_absorb2
% 0.26/0.48 thf(fact_278_sup__absorb2,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.26/0.48 => ( ( sup_sup_nat @ X @ Y )
% 0.26/0.48 = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_absorb2
% 0.26/0.48 thf(fact_279_sup__absorb1,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ Y @ X )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ X @ Y )
% 0.26/0.48 = X ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_absorb1
% 0.26/0.48 thf(fact_280_sup__absorb1,axiom,
% 0.26/0.48 ! [Y: nat,X: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y @ X )
% 0.26/0.48 => ( ( sup_sup_nat @ X @ Y )
% 0.26/0.48 = X ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_absorb1
% 0.26/0.48 thf(fact_281_sup_Oabsorb2,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ A @ B )
% 0.26/0.48 = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb2
% 0.26/0.48 thf(fact_282_sup_Oabsorb2,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ( ( sup_sup_nat @ A @ B )
% 0.26/0.48 = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb2
% 0.26/0.48 thf(fact_283_sup_Oabsorb1,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ A @ B )
% 0.26/0.48 = A ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb1
% 0.26/0.48 thf(fact_284_sup_Oabsorb1,axiom,
% 0.26/0.48 ! [B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ( ( sup_sup_nat @ A @ B )
% 0.26/0.48 = A ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.absorb1
% 0.26/0.48 thf(fact_285_sup__unique,axiom,
% 0.26/0.48 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X2 @ ( F @ X2 @ Y2 ) )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y2 @ ( F @ X2 @ Y2 ) )
% 0.26/0.48 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ Y2 @ X2 )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ Z2 @ X2 )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 0.26/0.48 => ( ( sup_su638957495lle_hf @ X @ Y )
% 0.26/0.48 = ( F @ X @ Y ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_unique
% 0.26/0.48 thf(fact_286_sup__unique,axiom,
% 0.26/0.48 ! [F: nat > nat > nat,X: nat,Y: nat] :
% 0.26/0.48 ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
% 0.26/0.48 => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
% 0.26/0.48 => ( ! [X2: nat,Y2: nat,Z2: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Z2 @ X2 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 0.26/0.48 => ( ( sup_sup_nat @ X @ Y )
% 0.26/0.48 = ( F @ X @ Y ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_unique
% 0.26/0.48 thf(fact_287_sup_OorderI,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ B ) )
% 0.26/0.48 => ( ord_le976219883lle_hf @ B @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.orderI
% 0.26/0.48 thf(fact_288_sup_OorderI,axiom,
% 0.26/0.48 ! [A: nat,B: nat] :
% 0.26/0.48 ( ( A
% 0.26/0.48 = ( sup_sup_nat @ A @ B ) )
% 0.26/0.48 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.orderI
% 0.26/0.48 thf(fact_289_sup_OorderE,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ( A
% 0.26/0.48 = ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.orderE
% 0.26/0.48 thf(fact_290_sup_OorderE,axiom,
% 0.26/0.48 ! [B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ( A
% 0.26/0.48 = ( sup_sup_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.orderE
% 0.26/0.48 thf(fact_291_le__iff__sup,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( sup_su638957495lle_hf @ X3 @ Y3 )
% 0.26/0.48 = Y3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_iff_sup
% 0.26/0.48 thf(fact_292_le__iff__sup,axiom,
% 0.26/0.48 ( ord_less_eq_nat
% 0.26/0.48 = ( ^ [X3: nat,Y3: nat] :
% 0.26/0.48 ( ( sup_sup_nat @ X3 @ Y3 )
% 0.26/0.48 = Y3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_iff_sup
% 0.26/0.48 thf(fact_293_sup__least,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ Y @ X )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ Z @ X )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_least
% 0.26/0.48 thf(fact_294_sup__least,axiom,
% 0.26/0.48 ! [Y: nat,X: nat,Z: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y @ X )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Z @ X )
% 0.26/0.48 => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_least
% 0.26/0.48 thf(fact_295_sup__mono,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ C )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ B @ D )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ ( sup_su638957495lle_hf @ C @ D ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_mono
% 0.26/0.48 thf(fact_296_sup__mono,axiom,
% 0.26/0.48 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ C )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B @ D )
% 0.26/0.48 => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_mono
% 0.26/0.48 thf(fact_297_sup_Omono,axiom,
% 0.26/0.48 ! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,D: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ C @ A )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ D @ B )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ C @ D ) @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.mono
% 0.26/0.48 thf(fact_298_sup_Omono,axiom,
% 0.26/0.48 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ C @ A )
% 0.26/0.48 => ( ( ord_less_eq_nat @ D @ B )
% 0.26/0.48 => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup.mono
% 0.26/0.48 thf(fact_299_le__supI2,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ B )
% 0.26/0.48 => ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI2
% 0.26/0.48 thf(fact_300_le__supI2,axiom,
% 0.26/0.48 ! [X: nat,B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ B )
% 0.26/0.48 => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI2
% 0.26/0.48 thf(fact_301_le__supI1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ X @ A )
% 0.26/0.48 => ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI1
% 0.26/0.48 thf(fact_302_le__supI1,axiom,
% 0.26/0.48 ! [X: nat,A: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X @ A )
% 0.26/0.48 => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI1
% 0.26/0.48 thf(fact_303_sup__ge2,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_ge2
% 0.26/0.48 thf(fact_304_sup__ge2,axiom,
% 0.26/0.48 ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_ge2
% 0.26/0.48 thf(fact_305_sup__ge1,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_ge1
% 0.26/0.48 thf(fact_306_sup__ge1,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % sup_ge1
% 0.26/0.48 thf(fact_307_le__supI,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A @ X )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ B @ X )
% 0.26/0.48 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI
% 0.26/0.48 thf(fact_308_le__supI,axiom,
% 0.26/0.48 ! [A: nat,X: nat,B: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A @ X )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B @ X )
% 0.26/0.48 => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supI
% 0.26/0.48 thf(fact_309_le__supE,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X )
% 0.26/0.48 => ~ ( ( ord_le976219883lle_hf @ A @ X )
% 0.26/0.48 => ~ ( ord_le976219883lle_hf @ B @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supE
% 0.26/0.48 thf(fact_310_le__supE,axiom,
% 0.26/0.48 ! [A: nat,B: nat,X: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ A @ X )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_supE
% 0.26/0.48 thf(fact_311_inf__sup__ord_I3_J,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(3)
% 0.26/0.48 thf(fact_312_inf__sup__ord_I3_J,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(3)
% 0.26/0.48 thf(fact_313_inf__sup__ord_I4_J,axiom,
% 0.26/0.48 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(4)
% 0.26/0.48 thf(fact_314_inf__sup__ord_I4_J,axiom,
% 0.26/0.48 ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % inf_sup_ord(4)
% 0.26/0.48 thf(fact_315_less__eq__hf__def,axiom,
% 0.26/0.48 ( ord_le976219883lle_hf
% 0.26/0.48 = ( ^ [A6: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.26/0.48 ! [X3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ X3 @ A6 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ X3 @ B6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % less_eq_hf_def
% 0.26/0.48 thf(fact_316_rev__hsubsetD,axiom,
% 0.26/0.48 ! [C: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_hmem @ C @ A4 )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % rev_hsubsetD
% 0.26/0.48 thf(fact_317_hsubsetCE,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 0.26/0.48 => ( ( hF_Mirabelle_hmem @ C @ A4 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hsubsetCE
% 0.26/0.48 thf(fact_318_hsubsetD,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 0.26/0.48 => ( ( hF_Mirabelle_hmem @ C @ A4 )
% 0.26/0.48 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hsubsetD
% 0.26/0.48 thf(fact_319_HF__Mirabelle__glliljednj_Ozero__le,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ zero_z189798548lle_hf @ X ) ).
% 0.26/0.48
% 0.26/0.48 % HF_Mirabelle_glliljednj.zero_le
% 0.26/0.48 thf(fact_320_hf__equalityE,axiom,
% 0.26/0.48 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( A4 = B4 )
% 0.26/0.48 => ~ ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 0.26/0.48 => ~ ( ord_le976219883lle_hf @ B4 @ A4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % hf_equalityE
% 0.26/0.48 thf(fact_321_distrib__inf__le,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) @ ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_inf_le
% 0.26/0.48 thf(fact_322_distrib__inf__le,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_inf_le
% 0.26/0.48 thf(fact_323_distrib__sup__le,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) @ ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_sup_le
% 0.26/0.48 thf(fact_324_distrib__sup__le,axiom,
% 0.26/0.48 ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % distrib_sup_le
% 0.26/0.48 thf(fact_325_less__eq__insert2__iff,axiom,
% 0.26/0.48 ! [Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ Z @ ( hF_Mirabelle_hinsert @ X @ Y ) )
% 0.26/0.48 = ( ( ord_le976219883lle_hf @ Z @ Y )
% 0.26/0.48 | ? [U2: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ( hF_Mirabelle_hinsert @ X @ U2 )
% 0.26/0.48 = Z )
% 0.26/0.48 & ~ ( hF_Mirabelle_hmem @ X @ U2 )
% 0.26/0.48 & ( ord_le976219883lle_hf @ U2 @ Y ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % less_eq_insert2_iff
% 0.26/0.48 thf(fact_326_order__refl,axiom,
% 0.26/0.48 ! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ X ) ).
% 0.26/0.48
% 0.26/0.48 % order_refl
% 0.26/0.48 thf(fact_327_order__refl,axiom,
% 0.26/0.48 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.26/0.48
% 0.26/0.48 % order_refl
% 0.26/0.48 thf(fact_328_le__numeral__extra_I3_J,axiom,
% 0.26/0.48 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.26/0.48
% 0.26/0.48 % le_numeral_extra(3)
% 0.26/0.48 thf(fact_329_HF__hfset,axiom,
% 0.26/0.48 ! [A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 0.26/0.48 = A ) ).
% 0.26/0.48
% 0.26/0.48 % HF_hfset
% 0.26/0.48 thf(fact_330_of__nat__le__0__iff,axiom,
% 0.26/0.48 ! [M: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 0.26/0.48 = ( M = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_le_0_iff
% 0.26/0.48 thf(fact_331_of__nat__eq__iff,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ( semiri1382578993at_nat @ M )
% 0.26/0.48 = ( semiri1382578993at_nat @ N ) )
% 0.26/0.48 = ( M = N ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_eq_iff
% 0.26/0.48 thf(fact_332_of__nat__0,axiom,
% 0.26/0.48 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.26/0.48 = zero_zero_nat ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_0
% 0.26/0.48 thf(fact_333_of__nat__0__eq__iff,axiom,
% 0.26/0.48 ! [N: nat] :
% 0.26/0.48 ( ( zero_zero_nat
% 0.26/0.48 = ( semiri1382578993at_nat @ N ) )
% 0.26/0.48 = ( zero_zero_nat = N ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_0_eq_iff
% 0.26/0.48 thf(fact_334_of__nat__eq__0__iff,axiom,
% 0.26/0.48 ! [M: nat] :
% 0.26/0.48 ( ( ( semiri1382578993at_nat @ M )
% 0.26/0.48 = zero_zero_nat )
% 0.26/0.48 = ( M = zero_zero_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_eq_0_iff
% 0.26/0.48 thf(fact_335_of__nat__le__iff,axiom,
% 0.26/0.48 ! [M: nat,N: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.26/0.48 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_le_iff
% 0.26/0.48 thf(fact_336_of__nat__0__le__iff,axiom,
% 0.26/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_0_le_iff
% 0.26/0.48 thf(fact_337_of__nat__mono,axiom,
% 0.26/0.48 ! [I: nat,J: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ I @ J )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_mono
% 0.26/0.48 thf(fact_338_of__nat__diff,axiom,
% 0.26/0.48 ! [N: nat,M: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.26/0.48 => ( ( semiri1382578993at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.26/0.48 = ( minus_minus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % of_nat_diff
% 0.26/0.48 thf(fact_339_dual__order_Oantisym,axiom,
% 0.26/0.48 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B @ A )
% 0.26/0.48 => ( ( ord_le976219883lle_hf @ A @ B )
% 0.26/0.48 => ( A = B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.antisym
% 0.26/0.48 thf(fact_340_dual__order_Oantisym,axiom,
% 0.26/0.48 ! [B: nat,A: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.26/0.48 => ( ( ord_less_eq_nat @ A @ B )
% 0.26/0.48 => ( A = B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.antisym
% 0.26/0.48 thf(fact_341_dual__order_Oeq__iff,axiom,
% 0.26/0.48 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 0.26/0.48 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.48 ( ( ord_le976219883lle_hf @ B3 @ A3 )
% 0.26/0.48 & ( ord_le976219883lle_hf @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.eq_iff
% 0.26/0.48 thf(fact_342_dual__order_Oeq__iff,axiom,
% 0.26/0.48 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 0.26/0.48 = ( ^ [A3: nat,B3: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.26/0.48 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.eq_iff
% 0.26/0.48
% 0.26/0.48 % Conjectures (1)
% 0.26/0.48 thf(conj_0,conjecture,
% 0.26/0.48 ( ( hF_Mirabelle_hpair @ x @ y )
% 0.26/0.48 != zero_z189798548lle_hf ) ).
% 0.26/0.48
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 ------------------------
% 2.61/1.56 ---- Embedded in THF ---
% 2.61/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.56 %%% Generated on Tue Feb 24 08:25:10 EST 2026
% 2.61/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.56 %%% Logic specification used:
% 2.61/1.56 %%% thf(spec, logic, $$dhol).
% 2.61/1.56
% 2.61/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2DTF1223.p
% See solution above
% 2.61/1.56 ------------------------
% 2.61/1.58 ---- Cleaned THF ---
% 2.61/1.59 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.61/1.59 set_HF_Mirabelle_hf: $tType ).
% 2.61/1.59
% 2.61/1.59 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 hF_Mirabelle_hf: $tType ).
% 2.61/1.59
% 2.61/1.59 thf(ty_n_t__Nat__Onat,type,
% 2.61/1.59 nat: $tType ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Groups_Ominus__class_Ominus_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.61/1.59 minus_minus_nat: nat > nat > nat ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.61/1.59 zero_zero_nat: nat ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 2.61/1.59 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 2.61/1.59 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 2.61/1.59 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 2.61/1.59 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 2.61/1.59 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 2.61/1.59 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 2.61/1.59 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 2.61/1.59 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 2.61/1.59 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 2.61/1.59 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 2.61/1.59 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 2.61/1.59 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 2.61/1.59 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 2.61/1.59 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 2.61/1.59 inf_inf_nat: nat > nat > nat ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Lattices_Osup__class_Osup_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 2.61/1.59 sup_sup_nat: nat > nat > nat ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 2.61/1.59 semiri1382578993at_nat: nat > nat ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.61/1.59 ord_less_eq_nat: nat > nat > $o ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.61/1.59 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 2.61/1.59
% 2.61/1.59 thf(sy_v_x,type,
% 2.61/1.59 x: hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(sy_v_y,type,
% 2.61/1.59 y: hF_Mirabelle_hf ).
% 2.61/1.59
% 2.61/1.59 thf(fact_0_hpair__iff,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 2.61/1.59 = ( hF_Mirabelle_hpair @ A2 @ B2 ) )
% 2.61/1.59 = ( ( A = A2 )
% 2.61/1.59 & ( B = B2 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_1_hpair__inject,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 2.61/1.59 = ( hF_Mirabelle_hpair @ A2 @ B2 ) )
% 2.61/1.59 => ~ ( ( A = A2 )
% 2.61/1.59 => ( B != B2 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_2_hpair__neq__fst,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hpair @ A @ B )
% 2.61/1.59 != A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_3_hpair__neq__snd,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hpair @ A @ B )
% 2.61/1.59 != B ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_4_hfst__conv,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_5_hsnd__conv,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
% 2.61/1.59 = B ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_6_HInter__hempty,axiom,
% 2.61/1.59 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_7_HCollect__hempty,axiom,
% 2.61/1.59 ! [P: hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_8_Replace__0,axiom,
% 2.61/1.59 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_9_HUnion__hempty,axiom,
% 2.61/1.59 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_10_zero__reorient,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( zero_z189798548lle_hf = X )
% 2.61/1.59 = ( X = zero_z189798548lle_hf ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_11_zero__reorient,axiom,
% 2.61/1.59 ! [X: nat] :
% 2.61/1.59 ( ( zero_zero_nat = X )
% 2.61/1.59 = ( X = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_12_RepFun__0,axiom,
% 2.61/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_13_hpair__def,axiom,
% 2.61/1.59 ( hF_Mirabelle_hpair
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_14_hpair__def_H,axiom,
% 2.61/1.59 ( hF_Mirabelle_hpair
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_15_singleton__eq__iff,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 2.61/1.59 = ( A = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_16_RepFun__hinsert,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A @ B ) @ F )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ ( F @ A ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_17_hinsert__commute,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y @ Z ) )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ Y @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_18_hinsert__nonempty,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hinsert @ A @ A4 )
% 2.61/1.59 != zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_19_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hinsert @ A @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 2.61/1.59 = ( ( ( A = C )
% 2.61/1.59 & ( B = D ) )
% 2.61/1.59 | ( ( A = D )
% 2.61/1.59 & ( B = C ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_20_hf__induct__ax,axiom,
% 2.61/1.59 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ zero_z189798548lle_hf )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ X2 )
% 2.61/1.59 => ! [Y2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ Y2 )
% 2.61/1.59 => ( P @ ( hF_Mirabelle_hinsert @ Y2 @ X2 ) ) ) )
% 2.61/1.59 => ( P @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_21_HInter__hinsert,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4 != zero_z189798548lle_hf )
% 2.61/1.59 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A4 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_22_Abs__hf__0,axiom,
% 2.61/1.59 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_23_HInter__iff,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4 != zero_z189798548lle_hf )
% 2.61/1.59 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A4 ) )
% 2.61/1.59 = ( ! [Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_24_HUnion__hinsert,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ ( hF_Mirabelle_HUnion @ A4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_25_hdiff__insert,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( minus_1232880740lle_hf @ A4 @ ( hF_Mirabelle_hinsert @ A @ B4 ) )
% 2.61/1.59 = ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A4 @ B4 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_26_hinsert__eq__sup,axiom,
% 2.61/1.59 ( hF_Mirabelle_hinsert
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_27_hf__induct,axiom,
% 2.61/1.59 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ zero_z189798548lle_hf )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ X2 )
% 2.61/1.59 => ( ( P @ Y2 )
% 2.61/1.59 => ( ~ ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 2.61/1.59 => ( P @ ( hF_Mirabelle_hinsert @ X2 @ Y2 ) ) ) ) )
% 2.61/1.59 => ( P @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_28_hf__cases,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( Y != zero_z189798548lle_hf )
% 2.61/1.59 => ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( Y
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_29_Replace__hunion,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ R )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A4 @ R ) @ ( hF_Mirabelle_Replace @ B4 @ R ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_30_HCollect__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A4 ) )
% 2.61/1.59 = ( ( P @ X )
% 2.61/1.59 & ( hF_Mirabelle_hmem @ X @ A4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_31_hf__equalityI,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X2 @ A )
% 2.61/1.59 = ( hF_Mirabelle_hmem @ X2 @ B ) )
% 2.61/1.59 => ( A = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_32_zero__diff,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 2.61/1.59 = zero_zero_nat ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_33_diff__zero,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ A @ A )
% 2.61/1.59 = zero_zero_nat ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_35_hmem__hinsert,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 2.61/1.59 = ( ( A = B )
% 2.61/1.59 | ( hF_Mirabelle_hmem @ A @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_36_hunion__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A @ B ) )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ X @ A )
% 2.61/1.59 | ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_37_hinter__iff,axiom,
% 2.61/1.59 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U @ ( inf_in956532509lle_hf @ X @ Y ) )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ U @ X )
% 2.61/1.59 & ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_38_RepFun__iff,axiom,
% 2.61/1.59 ! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
% 2.61/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.61/1.59 & ( V
% 2.61/1.59 = ( F @ U2 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_39_hunion__hempty__left,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A4 )
% 2.61/1.59 = A4 ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_40_hunion__hempty__right,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ A4 @ zero_z189798548lle_hf )
% 2.61/1.59 = A4 ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_41_HUnion__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A4 ) )
% 2.61/1.59 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 2.61/1.59 & ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_42_mem__Collect__eq,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 2.61/1.59 = ( P @ A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_43_Collect__mem__eq,axiom,
% 2.61/1.59 ! [A4: set_HF_Mirabelle_hf] :
% 2.61/1.59 ( ( collec2046588256lle_hf
% 2.61/1.59 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 ) )
% 2.61/1.59 = A4 ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_44_Collect__cong,axiom,
% 2.61/1.59 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( P @ X2 )
% 2.61/1.59 = ( Q @ X2 ) )
% 2.61/1.59 => ( ( collec2046588256lle_hf @ P )
% 2.61/1.59 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_45_hdiff__iff,axiom,
% 2.61/1.59 ! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U @ ( minus_1232880740lle_hf @ X @ Y ) )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ U @ X )
% 2.61/1.59 & ~ ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_46_hinter__hempty__left,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A4 )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_47_hinter__hempty__right,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ A4 @ zero_z189798548lle_hf )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_48_Replace__iff,axiom,
% 2.61/1.59 ! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_Replace @ A4 @ R ) )
% 2.61/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.61/1.59 & ( R @ U2 @ V )
% 2.61/1.59 & ! [Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( R @ U2 @ Y3 )
% 2.61/1.59 => ( Y3 = V ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_49_hdiff__zero,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( minus_1232880740lle_hf @ X @ zero_z189798548lle_hf )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_50_zero__hdiff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X )
% 2.61/1.59 = zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_51_RepFun__hunion,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ F )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A4 @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_52_HUnion__hunion,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A4 @ B4 ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A4 ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_53_diff__right__commute,axiom,
% 2.61/1.59 ! [A: nat,C: nat,B: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.61/1.59 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_54_replacement__fun,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ? [Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ! [V2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 2.61/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.61/1.59 & ( V2
% 2.61/1.59 = ( F @ U2 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_55_hmem__not__refl,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_56_comprehension,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.61/1.59 ? [Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ! [U3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 2.61/1.59 & ( P @ U3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_57_union__of__set,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ? [Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ! [U3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.61/1.59 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ Y3 @ X )
% 2.61/1.59 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_58_hmem__not__sym,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ~ ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.61/1.59 & ( hF_Mirabelle_hmem @ Y @ X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_59_binary__union,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ? [Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ! [U3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 2.61/1.59 | ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_60_replacement,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U4 @ X )
% 2.61/1.59 => ( ( R @ U4 @ V3 )
% 2.61/1.59 => ( ( R @ U4 @ V4 )
% 2.61/1.59 => ( V4 = V3 ) ) ) )
% 2.61/1.59 => ? [Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ! [V2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ V2 @ Z2 )
% 2.61/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.61/1.59 & ( R @ U2 @ V2 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_61_hmem__ne,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.61/1.59 => ( X != Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_62_hf__ext,axiom,
% 2.61/1.59 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.61/1.59 ! [X3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 2.61/1.59 = ( hF_Mirabelle_hmem @ X3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_63_hinter__hinsert__right,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) )
% 2.61/1.59 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_64_hinter__hinsert__left,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) )
% 2.61/1.59 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_65_foundation,axiom,
% 2.61/1.59 ! [Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( Z != zero_z189798548lle_hf )
% 2.61/1.59 => ? [W: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ W @ Z )
% 2.61/1.59 & ( ( inf_in956532509lle_hf @ W @ Z )
% 2.61/1.59 = zero_z189798548lle_hf ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_66_hinsert__hdiff__if,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 2.61/1.59 = ( minus_1232880740lle_hf @ A4 @ B4 ) ) )
% 2.61/1.59 & ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
% 2.61/1.59 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ X @ ( minus_1232880740lle_hf @ A4 @ B4 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_67_hinsert__iff,axiom,
% 2.61/1.59 ! [Z: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( Z
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ Y @ X ) )
% 2.61/1.59 = ( ! [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 2.61/1.59 | ( U2 = Y ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_68_hemptyE,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf] :
% 2.61/1.59 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_69_hempty__iff,axiom,
% 2.61/1.59 ! [Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( Z = zero_z189798548lle_hf )
% 2.61/1.59 = ( ! [X3: hF_Mirabelle_hf] :
% 2.61/1.59 ~ ( hF_Mirabelle_hmem @ X3 @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_70_hmem__hempty,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf] :
% 2.61/1.59 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_71_hunion__hinsert__right,axiom,
% 2.61/1.59 ! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_72_hunion__hinsert__left,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
% 2.61/1.59 = ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A4 @ B4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_73_RepFun__cong,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4 = B4 )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 2.61/1.59 => ( ( F @ X2 )
% 2.61/1.59 = ( G @ X2 ) ) )
% 2.61/1.59 => ( ( hF_Mirabelle_RepFun @ A4 @ F )
% 2.61/1.59 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_74_hmem__Sup__ne,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X @ Y )
% 2.61/1.59 => ( ( hF_Mirabelle_HUnion @ X )
% 2.61/1.59 != Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_75_Replace__cong,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.61/1.59 ( ( A4 = B4 )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X2 @ B4 )
% 2.61/1.59 => ( ( P @ X2 @ Y2 )
% 2.61/1.59 = ( Q @ X2 @ Y2 ) ) )
% 2.61/1.59 => ( ( hF_Mirabelle_Replace @ A4 @ P )
% 2.61/1.59 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_76_sup__inf__absorb,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_77_sup__inf__absorb,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_78_inf__sup__absorb,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_79_inf__sup__absorb,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_80_sup_Oright__idem,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
% 2.61/1.59 = ( sup_sup_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_81_sup_Oright__idem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ B )
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_82_sup__left__idem,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 2.61/1.59 = ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_83_sup__left__idem,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_84_sup_Oleft__idem,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
% 2.61/1.59 = ( sup_sup_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_85_sup_Oleft__idem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_86_sup__idem,axiom,
% 2.61/1.59 ! [X: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ X )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_87_sup__idem,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ X )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_88_sup_Oidem,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ A @ A )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_89_sup_Oidem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ A @ A )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_90_inf__right__idem,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
% 2.61/1.59 = ( inf_inf_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_91_inf__right__idem,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y )
% 2.61/1.59 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_92_inf_Oidem,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ A @ A )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_93_inf_Oidem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ A @ A )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_94_inf__idem,axiom,
% 2.61/1.59 ! [X: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ X )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_95_inf__idem,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ X )
% 2.61/1.59 = X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_96_inf_Oleft__idem,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
% 2.61/1.59 = ( inf_inf_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_97_inf_Oleft__idem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ A @ B ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_98_inf__left__idem,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 2.61/1.59 = ( inf_inf_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_99_inf__left__idem,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_100_inf_Oright__idem,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
% 2.61/1.59 = ( inf_inf_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_101_inf_Oright__idem,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_102_inf__sup__aci_I4_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
% 2.61/1.59 = ( inf_inf_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_103_inf__sup__aci_I4_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_104_inf__sup__aci_I3_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 2.61/1.59 = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_105_inf__sup__aci_I3_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_106_inf__sup__aci_I2_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
% 2.61/1.59 = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_107_inf__sup__aci_I2_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
% 2.61/1.59 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_108_inf__sup__aci_I1_J,axiom,
% 2.61/1.59 ( inf_inf_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_109_inf__sup__aci_I1_J,axiom,
% 2.61/1.59 ( inf_in956532509lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_110_boolean__algebra__cancel_Oinf1,axiom,
% 2.61/1.59 ! [A4: nat,K: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( A4
% 2.61/1.59 = ( inf_inf_nat @ K @ A ) )
% 2.61/1.59 => ( ( inf_inf_nat @ A4 @ B )
% 2.61/1.59 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_111_boolean__algebra__cancel_Oinf1,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4
% 2.61/1.59 = ( inf_in956532509lle_hf @ K @ A ) )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ A4 @ B )
% 2.61/1.59 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_112_boolean__algebra__cancel_Oinf2,axiom,
% 2.61/1.59 ! [B4: nat,K: nat,B: nat,A: nat] :
% 2.61/1.59 ( ( B4
% 2.61/1.59 = ( inf_inf_nat @ K @ B ) )
% 2.61/1.59 => ( ( inf_inf_nat @ A @ B4 )
% 2.61/1.59 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_113_boolean__algebra__cancel_Oinf2,axiom,
% 2.61/1.59 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( B4
% 2.61/1.59 = ( inf_in956532509lle_hf @ K @ B ) )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ A @ B4 )
% 2.61/1.59 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_114_inf_Oassoc,axiom,
% 2.61/1.59 ! [A: nat,B: nat,C: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
% 2.61/1.59 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_115_inf_Oassoc,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_116_inf__assoc,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
% 2.61/1.59 = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_117_inf__assoc,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
% 2.61/1.59 = ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_118_inf_Ocommute,axiom,
% 2.61/1.59 ( inf_inf_nat
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] : ( inf_inf_nat @ B3 @ A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_119_inf_Ocommute,axiom,
% 2.61/1.59 ( inf_in956532509lle_hf
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_120_inf__commute,axiom,
% 2.61/1.59 ( inf_inf_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_121_inf__commute,axiom,
% 2.61/1.59 ( inf_in956532509lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_122_inf_Oleft__commute,axiom,
% 2.61/1.59 ! [B: nat,A: nat,C: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
% 2.61/1.59 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_123_inf_Oleft__commute,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A @ C ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_124_inf__left__commute,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 2.61/1.59 = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_125_inf__left__commute,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_126_inf__sup__aci_I8_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
% 2.61/1.59 = ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_127_inf__sup__aci_I8_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_128_inf__sup__aci_I7_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 2.61/1.59 = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_129_inf__sup__aci_I7_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_130_inf__sup__aci_I6_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 2.61/1.59 = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_131_inf__sup__aci_I6_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 2.61/1.59 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_132_inf__sup__aci_I5_J,axiom,
% 2.61/1.59 ( sup_sup_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_133_inf__sup__aci_I5_J,axiom,
% 2.61/1.59 ( sup_su638957495lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_134_boolean__algebra__cancel_Osup1,axiom,
% 2.61/1.59 ! [A4: nat,K: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( A4
% 2.61/1.59 = ( sup_sup_nat @ K @ A ) )
% 2.61/1.59 => ( ( sup_sup_nat @ A4 @ B )
% 2.61/1.59 = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_135_boolean__algebra__cancel_Osup1,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4
% 2.61/1.59 = ( sup_su638957495lle_hf @ K @ A ) )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ A4 @ B )
% 2.61/1.59 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_136_boolean__algebra__cancel_Osup2,axiom,
% 2.61/1.59 ! [B4: nat,K: nat,B: nat,A: nat] :
% 2.61/1.59 ( ( B4
% 2.61/1.59 = ( sup_sup_nat @ K @ B ) )
% 2.61/1.59 => ( ( sup_sup_nat @ A @ B4 )
% 2.61/1.59 = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_137_boolean__algebra__cancel_Osup2,axiom,
% 2.61/1.59 ! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( B4
% 2.61/1.59 = ( sup_su638957495lle_hf @ K @ B ) )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ A @ B4 )
% 2.61/1.59 = ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_138_sup_Oassoc,axiom,
% 2.61/1.59 ! [A: nat,B: nat,C: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
% 2.61/1.59 = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_139_sup_Oassoc,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ C )
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_140_sup__assoc,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 2.61/1.59 = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_141_sup__assoc,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 2.61/1.59 = ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_142_sup_Ocommute,axiom,
% 2.61/1.59 ( sup_sup_nat
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_143_sup_Ocommute,axiom,
% 2.61/1.59 ( sup_su638957495lle_hf
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_144_sup__commute,axiom,
% 2.61/1.59 ( sup_sup_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_145_sup__commute,axiom,
% 2.61/1.59 ( sup_su638957495lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_146_sup_Oleft__commute,axiom,
% 2.61/1.59 ! [B: nat,A: nat,C: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
% 2.61/1.59 = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_147_sup_Oleft__commute,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A @ C ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_148_sup__left__commute,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 2.61/1.59 = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_149_sup__left__commute,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_150_distrib__imp1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ! [X2: nat,Y2: nat,Z2: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) )
% 2.61/1.59 = ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) )
% 2.61/1.59 => ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 2.61/1.59 = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_151_distrib__imp1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X2 @ ( sup_su638957495lle_hf @ Y2 @ Z2 ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ ( inf_in956532509lle_hf @ X2 @ Z2 ) ) )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_152_distrib__imp2,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ! [X2: nat,Y2: nat,Z2: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
% 2.61/1.59 = ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) )
% 2.61/1.59 => ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 2.61/1.59 = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_153_distrib__imp2,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z2 ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X2 @ Y2 ) @ ( sup_su638957495lle_hf @ X2 @ Z2 ) ) )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_154_inf__sup__distrib1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
% 2.61/1.59 = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_155_inf__sup__distrib1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_156_inf__sup__distrib2,axiom,
% 2.61/1.59 ! [Y: nat,Z: nat,X: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
% 2.61/1.59 = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_157_inf__sup__distrib2,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X )
% 2.61/1.59 = ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_158_sup__inf__distrib1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 2.61/1.59 = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_159_sup__inf__distrib1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_160_sup__inf__distrib2,axiom,
% 2.61/1.59 ! [Y: nat,Z: nat,X: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
% 2.61/1.59 = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_161_sup__inf__distrib2,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ Z ) @ X )
% 2.61/1.59 = ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_162_diff__self__eq__0,axiom,
% 2.61/1.59 ! [M: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ M @ M )
% 2.61/1.59 = zero_zero_nat ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_163_diff__0__eq__0,axiom,
% 2.61/1.59 ! [N: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 2.61/1.59 = zero_zero_nat ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_164_PrimReplace__iff,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U4 @ A4 )
% 2.61/1.59 => ( ( R @ U4 @ V3 )
% 2.61/1.59 => ( ( R @ U4 @ V4 )
% 2.61/1.59 => ( V4 = V3 ) ) ) )
% 2.61/1.59 => ( ( hF_Mirabelle_hmem @ V @ ( hF_Mir1248913145eplace @ A4 @ R ) )
% 2.61/1.59 = ( ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.61/1.59 & ( R @ U2 @ V ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_165_diffs0__imp__equal,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ( minus_minus_nat @ M @ N )
% 2.61/1.59 = zero_zero_nat )
% 2.61/1.59 => ( ( ( minus_minus_nat @ N @ M )
% 2.61/1.59 = zero_zero_nat )
% 2.61/1.59 => ( M = N ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_166_minus__nat_Odiff__0,axiom,
% 2.61/1.59 ! [M: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 2.61/1.59 = M ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_167_zero__natural_Orsp,axiom,
% 2.61/1.59 zero_zero_nat = zero_zero_nat ).
% 2.61/1.59
% 2.61/1.59 thf(fact_168_diff__commute,axiom,
% 2.61/1.59 ! [I: nat,J: nat,K: nat] :
% 2.61/1.59 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 2.61/1.59 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_169_less__eq__insert1__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ ( hF_Mirabelle_hinsert @ X @ Y ) @ Z )
% 2.61/1.59 = ( ( hF_Mirabelle_hmem @ X @ Z )
% 2.61/1.59 & ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_170_hmem__def,axiom,
% 2.61/1.59 ( hF_Mirabelle_hmem
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_171_diff__is__0__eq,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ( minus_minus_nat @ M @ N )
% 2.61/1.59 = zero_zero_nat )
% 2.61/1.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_172_le__zero__eq,axiom,
% 2.61/1.59 ! [N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.61/1.59 = ( N = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_173_inf_Obounded__iff,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 2.61/1.59 = ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 & ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_174_inf_Obounded__iff,axiom,
% 2.61/1.59 ! [A: nat,B: nat,C: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 2.61/1.59 = ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 & ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_175_le__inf__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
% 2.61/1.59 = ( ( ord_le976219883lle_hf @ X @ Y )
% 2.61/1.59 & ( ord_le976219883lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_176_le__inf__iff,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
% 2.61/1.59 = ( ( ord_less_eq_nat @ X @ Y )
% 2.61/1.59 & ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_177_sup_Obounded__iff,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
% 2.61/1.59 = ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 & ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_178_sup_Obounded__iff,axiom,
% 2.61/1.59 ! [B: nat,C: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
% 2.61/1.59 = ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_179_le__sup__iff,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
% 2.61/1.59 = ( ( ord_le976219883lle_hf @ X @ Z )
% 2.61/1.59 & ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_180_le__sup__iff,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
% 2.61/1.59 = ( ( ord_less_eq_nat @ X @ Z )
% 2.61/1.59 & ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_181_bot__nat__0_Oextremum,axiom,
% 2.61/1.59 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_182_le0,axiom,
% 2.61/1.59 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_183_diff__diff__cancel,axiom,
% 2.61/1.59 ! [I: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ I @ N )
% 2.61/1.59 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 2.61/1.59 = I ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_184_hsubsetI,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X2 @ A4 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ X2 @ B4 ) )
% 2.61/1.59 => ( ord_le976219883lle_hf @ A4 @ B4 ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_185_less__eq__hempty,axiom,
% 2.61/1.59 ! [U: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ U @ zero_z189798548lle_hf )
% 2.61/1.59 = ( U = zero_z189798548lle_hf ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_186_diff__is__0__eq_H,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ M @ N )
% 2.61/1.59 => ( ( minus_minus_nat @ M @ N )
% 2.61/1.59 = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_187_diff__le__mono2,axiom,
% 2.61/1.59 ! [M: nat,N: nat,L: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ M @ N )
% 2.61/1.59 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_188_le__diff__iff_H,axiom,
% 2.61/1.59 ! [A: nat,C: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ C )
% 2.61/1.59 => ( ( ord_less_eq_nat @ B @ C )
% 2.61/1.59 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.61/1.59 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_189_diff__le__self,axiom,
% 2.61/1.59 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_190_diff__le__mono,axiom,
% 2.61/1.59 ! [M: nat,N: nat,L: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ M @ N )
% 2.61/1.59 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_191_Nat_Odiff__diff__eq,axiom,
% 2.61/1.59 ! [K: nat,M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ K @ M )
% 2.61/1.59 => ( ( ord_less_eq_nat @ K @ N )
% 2.61/1.59 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.61/1.59 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_192_le__diff__iff,axiom,
% 2.61/1.59 ! [K: nat,M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ K @ M )
% 2.61/1.59 => ( ( ord_less_eq_nat @ K @ N )
% 2.61/1.59 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.61/1.59 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_193_eq__diff__iff,axiom,
% 2.61/1.59 ! [K: nat,M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ K @ M )
% 2.61/1.59 => ( ( ord_less_eq_nat @ K @ N )
% 2.61/1.59 => ( ( ( minus_minus_nat @ M @ K )
% 2.61/1.59 = ( minus_minus_nat @ N @ K ) )
% 2.61/1.59 = ( M = N ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_194_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.61/1.59 => ( A = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_195_bot__nat__0_Oextremum__unique,axiom,
% 2.61/1.59 ! [A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.61/1.59 = ( A = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_196_le__0__eq,axiom,
% 2.61/1.59 ! [N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.61/1.59 = ( N = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_197_less__eq__nat_Osimps_I1_J,axiom,
% 2.61/1.59 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_198_Nat_Oex__has__greatest__nat,axiom,
% 2.61/1.59 ! [P: nat > $o,K: nat,B: nat] :
% 2.61/1.59 ( ( P @ K )
% 2.61/1.59 => ( ! [Y2: nat] :
% 2.61/1.59 ( ( P @ Y2 )
% 2.61/1.59 => ( ord_less_eq_nat @ Y2 @ B ) )
% 2.61/1.59 => ? [X2: nat] :
% 2.61/1.59 ( ( P @ X2 )
% 2.61/1.59 & ! [Y5: nat] :
% 2.61/1.59 ( ( P @ Y5 )
% 2.61/1.59 => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_199_nat__le__linear,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ M @ N )
% 2.61/1.59 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_200_le__antisym,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ M @ N )
% 2.61/1.59 => ( ( ord_less_eq_nat @ N @ M )
% 2.61/1.59 => ( M = N ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_201_eq__imp__le,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( M = N )
% 2.61/1.59 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_202_le__trans,axiom,
% 2.61/1.59 ! [I: nat,J: nat,K: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ I @ J )
% 2.61/1.59 => ( ( ord_less_eq_nat @ J @ K )
% 2.61/1.59 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_203_le__refl,axiom,
% 2.61/1.59 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_204_canonically__ordered__monoid__add__class_Ozero__le,axiom,
% 2.61/1.59 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_205_inf_OcoboundedI2,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ C )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_206_inf_OcoboundedI2,axiom,
% 2.61/1.59 ! [B: nat,C: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ C )
% 2.61/1.59 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_207_inf_OcoboundedI1,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ C )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_208_inf_OcoboundedI1,axiom,
% 2.61/1.59 ! [A: nat,C: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ C )
% 2.61/1.59 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_209_inf_Oabsorb__iff2,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ A3 @ B3 )
% 2.61/1.59 = B3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_210_inf_Oabsorb__iff2,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [B3: nat,A3: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ A3 @ B3 )
% 2.61/1.59 = B3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_211_inf_Oabsorb__iff1,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ A3 @ B3 )
% 2.61/1.59 = A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_212_inf_Oabsorb__iff1,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ A3 @ B3 )
% 2.61/1.59 = A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_213_inf_Ocobounded2,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_214_inf_Ocobounded2,axiom,
% 2.61/1.59 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_215_inf_Ocobounded1,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_216_inf_Ocobounded1,axiom,
% 2.61/1.59 ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_217_inf_Oorder__iff,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.61/1.59 ( A3
% 2.61/1.59 = ( inf_in956532509lle_hf @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_218_inf_Oorder__iff,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] :
% 2.61/1.59 ( A3
% 2.61/1.59 = ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_219_inf__greatest,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ Y )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ X @ Z )
% 2.61/1.59 => ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_220_inf__greatest,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ Y )
% 2.61/1.59 => ( ( ord_less_eq_nat @ X @ Z )
% 2.61/1.59 => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_221_inf_OboundedI,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ A @ C )
% 2.61/1.59 => ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_222_inf_OboundedI,axiom,
% 2.61/1.59 ! [A: nat,B: nat,C: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ( ( ord_less_eq_nat @ A @ C )
% 2.61/1.59 => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_223_inf_OboundedE,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 2.61/1.59 => ~ ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ~ ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_224_inf_OboundedE,axiom,
% 2.61/1.59 ! [A: nat,B: nat,C: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 2.61/1.59 => ~ ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_225_inf__absorb2,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ Y @ X )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ X @ Y )
% 2.61/1.59 = Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_226_inf__absorb2,axiom,
% 2.61/1.59 ! [Y: nat,X: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ Y @ X )
% 2.61/1.59 => ( ( inf_inf_nat @ X @ Y )
% 2.61/1.59 = Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_227_inf__absorb1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ Y )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ X @ Y )
% 2.61/1.59 = X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_228_inf__absorb1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ Y )
% 2.61/1.59 => ( ( inf_inf_nat @ X @ Y )
% 2.61/1.59 = X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_229_inf_Oabsorb2,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ A @ B )
% 2.61/1.59 = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_230_inf_Oabsorb2,axiom,
% 2.61/1.59 ! [B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ( ( inf_inf_nat @ A @ B )
% 2.61/1.59 = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_231_inf_Oabsorb1,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ A @ B )
% 2.61/1.59 = A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_232_inf_Oabsorb1,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ( ( inf_inf_nat @ A @ B )
% 2.61/1.59 = A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_233_le__iff__inf,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( inf_in956532509lle_hf @ X3 @ Y3 )
% 2.61/1.59 = X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_234_le__iff__inf,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] :
% 2.61/1.59 ( ( inf_inf_nat @ X3 @ Y3 )
% 2.61/1.59 = X3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_235_inf__unique,axiom,
% 2.61/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ X2 )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ Y2 )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X2 @ Y2 )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ X2 @ Z2 )
% 2.61/1.59 => ( ord_le976219883lle_hf @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 2.61/1.59 => ( ( inf_in956532509lle_hf @ X @ Y )
% 2.61/1.59 = ( F @ X @ Y ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_236_inf__unique,axiom,
% 2.61/1.59 ! [F: nat > nat > nat,X: nat,Y: nat] :
% 2.61/1.59 ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
% 2.61/1.59 => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
% 2.61/1.59 => ( ! [X2: nat,Y2: nat,Z2: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.61/1.59 => ( ( ord_less_eq_nat @ X2 @ Z2 )
% 2.61/1.59 => ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
% 2.61/1.59 => ( ( inf_inf_nat @ X @ Y )
% 2.61/1.59 = ( F @ X @ Y ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_237_inf_OorderI,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ B ) )
% 2.61/1.59 => ( ord_le976219883lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_238_inf_OorderI,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( A
% 2.61/1.59 = ( inf_inf_nat @ A @ B ) )
% 2.61/1.59 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_239_inf_OorderE,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ( A
% 2.61/1.59 = ( inf_in956532509lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_240_inf_OorderE,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ( A
% 2.61/1.59 = ( inf_inf_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_241_le__infI2,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ X )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_242_le__infI2,axiom,
% 2.61/1.59 ! [B: nat,X: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ X )
% 2.61/1.59 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_243_le__infI1,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ X )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_244_le__infI1,axiom,
% 2.61/1.59 ! [A: nat,X: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ X )
% 2.61/1.59 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_245_inf__mono,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ C )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ B @ D )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ ( inf_in956532509lle_hf @ C @ D ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_246_inf__mono,axiom,
% 2.61/1.59 ! [A: nat,C: nat,B: nat,D: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ C )
% 2.61/1.59 => ( ( ord_less_eq_nat @ B @ D )
% 2.61/1.59 => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_247_le__infI,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ A )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ X @ B )
% 2.61/1.59 => ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_248_le__infI,axiom,
% 2.61/1.59 ! [X: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ A )
% 2.61/1.59 => ( ( ord_less_eq_nat @ X @ B )
% 2.61/1.59 => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_249_le__infE,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) )
% 2.61/1.59 => ~ ( ( ord_le976219883lle_hf @ X @ A )
% 2.61/1.59 => ~ ( ord_le976219883lle_hf @ X @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_250_le__infE,axiom,
% 2.61/1.59 ! [X: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
% 2.61/1.59 => ~ ( ( ord_less_eq_nat @ X @ A )
% 2.61/1.59 => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_251_inf__le2,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_252_inf__le2,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_253_inf__le1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_254_inf__le1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_255_inf__sup__ord_I1_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_256_inf__sup__ord_I1_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_257_inf__sup__ord_I2_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_258_inf__sup__ord_I2_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_259_sup_OcoboundedI2,axiom,
% 2.61/1.59 ! [C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ C @ B )
% 2.61/1.59 => ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_260_sup_OcoboundedI2,axiom,
% 2.61/1.59 ! [C: nat,B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ C @ B )
% 2.61/1.59 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_261_sup_OcoboundedI1,axiom,
% 2.61/1.59 ! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ C @ A )
% 2.61/1.59 => ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_262_sup_OcoboundedI1,axiom,
% 2.61/1.59 ! [C: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ C @ A )
% 2.61/1.59 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_263_sup_Oabsorb__iff2,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ A3 @ B3 )
% 2.61/1.59 = B3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_264_sup_Oabsorb__iff2,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ A3 @ B3 )
% 2.61/1.59 = B3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_265_sup_Oabsorb__iff1,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ A3 @ B3 )
% 2.61/1.59 = A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_266_sup_Oabsorb__iff1,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [B3: nat,A3: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ A3 @ B3 )
% 2.61/1.59 = A3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_267_sup_Ocobounded2,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ B @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_268_sup_Ocobounded2,axiom,
% 2.61/1.59 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_269_sup_Ocobounded1,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_270_sup_Ocobounded1,axiom,
% 2.61/1.59 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_271_sup_Oorder__iff,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.61/1.59 ( A3
% 2.61/1.59 = ( sup_su638957495lle_hf @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_272_sup_Oorder__iff,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [B3: nat,A3: nat] :
% 2.61/1.59 ( A3
% 2.61/1.59 = ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_273_sup_OboundedI,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ C @ A )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_274_sup_OboundedI,axiom,
% 2.61/1.59 ! [B: nat,A: nat,C: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ( ( ord_less_eq_nat @ C @ A )
% 2.61/1.59 => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_275_sup_OboundedE,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
% 2.61/1.59 => ~ ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ~ ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_276_sup_OboundedE,axiom,
% 2.61/1.59 ! [B: nat,C: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
% 2.61/1.59 => ~ ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_277_sup__absorb2,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ Y )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ X @ Y )
% 2.61/1.59 = Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_278_sup__absorb2,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ Y )
% 2.61/1.59 => ( ( sup_sup_nat @ X @ Y )
% 2.61/1.59 = Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_279_sup__absorb1,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ Y @ X )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ X @ Y )
% 2.61/1.59 = X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_280_sup__absorb1,axiom,
% 2.61/1.59 ! [Y: nat,X: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ Y @ X )
% 2.61/1.59 => ( ( sup_sup_nat @ X @ Y )
% 2.61/1.59 = X ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_281_sup_Oabsorb2,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ A @ B )
% 2.61/1.59 = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_282_sup_Oabsorb2,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ( ( sup_sup_nat @ A @ B )
% 2.61/1.59 = B ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_283_sup_Oabsorb1,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ A @ B )
% 2.61/1.59 = A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_284_sup_Oabsorb1,axiom,
% 2.61/1.59 ! [B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ( ( sup_sup_nat @ A @ B )
% 2.61/1.59 = A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_285_sup__unique,axiom,
% 2.61/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X2 @ ( F @ X2 @ Y2 ) )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y2 @ ( F @ X2 @ Y2 ) )
% 2.61/1.59 => ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ Y2 @ X2 )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ Z2 @ X2 )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 2.61/1.59 => ( ( sup_su638957495lle_hf @ X @ Y )
% 2.61/1.59 = ( F @ X @ Y ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_286_sup__unique,axiom,
% 2.61/1.59 ! [F: nat > nat > nat,X: nat,Y: nat] :
% 2.61/1.59 ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
% 2.61/1.59 => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
% 2.61/1.59 => ( ! [X2: nat,Y2: nat,Z2: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 2.61/1.59 => ( ( ord_less_eq_nat @ Z2 @ X2 )
% 2.61/1.59 => ( ord_less_eq_nat @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
% 2.61/1.59 => ( ( sup_sup_nat @ X @ Y )
% 2.61/1.59 = ( F @ X @ Y ) ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_287_sup_OorderI,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ B ) )
% 2.61/1.59 => ( ord_le976219883lle_hf @ B @ A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_288_sup_OorderI,axiom,
% 2.61/1.59 ! [A: nat,B: nat] :
% 2.61/1.59 ( ( A
% 2.61/1.59 = ( sup_sup_nat @ A @ B ) )
% 2.61/1.59 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_289_sup_OorderE,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ( A
% 2.61/1.59 = ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_290_sup_OorderE,axiom,
% 2.61/1.59 ! [B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ( A
% 2.61/1.59 = ( sup_sup_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_291_le__iff__sup,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( sup_su638957495lle_hf @ X3 @ Y3 )
% 2.61/1.59 = Y3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_292_le__iff__sup,axiom,
% 2.61/1.59 ( ord_less_eq_nat
% 2.61/1.59 = ( ^ [X3: nat,Y3: nat] :
% 2.61/1.59 ( ( sup_sup_nat @ X3 @ Y3 )
% 2.61/1.59 = Y3 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_293_sup__least,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ Y @ X )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ Z @ X )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_294_sup__least,axiom,
% 2.61/1.59 ! [Y: nat,X: nat,Z: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ Y @ X )
% 2.61/1.59 => ( ( ord_less_eq_nat @ Z @ X )
% 2.61/1.59 => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_295_sup__mono,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ C )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ B @ D )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ ( sup_su638957495lle_hf @ C @ D ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_296_sup__mono,axiom,
% 2.61/1.59 ! [A: nat,C: nat,B: nat,D: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ C )
% 2.61/1.59 => ( ( ord_less_eq_nat @ B @ D )
% 2.61/1.59 => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_297_sup_Omono,axiom,
% 2.61/1.59 ! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,D: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ C @ A )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ D @ B )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ C @ D ) @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_298_sup_Omono,axiom,
% 2.61/1.59 ! [C: nat,A: nat,D: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ C @ A )
% 2.61/1.59 => ( ( ord_less_eq_nat @ D @ B )
% 2.61/1.59 => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_299_le__supI2,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ B )
% 2.61/1.59 => ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_300_le__supI2,axiom,
% 2.61/1.59 ! [X: nat,B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ B )
% 2.61/1.59 => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_301_le__supI1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ X @ A )
% 2.61/1.59 => ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_302_le__supI1,axiom,
% 2.61/1.59 ! [X: nat,A: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ X @ A )
% 2.61/1.59 => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_303_sup__ge2,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_304_sup__ge2,axiom,
% 2.61/1.59 ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_305_sup__ge1,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_306_sup__ge1,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_307_le__supI,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A @ X )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ B @ X )
% 2.61/1.59 => ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_308_le__supI,axiom,
% 2.61/1.59 ! [A: nat,X: nat,B: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ A @ X )
% 2.61/1.59 => ( ( ord_less_eq_nat @ B @ X )
% 2.61/1.59 => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_309_le__supE,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X )
% 2.61/1.59 => ~ ( ( ord_le976219883lle_hf @ A @ X )
% 2.61/1.59 => ~ ( ord_le976219883lle_hf @ B @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_310_le__supE,axiom,
% 2.61/1.59 ! [A: nat,B: nat,X: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
% 2.61/1.59 => ~ ( ( ord_less_eq_nat @ A @ X )
% 2.61/1.59 => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_311_inf__sup__ord_I3_J,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_312_inf__sup__ord_I3_J,axiom,
% 2.61/1.59 ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_313_inf__sup__ord_I4_J,axiom,
% 2.61/1.59 ! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_314_inf__sup__ord_I4_J,axiom,
% 2.61/1.59 ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_315_less__eq__hf__def,axiom,
% 2.61/1.59 ( ord_le976219883lle_hf
% 2.61/1.59 = ( ^ [A6: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 2.61/1.59 ! [X3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ X3 @ A6 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ X3 @ B6 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_316_rev__hsubsetD,axiom,
% 2.61/1.59 ! [C: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_hmem @ C @ A4 )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_317_hsubsetCE,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 2.61/1.59 => ( ( hF_Mirabelle_hmem @ C @ A4 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_318_hsubsetD,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 2.61/1.59 => ( ( hF_Mirabelle_hmem @ C @ A4 )
% 2.61/1.59 => ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_319_HF__Mirabelle__glliljednj_Ozero__le,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ zero_z189798548lle_hf @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_320_hf__equalityE,axiom,
% 2.61/1.59 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( A4 = B4 )
% 2.61/1.59 => ~ ( ( ord_le976219883lle_hf @ A4 @ B4 )
% 2.61/1.59 => ~ ( ord_le976219883lle_hf @ B4 @ A4 ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_321_distrib__inf__le,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) @ ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_322_distrib__inf__le,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_323_distrib__sup__le,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) @ ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_324_distrib__sup__le,axiom,
% 2.61/1.59 ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_325_less__eq__insert2__iff,axiom,
% 2.61/1.59 ! [Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ Z @ ( hF_Mirabelle_hinsert @ X @ Y ) )
% 2.61/1.59 = ( ( ord_le976219883lle_hf @ Z @ Y )
% 2.61/1.59 | ? [U2: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ( hF_Mirabelle_hinsert @ X @ U2 )
% 2.61/1.59 = Z )
% 2.61/1.59 & ~ ( hF_Mirabelle_hmem @ X @ U2 )
% 2.61/1.59 & ( ord_le976219883lle_hf @ U2 @ Y ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_326_order__refl,axiom,
% 2.61/1.59 ! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_327_order__refl,axiom,
% 2.61/1.59 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_328_le__numeral__extra_I3_J,axiom,
% 2.61/1.59 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.61/1.59
% 2.61/1.59 thf(fact_329_HF__hfset,axiom,
% 2.61/1.59 ! [A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 2.61/1.59 = A ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_330_of__nat__le__0__iff,axiom,
% 2.61/1.59 ! [M: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
% 2.61/1.59 = ( M = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_331_of__nat__eq__iff,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ( semiri1382578993at_nat @ M )
% 2.61/1.59 = ( semiri1382578993at_nat @ N ) )
% 2.61/1.59 = ( M = N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_332_of__nat__0,axiom,
% 2.61/1.59 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 2.61/1.59 = zero_zero_nat ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_333_of__nat__0__eq__iff,axiom,
% 2.61/1.59 ! [N: nat] :
% 2.61/1.59 ( ( zero_zero_nat
% 2.61/1.59 = ( semiri1382578993at_nat @ N ) )
% 2.61/1.59 = ( zero_zero_nat = N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_334_of__nat__eq__0__iff,axiom,
% 2.61/1.59 ! [M: nat] :
% 2.61/1.59 ( ( ( semiri1382578993at_nat @ M )
% 2.61/1.59 = zero_zero_nat )
% 2.61/1.59 = ( M = zero_zero_nat ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_335_of__nat__le__iff,axiom,
% 2.61/1.59 ! [M: nat,N: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.61/1.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_336_of__nat__0__le__iff,axiom,
% 2.61/1.59 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_337_of__nat__mono,axiom,
% 2.61/1.59 ! [I: nat,J: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ I @ J )
% 2.61/1.59 => ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_338_of__nat__diff,axiom,
% 2.61/1.59 ! [N: nat,M: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ N @ M )
% 2.61/1.59 => ( ( semiri1382578993at_nat @ ( minus_minus_nat @ M @ N ) )
% 2.61/1.59 = ( minus_minus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_339_dual__order_Oantisym,axiom,
% 2.61/1.59 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B @ A )
% 2.61/1.59 => ( ( ord_le976219883lle_hf @ A @ B )
% 2.61/1.59 => ( A = B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_340_dual__order_Oantisym,axiom,
% 2.61/1.59 ! [B: nat,A: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B @ A )
% 2.61/1.59 => ( ( ord_less_eq_nat @ A @ B )
% 2.61/1.59 => ( A = B ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_341_dual__order_Oeq__iff,axiom,
% 2.61/1.59 ( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
% 2.61/1.59 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.61/1.59 ( ( ord_le976219883lle_hf @ B3 @ A3 )
% 2.61/1.59 & ( ord_le976219883lle_hf @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(fact_342_dual__order_Oeq__iff,axiom,
% 2.61/1.59 ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
% 2.61/1.59 = ( ^ [A3: nat,B3: nat] :
% 2.61/1.59 ( ( ord_less_eq_nat @ B3 @ A3 )
% 2.61/1.59 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 2.61/1.59
% 2.61/1.59 thf(conj_0,conjecture,
% 2.61/1.59 ( ( hF_Mirabelle_hpair @ x @ y )
% 2.61/1.59 != zero_z189798548lle_hf ) ).
% 2.61/1.59 ------------------------
% 2.74/1.61 % (1410)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/18Mi)
% 2.74/1.61 % (1404)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/183Mi)
% 2.74/1.61 % (1406)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_1223 for (2999ds/27Mi)
% 2.74/1.61 % (1407)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/2Mi)
% 2.74/1.61 % (1409)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_1223 for (2999ds/275Mi)
% 2.74/1.62 % (1408)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_1223 for (2999ds/2Mi)
% 2.74/1.62 % (1407)Instruction limit reached!
% 2.74/1.62 % (1407)------------------------------
% 2.74/1.62 % (1407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.62 % (1407)Termination reason: Unknown
% 2.74/1.62 % (1407)Termination phase: shuffling
% 2.74/1.62
% 2.74/1.62 % (1407)Memory used [KB]: 1279
% 2.74/1.62 % (1407)Time elapsed: 0.003 s
% 2.74/1.62 % (1407)Instructions burned: 3 (million)
% 2.74/1.62 % (1407)------------------------------
% 2.74/1.62 % (1407)------------------------------
% 2.74/1.62 % (1405)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_1223 for (2999ds/4Mi)
% 2.74/1.62 % (1408)Instruction limit reached!
% 2.74/1.62 % (1408)------------------------------
% 2.74/1.62 % (1408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.62 % (1408)Termination reason: Unknown
% 2.74/1.62 % (1408)Termination phase: shuffling
% 2.74/1.62
% 2.74/1.62 % (1408)Memory used [KB]: 1279
% 2.74/1.62 % (1410)Instruction limit reached!
% 2.74/1.62 % (1410)------------------------------
% 2.74/1.62 % (1410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.62 % (1410)Termination reason: Unknown
% 2.74/1.62 % (1410)Termination phase: Property scanning
% 2.74/1.62
% 2.74/1.62 % (1410)Memory used [KB]: 1535
% 2.74/1.62 % (1410)Time elapsed: 0.009 s
% 2.74/1.62 % (1410)Instructions burned: 21 (million)
% 2.74/1.62 % (1410)------------------------------
% 2.74/1.62 % (1410)------------------------------
% 2.74/1.62 % (1408)Time elapsed: 0.003 s
% 2.74/1.62 % (1408)Instructions burned: 2 (million)
% 2.74/1.62 % (1408)------------------------------
% 2.74/1.62 % (1408)------------------------------
% 2.74/1.62 % (1405)Instruction limit reached!
% 2.74/1.62 % (1405)------------------------------
% 2.74/1.62 % (1405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.62 % (1405)Termination reason: Unknown
% 2.74/1.62 % (1405)Termination phase: shuffling
% 2.74/1.62
% 2.74/1.62 % (1405)Memory used [KB]: 1279
% 2.74/1.62 % (1405)Time elapsed: 0.004 s
% 2.74/1.62 % (1405)Instructions burned: 4 (million)
% 2.74/1.62 % (1405)------------------------------
% 2.74/1.62 % (1405)------------------------------
% 2.74/1.63 % (1413)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_1223 for (2999ds/37Mi)
% 2.74/1.63 % (1406)Instruction limit reached!
% 2.74/1.63 % (1406)------------------------------
% 2.74/1.63 % (1406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.63 % (1406)Termination reason: Unknown
% 2.74/1.63 % (1406)Termination phase: Property scanning
% 2.74/1.63
% 2.74/1.63 % (1406)Memory used [KB]: 1535
% 2.74/1.63 % (1406)Time elapsed: 0.015 s
% 2.74/1.63 % (1406)Instructions burned: 28 (million)
% 2.74/1.63 % (1406)------------------------------
% 2.74/1.63 % (1406)------------------------------
% 2.74/1.63 % (1412)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/3Mi)
% 2.74/1.63 % (1412)Instruction limit reached!
% 2.74/1.63 % (1412)------------------------------
% 2.74/1.63 % (1412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.63 % (1412)Termination reason: Unknown
% 2.74/1.63 % (1412)Termination phase: shuffling
% 2.74/1.63
% 2.74/1.63 % (1412)Memory used [KB]: 1279
% 2.74/1.63 % (1412)Time elapsed: 0.003 s
% 2.74/1.63 % (1412)Instructions burned: 3 (million)
% 2.74/1.63 % (1412)------------------------------
% 2.74/1.63 % (1412)------------------------------
% 2.74/1.63 % (1414)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_1223 for (2999ds/15Mi)
% 2.74/1.64 % (1413)Instruction limit reached!
% 2.74/1.64 % (1413)------------------------------
% 2.74/1.64 % (1413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.64 % (1413)Termination reason: Unknown
% 2.74/1.64 % (1413)Termination phase: shuffling
% 2.74/1.64
% 2.74/1.64 % (1413)Memory used [KB]: 2046
% 2.74/1.64 % (1413)Time elapsed: 0.013 s
% 2.74/1.64 % (1413)Instructions burned: 38 (million)
% 2.74/1.64 % (1413)------------------------------
% 2.74/1.64 % (1413)------------------------------
% 2.74/1.64 % (1415)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_1223 for (2999ds/3Mi)
% 2.74/1.64 % (1415)Instruction limit reached!
% 2.74/1.64 % (1415)------------------------------
% 2.74/1.64 % (1415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.64 % (1415)Termination reason: Unknown
% 2.74/1.64 % (1415)Termination phase: shuffling
% 2.74/1.64
% 2.74/1.64 % (1415)Memory used [KB]: 1279
% 2.74/1.64 % (1415)Time elapsed: 0.003 s
% 2.74/1.64 % (1415)Instructions burned: 3 (million)
% 2.74/1.64 % (1415)------------------------------
% 2.74/1.64 % (1415)------------------------------
% 2.74/1.64 % (1414)Instruction limit reached!
% 2.74/1.64 % (1414)------------------------------
% 2.74/1.64 % (1414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.64 % (1414)Termination reason: Unknown
% 2.74/1.64 % (1414)Termination phase: shuffling
% 2.74/1.64
% 2.74/1.64 % (1414)Memory used [KB]: 1535
% 2.74/1.64 % (1414)Time elapsed: 0.010 s
% 2.74/1.64 % (1414)Instructions burned: 15 (million)
% 2.74/1.64 % (1414)------------------------------
% 2.74/1.64 % (1414)------------------------------
% 2.74/1.65 % (1416)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_1223 for (2999ds/1041Mi)
% 2.74/1.65 % (1417)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/7Mi)
% 2.74/1.65 % (1417)Instruction limit reached!
% 2.74/1.65 % (1417)------------------------------
% 2.74/1.65 % (1417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.65 % (1417)Termination reason: Unknown
% 2.74/1.65 % (1417)Termination phase: shuffling
% 2.74/1.65
% 2.74/1.65 % (1417)Memory used [KB]: 1407
% 2.74/1.65 % (1417)Time elapsed: 0.004 s
% 2.74/1.65 % (1417)Instructions burned: 7 (million)
% 2.74/1.65 % (1417)------------------------------
% 2.74/1.65 % (1417)------------------------------
% 2.74/1.66 % (1419)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/16Mi)
% 2.74/1.66 % (1420)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_1223 for (2999ds/3Mi)
% 2.74/1.66 % (1421)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_1223 for (2999ds/3Mi)
% 2.74/1.66 % (1422)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/7Mi)
% 2.74/1.66 % (1420)Instruction limit reached!
% 2.74/1.66 % (1420)------------------------------
% 2.74/1.66 % (1420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.66 % (1420)Termination reason: Unknown
% 2.74/1.66 % (1420)Termination phase: shuffling
% 2.74/1.66
% 2.74/1.66 % (1420)Memory used [KB]: 1279
% 2.74/1.66 % (1420)Time elapsed: 0.003 s
% 2.74/1.66 % (1420)Instructions burned: 3 (million)
% 2.74/1.66 % (1420)------------------------------
% 2.74/1.66 % (1420)------------------------------
% 2.74/1.66 % (1421)Instruction limit reached!
% 2.74/1.66 % (1421)------------------------------
% 2.74/1.66 % (1421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.66 % (1421)Termination reason: Unknown
% 2.74/1.66 % (1421)Termination phase: shuffling
% 2.74/1.66
% 2.74/1.66 % (1421)Memory used [KB]: 1279
% 2.74/1.66 % (1421)Time elapsed: 0.004 s
% 2.74/1.66 % (1421)Instructions burned: 4 (million)
% 2.74/1.66 % (1421)------------------------------
% 2.74/1.66 % (1421)------------------------------
% 2.74/1.66 % (1422)Instruction limit reached!
% 2.74/1.66 % (1422)------------------------------
% 2.74/1.66 % (1422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.66 % (1422)Termination reason: Unknown
% 2.74/1.66 % (1422)Termination phase: shuffling
% 2.74/1.66
% 2.74/1.66 % (1422)Memory used [KB]: 1407
% 2.74/1.66 % (1422)Time elapsed: 0.005 s
% 2.74/1.66 % (1422)Instructions burned: 9 (million)
% 2.74/1.66 % (1422)------------------------------
% 2.74/1.66 % (1422)------------------------------
% 2.74/1.67 % (1419)Instruction limit reached!
% 2.74/1.67 % (1419)------------------------------
% 2.74/1.67 % (1419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.67 % (1419)Termination reason: Unknown
% 2.74/1.67 % (1419)Termination phase: shuffling
% 2.74/1.67
% 2.74/1.67 % (1419)Memory used [KB]: 1535
% 2.74/1.67 % (1419)Time elapsed: 0.011 s
% 2.74/1.67 % (1419)Instructions burned: 17 (million)
% 2.74/1.67 % (1419)------------------------------
% 2.74/1.67 % (1419)------------------------------
% 2.74/1.67 % (1425)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/4Mi)
% 2.74/1.68 % (1425)Instruction limit reached!
% 2.74/1.68 % (1425)------------------------------
% 2.74/1.68 % (1425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.68 % (1425)Termination reason: Unknown
% 2.74/1.68 % (1425)Termination phase: shuffling
% 2.74/1.68
% 2.74/1.68 % (1425)Memory used [KB]: 1407
% 2.74/1.68 % (1425)Time elapsed: 0.004 s
% 2.74/1.68 % (1425)Instructions burned: 6 (million)
% 2.74/1.68 % (1425)------------------------------
% 2.74/1.68 % (1425)------------------------------
% 2.74/1.68 % (1424)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/3Mi)
% 2.74/1.68 % (1424)Instruction limit reached!
% 2.74/1.68 % (1424)------------------------------
% 2.74/1.68 % (1424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.68 % (1424)Termination reason: Unknown
% 2.74/1.68 % (1424)Termination phase: shuffling
% 2.74/1.68
% 2.74/1.68 % (1424)Memory used [KB]: 1279
% 2.74/1.68 % (1424)Time elapsed: 0.003 s
% 2.74/1.68 % (1424)Instructions burned: 3 (million)
% 2.74/1.68 % (1424)------------------------------
% 2.74/1.68 % (1424)------------------------------
% 2.74/1.68 % (1426)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/18Mi)
% 2.74/1.68 % (1427)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/710Mi)
% 2.74/1.69 % (1428)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/6Mi)
% 2.74/1.69 % (1428)Instruction limit reached!
% 2.74/1.69 % (1428)------------------------------
% 2.74/1.69 % (1428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.69 % (1428)Termination reason: Unknown
% 2.74/1.69 % (1428)Termination phase: shuffling
% 2.74/1.69
% 2.74/1.69 % (1428)Memory used [KB]: 1407
% 2.74/1.69 % (1428)Time elapsed: 0.004 s
% 2.74/1.69 % (1428)Instructions burned: 6 (million)
% 2.74/1.69 % (1428)------------------------------
% 2.74/1.69 % (1428)------------------------------
% 2.74/1.69 % (1426)Instruction limit reached!
% 2.74/1.69 % (1426)------------------------------
% 2.74/1.69 % (1426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.69 % (1426)Termination reason: Unknown
% 2.74/1.69 % (1426)Termination phase: Property scanning
% 2.74/1.69
% 2.74/1.69 % (1426)Memory used [KB]: 1535
% 2.74/1.69 % (1426)Time elapsed: 0.011 s
% 2.74/1.69 % (1426)Instructions burned: 19 (million)
% 2.74/1.69 % (1426)------------------------------
% 2.74/1.69 % (1426)------------------------------
% 2.74/1.70 % (1404)Instruction limit reached!
% 2.74/1.70 % (1404)------------------------------
% 2.74/1.70 % (1404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.70 % (1404)Termination reason: Unknown
% 2.74/1.70 % (1404)Termination phase: Saturation
% 2.74/1.70
% 2.74/1.70 % (1404)Memory used [KB]: 7164
% 2.74/1.70 % (1404)Time elapsed: 0.088 s
% 2.74/1.70 % (1404)Instructions burned: 184 (million)
% 2.74/1.70 % (1404)------------------------------
% 2.74/1.70 % (1404)------------------------------
% 2.74/1.70 % (1429)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_1223 for (2999ds/902Mi)
% 2.74/1.70 % (1430)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/21Mi)
% 2.74/1.71 % (1431)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_1223 for (2999ds/5Mi)
% 2.74/1.71 % (1431)Instruction limit reached!
% 2.74/1.71 % (1431)------------------------------
% 2.74/1.71 % (1431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.71 % (1431)Termination reason: Unknown
% 2.74/1.71 % (1431)Termination phase: shuffling
% 2.74/1.71
% 2.74/1.71 % (1431)Memory used [KB]: 1407
% 2.74/1.71 % (1431)Time elapsed: 0.004 s
% 2.74/1.71 % (1431)Instructions burned: 6 (million)
% 2.74/1.71 % (1431)------------------------------
% 2.74/1.71 % (1431)------------------------------
% 2.74/1.72 % (1430)Instruction limit reached!
% 2.74/1.72 % (1430)------------------------------
% 2.74/1.72 % (1430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.72 % (1430)Termination reason: Unknown
% 2.74/1.72 % (1430)Termination phase: shuffling
% 2.74/1.72
% 2.74/1.72 % (1430)Memory used [KB]: 1535
% 2.74/1.72 % (1430)Time elapsed: 0.012 s
% 2.74/1.72 % (1430)Instructions burned: 21 (million)
% 2.74/1.72 % (1430)------------------------------
% 2.74/1.72 % (1430)------------------------------
% 2.74/1.72 % (1432)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_1223 for (2999ds/6Mi)
% 2.74/1.72 % (1432)Instruction limit reached!
% 2.74/1.72 % (1432)------------------------------
% 2.74/1.72 % (1432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.72 % (1432)Termination reason: Unknown
% 2.74/1.72 % (1432)Termination phase: shuffling
% 2.74/1.72
% 2.74/1.72 % (1432)Memory used [KB]: 1407
% 2.74/1.72 % (1432)Time elapsed: 0.005 s
% 2.74/1.72 % (1432)Instructions burned: 6 (million)
% 2.74/1.72 % (1432)------------------------------
% 2.74/1.72 % (1432)------------------------------
% 2.74/1.73 % (1434)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_1223 for (2998ds/377Mi)
% 2.74/1.73 % (1435)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_1223 for (2998ds/779Mi)
% 2.74/1.74 % (1437)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_1223 for (2998ds/19Mi)
% 2.74/1.74 % (1409)Instruction limit reached!
% 2.74/1.74 % (1409)------------------------------
% 2.74/1.74 % (1409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.74 % (1409)Termination reason: Unknown
% 2.74/1.74 % (1409)Termination phase: Saturation
% 2.74/1.74
% 2.74/1.74 % (1409)Memory used [KB]: 6908
% 2.74/1.74 % (1409)Time elapsed: 0.125 s
% 2.74/1.74 % (1409)Instructions burned: 275 (million)
% 2.74/1.74 % (1409)------------------------------
% 2.74/1.74 % (1409)------------------------------
% 2.74/1.75 % (1437)Instruction limit reached!
% 2.74/1.75 % (1437)------------------------------
% 2.74/1.75 % (1437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.75 % (1437)Termination reason: Unknown
% 2.74/1.75 % (1437)Termination phase: shuffling
% 2.74/1.75
% 2.74/1.75 % (1437)Memory used [KB]: 1663
% 2.74/1.75 % (1437)Time elapsed: 0.010 s
% 2.74/1.75 % (1437)Instructions burned: 19 (million)
% 2.74/1.75 % (1437)------------------------------
% 2.74/1.75 % (1437)------------------------------
% 2.74/1.75 % (1440)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_1223 for (2998ds/879Mi)
% 2.74/1.76 % (1441)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_1223 for (2998ds/17Mi)
% 2.74/1.77 % (1440)First to succeed.
% 2.74/1.77 % (1440)Refutation found. Thanks to Tanya!
% 2.74/1.77 % SZS status Theorem for DTF2THF_1223
% 2.74/1.77 % SZS output start Proof for DTF2THF_1223
% 2.74/1.77 thf(type_def_5, type, hF_Mirabelle_hf: $tType).
% 2.74/1.77 thf(type_def_7, type, nat: $tType).
% 2.74/1.77 thf(type_def_8, type, set_HF_Mirabelle_hf: $tType).
% 2.74/1.77 thf(func_def_0, type, set_HF_Mirabelle_hf: $tType).
% 2.74/1.77 thf(func_def_1, type, hF_Mirabelle_hf: $tType).
% 2.74/1.77 thf(func_def_2, type, nat: $tType).
% 2.74/1.77 thf(func_def_3, type, minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_4, type, minus_minus_nat: nat > nat > nat).
% 2.74/1.77 thf(func_def_5, type, zero_z189798548lle_hf: hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_6, type, zero_zero_nat: nat).
% 2.74/1.77 thf(func_def_7, type, hF_Mir818139703ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_8, type, hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_9, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_10, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_11, type, hF_Mir1248913145eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_12, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_13, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_14, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_15, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_16, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_17, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_18, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.74/1.77 thf(func_def_19, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_20, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_21, type, inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_22, type, inf_inf_nat: nat > nat > nat).
% 2.74/1.77 thf(func_def_23, type, sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_24, type, sup_sup_nat: nat > nat > nat).
% 2.74/1.77 thf(func_def_25, type, semiri1382578993at_nat: nat > nat).
% 2.74/1.77 thf(func_def_26, type, ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.74/1.77 thf(func_def_27, type, ord_less_eq_nat: nat > nat > $o).
% 2.74/1.77 thf(func_def_28, type, collec2046588256lle_hf: (hF_Mirabelle_hf > $o) > set_HF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_29, type, member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 2.74/1.77 thf(func_def_30, type, x: hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_31, type, y: hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_46, type, sK0: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_47, type, sK1: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_48, type, sK2: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_49, type, sK3: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_50, type, sK4: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_51, type, sK5: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_52, type, sK6: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_54, type, sK8: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(func_def_55, type, sK9: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.74/1.77 thf(f1002,plain,(
% 2.74/1.77 $false),
% 2.74/1.77 inference(trivial_inequality_removal,[status(thm)],[f1001])).
% 2.74/1.77 thf(f1001,plain,(
% 2.74/1.77 ((hF_Mirabelle_Abs_hf @ zero_zero_nat) != (hF_Mirabelle_Abs_hf @ zero_zero_nat))),
% 2.74/1.77 inference(superposition,[status(thm)],[f921,f967])).
% 2.74/1.77 thf(f967,plain,(
% 2.74/1.77 ((hF_Mirabelle_Abs_hf @ zero_zero_nat) = (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ x @ (hF_Mirabelle_Abs_hf @ zero_zero_nat)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ x @ (hF_Mirabelle_hinsert @ y @ (hF_Mirabelle_Abs_hf @ zero_zero_nat))) @ (hF_Mirabelle_Abs_hf @ zero_zero_nat))))),
% 2.74/1.77 inference(beta_eta_normalization,[status(thm)],[f905])).
% 2.74/1.77 thf(f905,plain,(
% 2.74/1.77 ((hF_Mirabelle_Abs_hf @ zero_zero_nat) = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_Abs_hf @ zero_zero_nat)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ (hF_Mirabelle_Abs_hf @ zero_zero_nat))) @ (hF_Mirabelle_Abs_hf @ zero_zero_nat)))))) @ x @ y))),
% 2.74/1.77 inference(definition_unfolding,[status(thm)],[f860,f851,f880])).
% 2.74/1.77 thf(f880,plain,(
% 2.74/1.77 (hF_Mirabelle_hpair = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_Abs_hf @ zero_zero_nat)) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ (hF_Mirabelle_Abs_hf @ zero_zero_nat))) @ (hF_Mirabelle_Abs_hf @ zero_zero_nat)))))))),
% 2.74/1.77 inference(definition_unfolding,[status(thm)],[f834,f851,f851,f851])).
% 2.74/1.77 thf(f834,plain,(
% 2.74/1.77 (hF_Mirabelle_hpair = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ zero_z189798548lle_hf) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ zero_z189798548lle_hf)) @ zero_z189798548lle_hf))))))),
% 2.74/1.77 inference(cnf_transformation,[status(thm)],[f364])).
% 2.74/1.77 thf(f364,plain,(
% 2.74/1.77 (hF_Mirabelle_hpair = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ zero_z189798548lle_hf) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ Y0 @ (hF_Mirabelle_hinsert @ Y1 @ zero_z189798548lle_hf)) @ zero_z189798548lle_hf))))))),
% 2.74/1.77 inference(fool_elimination,[status(thm)],[f363])).
% 2.74/1.77 thf(f363,plain,(
% 2.74/1.77 (hF_Mirabelle_hpair = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X0 @ zero_z189798548lle_hf) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X0 @ (hF_Mirabelle_hinsert @ X1 @ zero_z189798548lle_hf)) @ zero_z189798548lle_hf))))),
% 2.74/1.77 inference(rectify,[status(thm)],[f14])).
% 2.74/1.77 thf(f14,axiom,(
% 2.74/1.77 (hF_Mirabelle_hpair = (^[X8 : hF_Mirabelle_hf, X9 : hF_Mirabelle_hf] : (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X8 @ zero_z189798548lle_hf) @ (hF_Mirabelle_hinsert @ (hF_Mirabelle_hinsert @ X8 @ (hF_Mirabelle_hinsert @ X9 @ zero_z189798548lle_hf)) @ zero_z189798548lle_hf))))),
% 2.74/1.77 file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_13_hpair__def)).
% 2.74/1.77 thf(f851,plain,(
% 2.74/1.77 (zero_z189798548lle_hf = (hF_Mirabelle_Abs_hf @ zero_zero_nat))),
% 2.74/1.77 inference(cnf_transformation,[status(thm)],[f23])).
% 2.74/1.77 thf(f23,axiom,(
% 2.74/1.77 (zero_z189798548lle_hf = (hF_Mirabelle_Abs_hf @ zero_zero_nat))),
% 2.74/1.77 file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_22_Abs__hf__0)).
% 2.74/1.77 thf(f860,plain,(
% 2.74/1.77 (zero_z189798548lle_hf = (hF_Mirabelle_hpair @ x @ y))),
% 2.74/1.77 inference(cnf_transformation,[status(thm)],[f796])).
% 2.74/1.77 thf(f796,plain,(
% 2.74/1.77 (zero_z189798548lle_hf = (hF_Mirabelle_hpair @ x @ y))),
% 2.74/1.77 inference(flattening,[status(thm)],[f345])).
% 2.74/1.77 thf(f345,negated_conjecture,(
% 2.74/1.77 ~(zero_z189798548lle_hf != (hF_Mirabelle_hpair @ x @ y))),
% 2.74/1.77 inference(negated_conjecture,[status(cth)],[f344])).
% 2.74/1.77 thf(f344,conjecture,(
% 2.74/1.77 (zero_z189798548lle_hf != (hF_Mirabelle_hpair @ x @ y))),
% 2.74/1.77 file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',conj_0)).
% 2.74/1.77 thf(f921,plain,(
% 2.74/1.77 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X0 @ X1) != (hF_Mirabelle_Abs_hf @ zero_zero_nat))) )),
% 2.74/1.77 inference(definition_unfolding,[status(thm)],[f876,f851])).
% 2.74/1.77 thf(f876,plain,(
% 2.74/1.77 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((zero_z189798548lle_hf != (hF_Mirabelle_hinsert @ X0 @ X1))) )),
% 2.74/1.77 inference(cnf_transformation,[status(thm)],[f797])).
% 2.74/1.77 thf(f797,plain,(
% 2.74/1.77 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (zero_z189798548lle_hf != (hF_Mirabelle_hinsert @ X0 @ X1))),
% 2.74/1.77 inference(rectify,[status(thm)],[f19])).
% 2.74/1.77 thf(f19,axiom,(
% 2.74/1.77 ! [X0 : hF_Mirabelle_hf,X12 : hF_Mirabelle_hf] : (zero_z189798548lle_hf != (hF_Mirabelle_hinsert @ X0 @ X12))),
% 2.74/1.77 file('/export/starexec/sandbox2/tmp/tmp.I48mCQMStL/DTF2THF_1223.p',fact_18_hinsert__nonempty)).
% 2.74/1.77 % SZS output end Proof for DTF2THF_1223
% 2.74/1.77 % (1440)------------------------------
% 2.74/1.77 % (1440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.74/1.77 % (1440)Termination reason: Refutation
% 2.74/1.77
% 2.74/1.77 % (1440)Memory used [KB]: 6268
% 2.74/1.77 % (1440)Time elapsed: 0.021 s
% 2.74/1.77 % (1440)Instructions burned: 63 (million)
% 2.74/1.77 % (1440)------------------------------
% 2.74/1.77 % (1440)------------------------------
% 2.74/1.77 % (1402)Success in time 0.164 s
%------------------------------------------------------------------------------