%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP073^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:19 AM UTC 2026
% Result : Theorem 2.94s 1.78s
% Output : Refutation 2.94s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 4
% Syntax : Number of formulae : 32 ( 24 unt; 0 typ; 0 def)
% Number of atoms : 118 ( 32 equ; 0 cnn)
% Maximal formula atoms : 2 ( 3 avg)
% Number of connectives : 173 ( 12 ~; 5 |; 0 &; 139 @)
% ( 0 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 172 ( 168 usr; 8 con; 0-3 aty)
% ( 5 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 47 ( 38 ^; 9 !; 0 ?; 47 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_HF_Mirabelle_hf: $tType ).
thf(type_def_6,type,
nat: $tType ).
thf(type_def_8,type,
set_nat: $tType ).
thf(type_def_9,type,
set_se933006839lle_hf: $tType ).
thf(type_def_10,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_0,type,
set_se933006839lle_hf: $tType ).
thf(func_def_1,type,
set_HF_Mirabelle_hf: $tType ).
thf(func_def_2,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_3,type,
set_nat: $tType ).
thf(func_def_4,type,
nat: $tType ).
thf(func_def_5,type,
finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
thf(func_def_6,type,
finite_card_nat: set_nat > nat ).
thf(func_def_7,type,
finite90088345lle_hf: set_se933006839lle_hf > nat ).
thf(func_def_8,type,
finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
thf(func_def_9,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_10,type,
finite1450550360lle_hf: set_se933006839lle_hf > $o ).
thf(func_def_11,type,
inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
thf(func_def_12,type,
inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
thf(func_def_13,type,
inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
thf(func_def_14,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_15,type,
inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
thf(func_def_16,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(func_def_17,type,
zero_zero_nat: nat ).
thf(func_def_18,type,
hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_19,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_20,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_21,type,
hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_22,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_23,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_24,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_25,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_26,type,
ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_27,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_28,type,
ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_29,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_30,type,
ord_le2017230260lle_hf: ( $o > hF_Mirabelle_hf ) > ( $o > hF_Mirabelle_hf ) > $o ).
thf(func_def_31,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(func_def_32,type,
ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_33,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_34,type,
ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_35,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_36,type,
order_1640852850lle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_37,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_38,type,
order_690706045lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > $o ).
thf(func_def_39,type,
order_557194383hf_nat: ( hF_Mirabelle_hf > nat ) > $o ).
thf(func_def_40,type,
order_671905679lle_hf: ( nat > hF_Mirabelle_hf ) > $o ).
thf(func_def_41,type,
order_1631207636at_nat: ( nat > nat ) > $o ).
thf(func_def_42,type,
orderi1737556723lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > $o ).
thf(func_def_43,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_44,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(func_def_45,type,
collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
thf(func_def_46,type,
image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_47,type,
image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
thf(func_def_48,type,
image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
thf(func_def_49,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_50,type,
image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
thf(func_def_51,type,
set_or1544565540an_nat: nat > nat > set_nat ).
thf(func_def_52,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_53,type,
member_nat: nat > set_nat > $o ).
thf(func_def_54,type,
member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
thf(func_def_55,type,
a: hF_Mirabelle_hf ).
thf(func_def_56,type,
b: hF_Mirabelle_hf ).
thf(func_def_71,type,
sK0: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_72,type,
sK1: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_73,type,
sK2: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_74,type,
sK3: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_75,type,
sK4: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_76,type,
sK5: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_77,type,
sK6: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_78,type,
sK7: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_79,type,
sK8: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_80,type,
sK9: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_81,type,
sK10: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_82,type,
sK11: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_83,type,
sK12: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_84,type,
sK13: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_85,type,
sK14: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_86,type,
sK15: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_87,type,
sK16: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_88,type,
sK17: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_89,type,
sK18: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_90,type,
sK19: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_91,type,
sK20: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_92,type,
sK21: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_93,type,
sK22: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_94,type,
sK23: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_95,type,
sK24: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_96,type,
sK25: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_97,type,
sK26: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_98,type,
sK27: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_99,type,
sK28: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_100,type,
sK29: ( nat > hF_Mirabelle_hf ) > nat ).
thf(func_def_101,type,
sK30: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_102,type,
sK31: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_103,type,
sK32: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_104,type,
sK33: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_105,type,
sK34: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_106,type,
sK35: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_107,type,
sK36: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_108,type,
sK37: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_109,type,
sK38: ( hF_Mirabelle_hf > nat ) > hF_Mirabelle_hf ).
thf(func_def_110,type,
sK39: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_111,type,
sK40: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_112,type,
sK41: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_113,type,
sK42: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_115,type,
sK44: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_116,type,
sK45: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_117,type,
sK46: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_118,type,
sK47: hF_Mirabelle_hf ).
thf(func_def_119,type,
sK48: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_120,type,
sK49: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_121,type,
sK50: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_122,type,
sK51: nat > ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_123,type,
sK52: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_124,type,
sK53: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_125,type,
sK54: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_126,type,
sK55: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_127,type,
sK56: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_128,type,
sK57: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_129,type,
sK58: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_130,type,
sK59: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_131,type,
sK60: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_132,type,
sK61: ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_133,type,
sK62: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_134,type,
sK63: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_135,type,
sK64: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_136,type,
sK65: hF_Mirabelle_hf > hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_137,type,
sK66: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_138,type,
sK67: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_139,type,
sK68: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_140,type,
sK69: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_141,type,
sK70: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_142,type,
sK71: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_143,type,
sK72: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_144,type,
sK73: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_145,type,
sK74: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_146,type,
sK75: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_147,type,
sK76: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_148,type,
sK77: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_149,type,
sK78: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_150,type,
sK79: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_151,type,
sK80: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_152,type,
sK81: ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_153,type,
sK82: nat > ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_154,type,
sK83: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_155,type,
sK84: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_156,type,
sK85: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_157,type,
sK86: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_158,type,
sK87: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_159,type,
sK88: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_160,type,
sK89: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_161,type,
sK90: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_162,type,
sK91: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_163,type,
sK92: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_164,type,
sK93: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_165,type,
sK94: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_166,type,
sK95: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_167,type,
sK96: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_168,type,
sK97: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_169,type,
sK98: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_170,type,
sK99: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_171,type,
sK100: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_172,type,
sK101: ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_173,type,
sK102: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_174,type,
sK103: ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_175,type,
sK104: nat > ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_176,type,
sK105: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_177,type,
sK106: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_178,type,
sK107: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_179,type,
sK108: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_180,type,
sK109: hF_Mirabelle_hf > hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_181,type,
sK110: nat > ( nat > hF_Mirabelle_hf ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_182,type,
sK111: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_183,type,
sK112: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_184,type,
sK113: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_185,type,
sK114: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_186,type,
sK115: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_187,type,
sK116: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(f1920,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1919]) ).
thf(f1919,plain,
$true = $false,
inference(forward_demodulation,[status(thm)],[f1918,f1327]) ).
thf(f1327,plain,
( $false
= ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ b ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1326]) ).
thf(f1326,plain,
( $false
= ( ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ a ) )
=> ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ b ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1325]) ).
thf(f1325,plain,
( $false
= ( ^ [Y0: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ a ) )
=> ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ b ) ) )
@ sK47 ) ),
inference(sigma_clausification,[status(thm)],[f1324]) ).
thf(f1324,plain,
( $true
!= ( !! @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ a ) )
=> ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ b ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1243]) ).
thf(f1243,plain,
( $true
!= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( ^ [Y3: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y3 @ ( hF_Mirabelle_hfset @ Y4 ) )
@ Y2
@ Y0 )
=> ( ^ [Y3: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y3 @ ( hF_Mirabelle_hfset @ Y4 ) )
@ Y2
@ Y1 ) ) )
@ a
@ b ) ),
inference(definition_unfolding,[status(thm)],[f1179,f1203]) ).
thf(f1203,plain,
( ord_le976219883lle_hf
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( ^ [Y3: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y3 @ ( hF_Mirabelle_hfset @ Y4 ) )
@ Y2
@ Y0 )
=> ( ^ [Y3: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y3 @ ( hF_Mirabelle_hfset @ Y4 ) )
@ Y2
@ Y1 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1159,f1164,f1164]) ).
thf(f1164,plain,
( hF_Mirabelle_hmem
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f624]) ).
thf(f624,plain,
( hF_Mirabelle_hmem
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f623]) ).
thf(f623,plain,
( ( ^ [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ X1 ) ) )
= hF_Mirabelle_hmem ),
inference(rectify,[status(thm)],[f71]) ).
thf(f71,axiom,
( ( ^ [X5: hF_Mirabelle_hf,X6: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X5 @ ( hF_Mirabelle_hfset @ X6 ) ) )
= hF_Mirabelle_hmem ),
file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',fact_70_hmem__def) ).
thf(f1159,plain,
( ord_le976219883lle_hf
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y2 @ Y0 )
=> ( hF_Mirabelle_hmem @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f738]) ).
thf(f738,plain,
( ord_le976219883lle_hf
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] :
( !! @ hF_Mirabelle_hf
@ ^ [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y2 @ Y0 )
=> ( hF_Mirabelle_hmem @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f737]) ).
thf(f737,plain,
( ord_le976219883lle_hf
= ( ^ [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ X0 )
=> ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
( ord_le976219883lle_hf
= ( ^ [X20: hF_Mirabelle_hf,X21: hF_Mirabelle_hf] :
! [X7: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X7 @ X20 )
=> ( hF_Mirabelle_hmem @ X7 @ X21 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',fact_6_less__eq__hf__def) ).
thf(f1179,plain,
( ( ord_le976219883lle_hf @ a @ b )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1032]) ).
thf(f1032,plain,
( ( ord_le976219883lle_hf @ a @ b )
!= $true ),
inference(flattening,[status(thm)],[f670]) ).
thf(f670,plain,
( ( ord_le976219883lle_hf @ a @ b )
!= $true ),
inference(fool_elimination,[status(thm)],[f669]) ).
thf(f669,plain,
~ ( ord_le976219883lle_hf @ a @ b ),
inference(rectify,[status(thm)],[f350]) ).
thf(f350,negated_conjecture,
~ ( ord_le976219883lle_hf @ a @ b ),
inference(negated_conjecture,[status(cth)],[f349]) ).
thf(f349,conjecture,
ord_le976219883lle_hf @ a @ b,
file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',conj_1) ).
thf(f1918,plain,
( $true
= ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ b ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1917]) ).
thf(f1917,plain,
( ( $true != $true )
| ( $true
= ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ b ) ) ) ),
inference(superposition,[status(thm)],[f1773,f1328]) ).
thf(f1328,plain,
( $true
= ( member1367349282lle_hf @ sK47 @ ( hF_Mirabelle_hfset @ a ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1326]) ).
thf(f1773,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ a ) ) )
| ( ( member1367349282lle_hf @ X0 @ ( hF_Mirabelle_hfset @ b ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1241]) ).
thf(f1241,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X0
@ a ) )
| ( $true
= ( ^ [Y0: hF_Mirabelle_hf,Y1: hF_Mirabelle_hf] : ( member1367349282lle_hf @ Y0 @ ( hF_Mirabelle_hfset @ Y1 ) )
@ X0
@ b ) ) ),
inference(definition_unfolding,[status(thm)],[f1177,f1164,f1164]) ).
thf(f1177,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( $true
= ( hF_Mirabelle_hmem @ X0 @ b ) ) ),
inference(cnf_transformation,[status(thm)],[f1066]) ).
thf(f1066,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( $true
= ( hF_Mirabelle_hmem @ X0 @ b ) ) ),
inference(ennf_transformation,[status(thm)],[f867]) ).
thf(f867,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
= ( hF_Mirabelle_hmem @ X0 @ a ) )
=> ( $true
= ( hF_Mirabelle_hmem @ X0 @ b ) ) ),
inference(fool_elimination,[status(thm)],[f866]) ).
thf(f866,plain,
! [X0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X0 @ a )
=> ( hF_Mirabelle_hmem @ X0 @ b ) ),
inference(rectify,[status(thm)],[f348]) ).
thf(f348,axiom,
! [X46: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X46 @ a )
=> ( hF_Mirabelle_hmem @ X46 @ b ) ),
file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP073^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n001.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 08:25:50 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP073^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer HF problem prob_473__5331678_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : HF/prob_473__5331678_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 406 ( 92 unt; 57 typ; 0 def)
% 0.19/0.45 % Number of atoms : 1190 ( 228 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 12 ( 3 avg)
% 0.19/0.45 % Number of connectives : 3157 ( 104 ~; 22 |; 94 &;2290 @)
% 0.19/0.45 % ( 0 <=>; 647 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 18 ( 8 avg)
% 0.19/0.45 % Number of types : 6 ( 5 usr)
% 0.19/0.45 % Number of type conns : 294 ( 294 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 55 ( 52 usr; 6 con; 0-2 aty)
% 0.19/0.45 % Number of variables : 1113 ( 76 ^; 968 !; 69 ?;1113 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:37:18.035
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (5)
% 0.19/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 0.19/0.45 set_se933006839lle_hf: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 set_HF_Mirabelle_hf: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 hF_Mirabelle_hf: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.19/0.45 set_nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (52)
% 0.19/0.45 thf(sy_c_Finite__Set_Ocard_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.19/0.45 finite_card_nat: set_nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 finite90088345lle_hf: set_se933006839lle_hf > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Finite__Set_Ofinite_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.19/0.45 finite_finite_nat: set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 finite1450550360lle_hf: set_se933006839lle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.19/0.45 inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.19/0.45 zero_zero_nat: nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.19/0.45 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.19/0.45 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.19/0.45 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.19/0.45 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.19/0.45 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.19/0.45 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.19/0.45 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.19/0.45 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.19/0.45 ord_less_nat: nat > nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.19/0.45 ord_less_set_nat: set_nat > set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 ord_le2017230260lle_hf: ( $o > hF_Mirabelle_hf ) > ( $o > hF_Mirabelle_hf ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 0.19/0.45 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.19/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.19/0.45 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 order_1640852850lle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 0.19/0.45 order_Greatest_nat: ( nat > $o ) > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 order_690706045lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.19/0.45 order_557194383hf_nat: ( hF_Mirabelle_hf > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 order_671905679lle_hf: ( nat > hF_Mirabelle_hf ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 order_1631207636at_nat: ( nat > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oordering_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 orderi1737556723lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
% 0.19/0.45 ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 0.19/0.45 image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 0.19/0.45 set_or1544565540an_nat: nat > nat > set_nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.19/0.45 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001t__Nat__Onat,type,
% 0.19/0.45 member_nat: nat > set_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.19/0.45 member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_A,type,
% 0.19/0.45 a: hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_B,type,
% 0.19/0.45 b: hF_Mirabelle_hf ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (347)
% 0.19/0.45 thf(fact_0_hf__equalityI,axiom,
% 0.19/0.45 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.19/0.45 ( ! [X: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X @ A )
% 0.19/0.45 = ( hF_Mirabelle_hmem @ X @ B ) )
% 0.19/0.45 => ( A = B ) ) ).
% 0.19/0.45
% 0.19/0.45 % hf_equalityI
% 0.19/0.45 thf(fact_1_hf__ext,axiom,
% 0.19/0.45 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [X2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.19/0.45 = ( hF_Mirabelle_hmem @ X2 @ B2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % hf_ext
% 0.19/0.45 thf(fact_2_replacement,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.19/0.45 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U @ X3 )
% 0.19/0.45 => ( ( R @ U @ V )
% 0.19/0.45 => ( ( R @ U @ V2 )
% 0.19/0.45 => ( V2 = V ) ) ) )
% 0.19/0.45 => ? [Z2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [V3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.19/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U2 @ X3 )
% 0.19/0.45 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % replacement
% 0.19/0.45 thf(fact_3_binary__union,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.45 ? [Z2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.19/0.45 = ( ( hF_Mirabelle_hmem @ U3 @ X3 )
% 0.19/0.45 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binary_union
% 0.19/0.45 thf(fact_4_union__of__set,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf] :
% 0.19/0.45 ? [Z2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.19/0.45 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ Y3 @ X3 )
% 0.19/0.45 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % union_of_set
% 0.19/0.45 thf(fact_5_comprehension,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.19/0.45 ? [Z2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [U3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.19/0.45 = ( ( hF_Mirabelle_hmem @ U3 @ X3 )
% 0.19/0.45 & ( P @ U3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % comprehension
% 0.19/0.45 thf(fact_6_less__eq__hf__def,axiom,
% 0.19/0.45 ( ord_le976219883lle_hf
% 0.19/0.45 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.19/0.45 ! [X2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 0.19/0.45 => ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_eq_hf_def
% 0.19/0.45 thf(fact_7_replacement__fun,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.19/0.45 ? [Z2: hF_Mirabelle_hf] :
% 0.19/0.45 ! [V3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.19/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U2 @ X3 )
% 0.19/0.45 & ( V3
% 0.19/0.45 = ( F @ U2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % replacement_fun
% 0.19/0.45 thf(fact_8_order__refl,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X3 @ X3 ) ).
% 0.19/0.45
% 0.19/0.45 % order_refl
% 0.19/0.45 thf(fact_9_order__refl,axiom,
% 0.19/0.45 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.19/0.45
% 0.19/0.45 % order_refl
% 0.19/0.45 thf(fact_10_PrimReplace__iff,axiom,
% 0.19/0.45 ! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U @ A4 )
% 0.19/0.45 => ( ( R @ U @ V )
% 0.19/0.45 => ( ( R @ U @ V2 )
% 0.19/0.45 => ( V2 = V ) ) ) )
% 0.19/0.45 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A4 @ R ) )
% 0.19/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.19/0.45 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % PrimReplace_iff
% 0.19/0.45 thf(fact_11_HCollect__iff,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mir818139703ollect @ P @ A4 ) )
% 0.19/0.45 = ( ( P @ X3 )
% 0.19/0.45 & ( hF_Mirabelle_hmem @ X3 @ A4 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % HCollect_iff
% 0.19/0.45 thf(fact_12_HUnion__iff,axiom,
% 0.19/0.45 ! [X3: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HUnion @ A4 ) )
% 0.19/0.45 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.19/0.45 & ( hF_Mirabelle_hmem @ X3 @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % HUnion_iff
% 0.19/0.45 thf(fact_13_Replace__iff,axiom,
% 0.19/0.45 ! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A4 @ R ) )
% 0.19/0.45 = ( ? [U2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.19/0.45 & ( R @ U2 @ V4 )
% 0.19/0.45 & ! [Y3: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( R @ U2 @ Y3 )
% 0.19/0.45 => ( Y3 = V4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Replace_iff
% 0.19/0.45 thf(fact_14_order__subst1,axiom,
% 0.19/0.45 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.45 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst1
% 0.19/0.45 thf(fact_15_order__subst1,axiom,
% 0.19/0.45 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 0.19/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.45 => ( ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst1
% 0.19/0.45 thf(fact_16_order__subst1,axiom,
% 0.19/0.45 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.45 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst1
% 0.19/0.45 thf(fact_17_order__subst1,axiom,
% 0.19/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.19/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.45 => ( ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst1
% 0.19/0.45 thf(fact_18_order__subst2,axiom,
% 0.19/0.45 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 0.19/0.45 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst2
% 0.19/0.45 thf(fact_19_order__subst2,axiom,
% 0.19/0.45 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.19/0.45 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst2
% 0.19/0.45 thf(fact_20_order__subst2,axiom,
% 0.19/0.45 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 0.19/0.45 => ( ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst2
% 0.19/0.45 thf(fact_21_order__subst2,axiom,
% 0.19/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.19/0.45 => ( ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % order_subst2
% 0.19/0.45 thf(fact_22_verit__la__disequality,axiom,
% 0.19/0.45 ! [A: nat,B: nat] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 | ~ ( ord_less_eq_nat @ A @ B )
% 0.19/0.45 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 % verit_la_disequality
% 0.19/0.45 thf(fact_23_Replace__cong,axiom,
% 0.19/0.45 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.19/0.45 ( ( A4 = B4 )
% 0.19/0.45 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.19/0.45 => ( ( P @ X @ Y4 )
% 0.19/0.45 = ( Q @ X @ Y4 ) ) )
% 0.19/0.45 => ( ( hF_Mirabelle_Replace @ A4 @ P )
% 0.19/0.45 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Replace_cong
% 0.19/0.45 thf(fact_24_dual__order_Oantisym,axiom,
% 0.19/0.45 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ B @ A )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.45 => ( A = B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.antisym
% 0.19/0.45 thf(fact_25_dual__order_Oantisym,axiom,
% 0.19/0.45 ! [B: nat,A: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.45 => ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.45 => ( A = B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.antisym
% 0.19/0.45 thf(fact_26_dual__order_Oeq__iff,axiom,
% 0.19/0.45 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ B2 @ A2 )
% 0.19/0.45 & ( ord_le976219883lle_hf @ A2 @ B2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.eq_iff
% 0.19/0.45 thf(fact_27_dual__order_Oeq__iff,axiom,
% 0.19/0.45 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.19/0.45 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.19/0.45 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.eq_iff
% 0.19/0.45 thf(fact_28_dual__order_Otrans,axiom,
% 0.19/0.45 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.45 ( ( ord_le976219883lle_hf @ B @ A )
% 0.19/0.45 => ( ( ord_le976219883lle_hf @ C @ B )
% 0.19/0.45 => ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.trans
% 0.19/0.45 thf(fact_29_dual__order_Otrans,axiom,
% 0.19/0.45 ! [B: nat,A: nat,C: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.45 => ( ( ord_less_eq_nat @ C @ B )
% 0.19/0.45 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dual_order.trans
% 0.19/0.45 thf(fact_30_linorder__wlog,axiom,
% 0.19/0.45 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.45 ( ! [A5: nat,B5: nat] :
% 0.19/0.45 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.19/0.45 => ( P @ A5 @ B5 ) )
% 0.19/0.46 => ( ! [A5: nat,B5: nat] :
% 0.19/0.46 ( ( P @ B5 @ A5 )
% 0.19/0.46 => ( P @ A5 @ B5 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_31_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_32_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_33_order__trans,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ X3 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_34_order__trans,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat,Z3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.19/0.46 => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_35_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_36_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_37_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_38_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_39_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.46 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_40_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_41_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.19/0.46 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A2 @ B2 )
% 0.19/0.46 & ( ord_le976219883lle_hf @ B2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_42_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.19/0.46 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.19/0.46 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_43_antisym__conv,axiom,
% 0.19/0.46 ! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 = ( X3 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_44_antisym__conv,axiom,
% 0.19/0.46 ! [Y2: nat,X3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 = ( X3 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_45_le__cases3,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat,Z3: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_46_order_Otrans,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.46 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_47_order_Otrans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_48_le__cases,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_49_eq__refl,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( X3 = Y2 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_50_eq__refl,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( X3 = Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_51_linear,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_52_antisym,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 0.19/0.46 => ( X3 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_53_antisym,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.19/0.46 => ( X3 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_54_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.19/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 0.19/0.46 & ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_55_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.19/0.46 = ( ^ [X2: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.19/0.46 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_56_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_57_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_58_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_59_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_60_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_61_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_62_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_63_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_64_Greatest__equality,axiom,
% 0.19/0.46 ! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 => ( ! [Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ Y4 @ X3 ) )
% 0.19/0.46 => ( ( order_1640852850lle_hf @ P )
% 0.19/0.46 = X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Greatest_equality
% 0.19/0.46 thf(fact_65_Greatest__equality,axiom,
% 0.19/0.46 ! [P: nat > $o,X3: nat] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 => ( ! [Y4: nat] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y4 @ X3 ) )
% 0.19/0.46 => ( ( order_Greatest_nat @ P )
% 0.19/0.46 = X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Greatest_equality
% 0.19/0.46 thf(fact_66_GreatestI2__order,axiom,
% 0.19/0.46 ! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 => ( ! [Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ Y4 @ X3 ) )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 => ( ! [Y5: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ Y5 @ X ) )
% 0.19/0.46 => ( Q @ X ) ) )
% 0.19/0.46 => ( Q @ ( order_1640852850lle_hf @ P ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % GreatestI2_order
% 0.19/0.46 thf(fact_67_GreatestI2__order,axiom,
% 0.19/0.46 ! [P: nat > $o,X3: nat,Q: nat > $o] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 => ( ! [Y4: nat] :
% 0.19/0.46 ( ( P @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y4 @ X3 ) )
% 0.19/0.46 => ( ! [X: nat] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 => ( ! [Y5: nat] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y5 @ X ) )
% 0.19/0.46 => ( Q @ X ) ) )
% 0.19/0.46 => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % GreatestI2_order
% 0.19/0.46 thf(fact_68_le__rel__bool__arg__iff,axiom,
% 0.19/0.46 ( ord_le2017230260lle_hf
% 0.19/0.46 = ( ^ [X4: $o > hF_Mirabelle_hf,Y6: $o > hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 0.19/0.46 & ( ord_le976219883lle_hf @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_rel_bool_arg_iff
% 0.19/0.46 thf(fact_69_le__rel__bool__arg__iff,axiom,
% 0.19/0.46 ( ord_less_eq_o_nat
% 0.19/0.46 = ( ^ [X4: $o > nat,Y6: $o > nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 0.19/0.46 & ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_rel_bool_arg_iff
% 0.19/0.46 thf(fact_70_hmem__def,axiom,
% 0.19/0.46 ( hF_Mirabelle_hmem
% 0.19/0.46 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % hmem_def
% 0.19/0.46 thf(fact_71_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.19/0.46 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_72_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.19/0.46 ( ( member1490636632lle_hf @ A @ ( collec1758573718lle_hf @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_73_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.19/0.46 ( ( collec2046588256lle_hf
% 0.19/0.46 @ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A4 ) )
% 0.19/0.46 = A4 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_74_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A4: set_se933006839lle_hf] :
% 0.19/0.46 ( ( collec1758573718lle_hf
% 0.19/0.46 @ ^ [X2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X2 @ A4 ) )
% 0.19/0.46 = A4 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_75_Collect__cong,axiom,
% 0.19/0.46 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 0.19/0.46 ( ! [X: set_HF_Mirabelle_hf] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 = ( Q @ X ) )
% 0.19/0.46 => ( ( collec1758573718lle_hf @ P )
% 0.19/0.46 = ( collec1758573718lle_hf @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_76_RepFun__iff,axiom,
% 0.19/0.46 ! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.19/0.46 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
% 0.19/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.19/0.46 & ( V4
% 0.19/0.46 = ( F @ U2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % RepFun_iff
% 0.19/0.46 thf(fact_77_less__hf__def,axiom,
% 0.19/0.46 ( ord_le1310584031lle_hf
% 0.19/0.46 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ A3 @ B3 )
% 0.19/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_hf_def
% 0.19/0.46 thf(fact_78_antimono__def,axiom,
% 0.19/0.46 ( order_690706045lle_hf
% 0.19/0.46 = ( ^ [F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.19/0.46 ! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimono_def
% 0.19/0.46 thf(fact_79_antimono__def,axiom,
% 0.19/0.46 ( order_557194383hf_nat
% 0.19/0.46 = ( ^ [F2: hF_Mirabelle_hf > nat] :
% 0.19/0.46 ! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimono_def
% 0.19/0.46 thf(fact_80_antimono__def,axiom,
% 0.19/0.46 ( order_671905679lle_hf
% 0.19/0.46 = ( ^ [F2: nat > hF_Mirabelle_hf] :
% 0.19/0.46 ! [X2: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimono_def
% 0.19/0.46 thf(fact_81_antimono__def,axiom,
% 0.19/0.46 ( order_1631207636at_nat
% 0.19/0.46 = ( ^ [F2: nat > nat] :
% 0.19/0.46 ! [X2: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimono_def
% 0.19/0.46 thf(fact_82_antimonoI,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.19/0.46 ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.19/0.46 => ( order_690706045lle_hf @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoI
% 0.19/0.46 thf(fact_83_antimonoI,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > nat] :
% 0.19/0.46 ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.19/0.46 => ( order_557194383hf_nat @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoI
% 0.19/0.46 thf(fact_84_antimonoI,axiom,
% 0.19/0.46 ! [F: nat > hF_Mirabelle_hf] :
% 0.19/0.46 ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.19/0.46 => ( order_671905679lle_hf @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoI
% 0.19/0.46 thf(fact_85_antimonoI,axiom,
% 0.19/0.46 ! [F: nat > nat] :
% 0.19/0.46 ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.19/0.46 => ( order_1631207636at_nat @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoI
% 0.19/0.46 thf(fact_86_antimonoE,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( order_690706045lle_hf @ F )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoE
% 0.19/0.46 thf(fact_87_antimonoE,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( order_557194383hf_nat @ F )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoE
% 0.19/0.46 thf(fact_88_antimonoE,axiom,
% 0.19/0.46 ! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
% 0.19/0.46 ( ( order_671905679lle_hf @ F )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoE
% 0.19/0.46 thf(fact_89_antimonoE,axiom,
% 0.19/0.46 ! [F: nat > nat,X3: nat,Y2: nat] :
% 0.19/0.46 ( ( order_1631207636at_nat @ F )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoE
% 0.19/0.46 thf(fact_90_antimonoD,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( order_690706045lle_hf @ F )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoD
% 0.19/0.46 thf(fact_91_antimonoD,axiom,
% 0.19/0.46 ! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( order_557194383hf_nat @ F )
% 0.19/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoD
% 0.19/0.46 thf(fact_92_antimonoD,axiom,
% 0.19/0.46 ! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
% 0.19/0.46 ( ( order_671905679lle_hf @ F )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoD
% 0.19/0.46 thf(fact_93_antimonoD,axiom,
% 0.19/0.46 ! [F: nat > nat,X3: nat,Y2: nat] :
% 0.19/0.46 ( ( order_1631207636at_nat @ F )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antimonoD
% 0.19/0.46 thf(fact_94_verit__comp__simplify1_I1_J,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf] :
% 0.19/0.46 ~ ( ord_le1310584031lle_hf @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % verit_comp_simplify1(1)
% 0.19/0.46 thf(fact_95_verit__comp__simplify1_I1_J,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % verit_comp_simplify1(1)
% 0.19/0.46 thf(fact_96_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_97_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_98_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_99_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_100_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_101_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_102_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_103_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_104_order__less__subst1,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_105_order__less__subst1,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_106_order__less__subst1,axiom,
% 0.19/0.46 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_107_order__less__subst1,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_108_order__less__subst2,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_109_order__less__subst2,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_110_order__less__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_111_order__less__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: nat,Y4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_112_gt__ex,axiom,
% 0.19/0.46 ! [X3: nat] :
% 0.19/0.46 ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 0.19/0.46
% 0.19/0.46 % gt_ex
% 0.19/0.46 thf(fact_113_neqE,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( X3 != Y2 )
% 0.19/0.46 => ( ~ ( ord_less_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neqE
% 0.19/0.46 thf(fact_114_neq__iff,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( X3 != Y2 )
% 0.19/0.46 = ( ( ord_less_nat @ X3 @ Y2 )
% 0.19/0.46 | ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_iff
% 0.19/0.46 thf(fact_115_order_Oasym,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.asym
% 0.19/0.46 thf(fact_116_order_Oasym,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.asym
% 0.19/0.46 thf(fact_117_less__imp__neq,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( X3 != Y2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_neq
% 0.19/0.46 thf(fact_118_less__imp__neq,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.19/0.46 => ( X3 != Y2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_neq
% 0.19/0.46 thf(fact_119_less__asym,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym
% 0.19/0.46 thf(fact_120_less__asym,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.19/0.46 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym
% 0.19/0.46 thf(fact_121_less__asym_H,axiom,
% 0.19/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.19/0.46 => ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym'
% 0.19/0.46 thf(fact_122_less__asym_H,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym'
% 0.19/0.46 thf(fact_123_less__trans,axiom,
% 0.19/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.19/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
% 0.19/0.46 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_trans
% 0.19/0.46 thf(fact_124_less__trans,axiom,
% 0.19/0.46 ! [X3: nat,Y2: nat,Z3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.19/0.46 => ( ( ord_less_nat @ Y2 @ Z3 )
% 0.19/0.46 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_trans
% 0.31/0.46 thf(fact_125_less__linear,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 | ( X3 = Y2 )
% 0.31/0.46 | ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_linear
% 0.31/0.46 thf(fact_126_less__irrefl,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf] :
% 0.31/0.46 ~ ( ord_le1310584031lle_hf @ X3 @ X3 ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl
% 0.31/0.46 thf(fact_127_less__irrefl,axiom,
% 0.31/0.46 ! [X3: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ X3 @ X3 ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl
% 0.31/0.46 thf(fact_128_ord__eq__less__trans,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_trans
% 0.31/0.46 thf(fact_129_ord__eq__less__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_trans
% 0.31/0.46 thf(fact_130_ord__less__eq__trans,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_trans
% 0.31/0.46 thf(fact_131_ord__less__eq__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( B = C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_trans
% 0.31/0.46 thf(fact_132_dual__order_Oasym,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B @ A )
% 0.31/0.46 => ~ ( ord_le1310584031lle_hf @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.asym
% 0.31/0.46 thf(fact_133_dual__order_Oasym,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.asym
% 0.31/0.46 thf(fact_134_less__imp__not__eq,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( X3 != Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq
% 0.31/0.46 thf(fact_135_less__imp__not__eq,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( X3 != Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq
% 0.31/0.46 thf(fact_136_less__not__sym,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_sym
% 0.31/0.46 thf(fact_137_less__not__sym,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_sym
% 0.31/0.46 thf(fact_138_less__induct,axiom,
% 0.31/0.46 ! [P: nat > $o,A: nat] :
% 0.31/0.46 ( ! [X: nat] :
% 0.31/0.46 ( ! [Y5: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Y5 @ X )
% 0.31/0.46 => ( P @ Y5 ) )
% 0.31/0.46 => ( P @ X ) )
% 0.31/0.46 => ( P @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_induct
% 0.31/0.46 thf(fact_139_antisym__conv3,axiom,
% 0.31/0.46 ! [Y2: nat,X3: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ Y2 @ X3 )
% 0.31/0.46 => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 0.31/0.46 = ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv3
% 0.31/0.46 thf(fact_140_less__imp__not__eq2,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( Y2 != X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq2
% 0.31/0.46 thf(fact_141_less__imp__not__eq2,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( Y2 != X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq2
% 0.31/0.46 thf(fact_142_less__imp__triv,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,P: $o] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ Y2 @ X3 )
% 0.31/0.46 => P ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_triv
% 0.31/0.46 thf(fact_143_less__imp__triv,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat,P: $o] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_less_nat @ Y2 @ X3 )
% 0.31/0.46 => P ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_triv
% 0.31/0.46 thf(fact_144_linorder__cases,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( X3 != Y2 )
% 0.31/0.46 => ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_cases
% 0.31/0.46 thf(fact_145_dual__order_Oirrefl,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf] :
% 0.31/0.46 ~ ( ord_le1310584031lle_hf @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.irrefl
% 0.31/0.46 thf(fact_146_dual__order_Oirrefl,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.irrefl
% 0.31/0.46 thf(fact_147_order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans
% 0.31/0.46 thf(fact_148_order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans
% 0.31/0.46 thf(fact_149_less__imp__not__less,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_less
% 0.31/0.46 thf(fact_150_less__imp__not__less,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_less
% 0.31/0.46 thf(fact_151_exists__least__iff,axiom,
% 0.31/0.46 ( ( ^ [P2: nat > $o] :
% 0.31/0.46 ? [X5: nat] : ( P2 @ X5 ) )
% 0.31/0.46 = ( ^ [P3: nat > $o] :
% 0.31/0.46 ? [N: nat] :
% 0.31/0.46 ( ( P3 @ N )
% 0.31/0.46 & ! [M: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M @ N )
% 0.31/0.46 => ~ ( P3 @ M ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % exists_least_iff
% 0.31/0.46 thf(fact_152_linorder__less__wlog,axiom,
% 0.31/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.31/0.46 ( ! [A5: nat,B5: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A5 @ B5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 0.31/0.46 => ( ! [A5: nat,B5: nat] :
% 0.31/0.46 ( ( P @ B5 @ A5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_less_wlog
% 0.31/0.46 thf(fact_153_dual__order_Ostrict__trans,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B @ A )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ C @ B )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans
% 0.31/0.46 thf(fact_154_dual__order_Ostrict__trans,axiom,
% 0.31/0.46 ! [B: nat,A: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_nat @ C @ B )
% 0.31/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans
% 0.31/0.46 thf(fact_155_not__less__iff__gr__or__eq,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 0.31/0.46 = ( ( ord_less_nat @ Y2 @ X3 )
% 0.31/0.46 | ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_iff_gr_or_eq
% 0.31/0.46 thf(fact_156_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_not_eq
% 0.31/0.46 thf(fact_157_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_not_eq
% 0.31/0.46 thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B @ A )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_not_eq
% 0.31/0.46 thf(fact_159_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( A != B ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_not_eq
% 0.31/0.46 thf(fact_160_verit__comp__simplify1_I3_J,axiom,
% 0.31/0.46 ! [B6: nat,A6: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 0.31/0.46 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % verit_comp_simplify1(3)
% 0.31/0.46 thf(fact_161_leD,axiom,
% 0.31/0.46 ! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 0.31/0.46 => ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % leD
% 0.31/0.46 thf(fact_162_leD,axiom,
% 0.31/0.46 ! [Y2: nat,X3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 0.31/0.46 => ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % leD
% 0.31/0.46 thf(fact_163_leI,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % leI
% 0.31/0.46 thf(fact_164_le__less,axiom,
% 0.31/0.46 ( ord_le976219883lle_hf
% 0.31/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X2 @ Y3 )
% 0.31/0.46 | ( X2 = Y3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less
% 0.31/0.46 thf(fact_165_le__less,axiom,
% 0.31/0.46 ( ord_less_eq_nat
% 0.31/0.46 = ( ^ [X2: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X2 @ Y3 )
% 0.31/0.46 | ( X2 = Y3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less
% 0.31/0.46 thf(fact_166_less__le,axiom,
% 0.31/0.46 ( ord_le1310584031lle_hf
% 0.31/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 0.31/0.46 & ( X2 != Y3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le
% 0.31/0.46 thf(fact_167_less__le,axiom,
% 0.31/0.46 ( ord_less_nat
% 0.31/0.46 = ( ^ [X2: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.31/0.46 & ( X2 != Y3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le
% 0.31/0.46 thf(fact_168_order__le__less__subst1,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst1
% 0.31/0.46 thf(fact_169_order__le__less__subst1,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst1
% 0.31/0.46 thf(fact_170_order__le__less__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst1
% 0.31/0.46 thf(fact_171_order__le__less__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst1
% 0.31/0.46 thf(fact_172_order__le__less__subst2,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst2
% 0.31/0.46 thf(fact_173_order__le__less__subst2,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst2
% 0.31/0.46 thf(fact_174_order__le__less__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst2
% 0.31/0.46 thf(fact_175_order__le__less__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst2
% 0.31/0.46 thf(fact_176_order__less__le__subst1,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst1
% 0.31/0.46 thf(fact_177_order__less__le__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst1
% 0.31/0.46 thf(fact_178_order__less__le__subst1,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst1
% 0.31/0.46 thf(fact_179_order__less__le__subst1,axiom,
% 0.31/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst1
% 0.31/0.46 thf(fact_180_order__less__le__subst2,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst2
% 0.31/0.46 thf(fact_181_order__less__le__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst2
% 0.31/0.46 thf(fact_182_order__less__le__subst2,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst2
% 0.31/0.46 thf(fact_183_order__less__le__subst2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.31/0.46 => ( ! [X: nat,Y4: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Y4 )
% 0.31/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.31/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst2
% 0.31/0.46 thf(fact_184_not__le,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
% 0.31/0.46 = ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_le
% 0.31/0.46 thf(fact_185_not__less,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 0.31/0.46 = ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less
% 0.31/0.46 thf(fact_186_le__neq__trans,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.31/0.46 => ( ( A != B )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_neq_trans
% 0.31/0.46 thf(fact_187_le__neq__trans,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( A != B )
% 0.31/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_neq_trans
% 0.31/0.46 thf(fact_188_antisym__conv1,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.31/0.46 = ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv1
% 0.31/0.46 thf(fact_189_antisym__conv1,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.46 = ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv1
% 0.31/0.46 thf(fact_190_antisym__conv2,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) )
% 0.31/0.46 = ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv2
% 0.31/0.46 thf(fact_191_antisym__conv2,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 0.31/0.46 = ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv2
% 0.31/0.46 thf(fact_192_less__imp__le,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_le
% 0.31/0.46 thf(fact_193_less__imp__le,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_le
% 0.31/0.46 thf(fact_194_le__less__trans,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less_trans
% 0.31/0.46 thf(fact_195_le__less__trans,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat,Z3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_less_nat @ Y2 @ Z3 )
% 0.31/0.46 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less_trans
% 0.31/0.46 thf(fact_196_less__le__trans,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_trans
% 0.31/0.46 thf(fact_197_less__le__trans,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat,Z3: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.31/0.46 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_trans
% 0.31/0.46 thf(fact_198_le__less__linear,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.46 | ( ord_less_nat @ Y2 @ X3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less_linear
% 0.31/0.46 thf(fact_199_le__imp__less__or__eq,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 0.31/0.46 | ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_imp_less_or_eq
% 0.31/0.46 thf(fact_200_le__imp__less__or__eq,axiom,
% 0.31/0.46 ! [X3: nat,Y2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.31/0.46 => ( ( ord_less_nat @ X3 @ Y2 )
% 0.31/0.46 | ( X3 = Y2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_imp_less_or_eq
% 0.31/0.46 thf(fact_201_less__le__not__le,axiom,
% 0.31/0.46 ( ord_le1310584031lle_hf
% 0.31/0.46 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 0.31/0.46 & ~ ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_not_le
% 0.31/0.46 thf(fact_202_less__le__not__le,axiom,
% 0.31/0.46 ( ord_less_nat
% 0.31/0.46 = ( ^ [X2: nat,Y3: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.31/0.46 & ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_not_le
% 0.31/0.46 thf(fact_203_not__le__imp__less,axiom,
% 0.31/0.46 ! [Y2: nat,X3: nat] :
% 0.31/0.46 ( ~ ( ord_less_eq_nat @ Y2 @ X3 )
% 0.31/0.46 => ( ord_less_nat @ X3 @ Y2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_le_imp_less
% 0.31/0.46 thf(fact_204_order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ B @ C )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans1
% 0.31/0.46 thf(fact_205_order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans1
% 0.31/0.46 thf(fact_206_order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ B @ C )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans2
% 0.31/0.46 thf(fact_207_order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [A: nat,B: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.31/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans2
% 0.31/0.46 thf(fact_208_order_Oorder__iff__strict,axiom,
% 0.31/0.46 ( ord_le976219883lle_hf
% 0.31/0.46 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A2 @ B2 )
% 0.31/0.46 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.order_iff_strict
% 0.31/0.46 thf(fact_209_order_Oorder__iff__strict,axiom,
% 0.31/0.46 ( ord_less_eq_nat
% 0.31/0.46 = ( ^ [A2: nat,B2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A2 @ B2 )
% 0.31/0.46 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.order_iff_strict
% 0.31/0.46 thf(fact_210_order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_le1310584031lle_hf
% 0.31/0.46 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ A2 @ B2 )
% 0.31/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_iff_order
% 0.31/0.46 thf(fact_211_order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_less_nat
% 0.31/0.46 = ( ^ [A2: nat,B2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.31/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_iff_order
% 0.31/0.46 thf(fact_212_dual__order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ B @ A )
% 0.31/0.46 => ( ( ord_le1310584031lle_hf @ C @ B )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans1
% 0.31/0.46 thf(fact_213_dual__order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [B: nat,A: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_nat @ C @ B )
% 0.31/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans1
% 0.31/0.46 thf(fact_214_dual__order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B @ A )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ C @ B )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans2
% 0.31/0.46 thf(fact_215_dual__order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [B: nat,A: nat,C: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.31/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans2
% 0.31/0.46 thf(fact_216_order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ A @ B )
% 0.31/0.46 => ( ord_le976219883lle_hf @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_order
% 0.31/0.46 thf(fact_217_order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_order
% 0.31/0.46 thf(fact_218_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.46 ( ord_le976219883lle_hf
% 0.31/0.46 = ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B2 @ A2 )
% 0.31/0.46 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.order_iff_strict
% 0.31/0.46 thf(fact_219_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.46 ( ord_less_eq_nat
% 0.31/0.46 = ( ^ [B2: nat,A2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B2 @ A2 )
% 0.31/0.46 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.order_iff_strict
% 0.31/0.46 thf(fact_220_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_le1310584031lle_hf
% 0.31/0.46 = ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le976219883lle_hf @ B2 @ A2 )
% 0.31/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_iff_order
% 0.31/0.46 thf(fact_221_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.46 ( ord_less_nat
% 0.31/0.46 = ( ^ [B2: nat,A2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.31/0.46 & ( A2 != B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_iff_order
% 0.31/0.46 thf(fact_222_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1310584031lle_hf @ B @ A )
% 0.31/0.46 => ( ord_le976219883lle_hf @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_order
% 0.31/0.46 thf(fact_223_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [B: nat,A: nat] :
% 0.31/0.46 ( ( ord_less_nat @ B @ A )
% 0.31/0.46 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_order
% 0.31/0.46 thf(fact_224_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( A != B )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ A @ B )
% 0.31/0.46 => ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.not_eq_order_implies_strict
% 0.31/0.46 thf(fact_225_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.46 ! [A: nat,B: nat] :
% 0.31/0.46 ( ( A != B )
% 0.31/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.31/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.not_eq_order_implies_strict
% 0.31/0.46 thf(fact_226_RepFun__cong,axiom,
% 0.31/0.46 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( A4 = B4 )
% 0.31/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( hF_Mirabelle_hmem @ X @ B4 )
% 0.31/0.46 => ( ( F @ X )
% 0.31/0.46 = ( G @ X ) ) )
% 0.31/0.46 => ( ( hF_Mirabelle_RepFun @ A4 @ F )
% 0.31/0.46 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RepFun_cong
% 0.31/0.46 thf(fact_227_minf_I8_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(8)
% 0.31/0.46 thf(fact_228_minf_I6_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(6)
% 0.31/0.46 thf(fact_229_pinf_I8_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(8)
% 0.31/0.46 thf(fact_230_pinf_I6_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(6)
% 0.31/0.46 thf(fact_231_complete__interval,axiom,
% 0.31/0.46 ! [A: nat,B: nat,P: nat > $o] :
% 0.31/0.46 ( ( ord_less_nat @ A @ B )
% 0.31/0.46 => ( ( P @ A )
% 0.31/0.46 => ( ~ ( P @ B )
% 0.31/0.46 => ? [C2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ C2 )
% 0.31/0.46 & ( ord_less_eq_nat @ C2 @ B )
% 0.31/0.46 & ! [X6: nat] :
% 0.31/0.46 ( ( ( ord_less_eq_nat @ A @ X6 )
% 0.31/0.46 & ( ord_less_nat @ X6 @ C2 ) )
% 0.31/0.46 => ( P @ X6 ) )
% 0.31/0.46 & ! [D: nat] :
% 0.31/0.46 ( ! [X: nat] :
% 0.31/0.46 ( ( ( ord_less_eq_nat @ A @ X )
% 0.31/0.46 & ( ord_less_nat @ X @ D ) )
% 0.31/0.46 => ( P @ X ) )
% 0.31/0.46 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % complete_interval
% 0.31/0.46 thf(fact_232_HF__hfset,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 0.31/0.46 = A ) ).
% 0.31/0.46
% 0.31/0.46 % HF_hfset
% 0.31/0.46 thf(fact_233_minf_I7_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(7)
% 0.31/0.46 thf(fact_234_minf_I5_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( ord_less_nat @ X6 @ T ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(5)
% 0.31/0.46 thf(fact_235_minf_I4_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( X6 != T ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(4)
% 0.31/0.46 thf(fact_236_minf_I3_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( X6 != T ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(3)
% 0.31/0.46 thf(fact_237_minf_I2_J,axiom,
% 0.31/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.31/0.46 ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Z4 )
% 0.31/0.46 => ( ( P @ X )
% 0.31/0.46 = ( P4 @ X ) ) )
% 0.31/0.46 => ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Z4 )
% 0.31/0.46 => ( ( Q @ X )
% 0.31/0.46 = ( Q2 @ X ) ) )
% 0.31/0.46 => ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( ( ( P @ X6 )
% 0.31/0.46 | ( Q @ X6 ) )
% 0.31/0.46 = ( ( P4 @ X6 )
% 0.31/0.46 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(2)
% 0.31/0.46 thf(fact_238_minf_I1_J,axiom,
% 0.31/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.31/0.46 ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Z4 )
% 0.31/0.46 => ( ( P @ X )
% 0.31/0.46 = ( P4 @ X ) ) )
% 0.31/0.46 => ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X @ Z4 )
% 0.31/0.46 => ( ( Q @ X )
% 0.31/0.46 = ( Q2 @ X ) ) )
% 0.31/0.46 => ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.31/0.46 => ( ( ( P @ X6 )
% 0.31/0.46 & ( Q @ X6 ) )
% 0.31/0.46 = ( ( P4 @ X6 )
% 0.31/0.46 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % minf(1)
% 0.31/0.46 thf(fact_239_pinf_I7_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( ord_less_nat @ T @ X6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(7)
% 0.31/0.46 thf(fact_240_pinf_I5_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(5)
% 0.31/0.46 thf(fact_241_pinf_I4_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( X6 != T ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(4)
% 0.31/0.46 thf(fact_242_pinf_I3_J,axiom,
% 0.31/0.46 ! [T: nat] :
% 0.31/0.46 ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( X6 != T ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(3)
% 0.31/0.46 thf(fact_243_pinf_I2_J,axiom,
% 0.31/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.31/0.46 ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z4 @ X )
% 0.31/0.46 => ( ( P @ X )
% 0.31/0.46 = ( P4 @ X ) ) )
% 0.31/0.46 => ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z4 @ X )
% 0.31/0.46 => ( ( Q @ X )
% 0.31/0.46 = ( Q2 @ X ) ) )
% 0.31/0.46 => ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( ( ( P @ X6 )
% 0.31/0.46 | ( Q @ X6 ) )
% 0.31/0.46 = ( ( P4 @ X6 )
% 0.31/0.46 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(2)
% 0.31/0.46 thf(fact_244_pinf_I1_J,axiom,
% 0.31/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.31/0.46 ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z4 @ X )
% 0.31/0.46 => ( ( P @ X )
% 0.31/0.46 = ( P4 @ X ) ) )
% 0.31/0.46 => ( ? [Z4: nat] :
% 0.31/0.46 ! [X: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z4 @ X )
% 0.31/0.46 => ( ( Q @ X )
% 0.31/0.46 = ( Q2 @ X ) ) )
% 0.31/0.46 => ? [Z2: nat] :
% 0.31/0.46 ! [X6: nat] :
% 0.31/0.46 ( ( ord_less_nat @ Z2 @ X6 )
% 0.31/0.46 => ( ( ( P @ X6 )
% 0.31/0.46 & ( Q @ X6 ) )
% 0.31/0.46 = ( ( P4 @ X6 )
% 0.31/0.46 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pinf(1)
% 0.31/0.46 thf(fact_245_hfset__HF,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
% 0.31/0.46 = A4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % hfset_HF
% 0.31/0.46 thf(fact_246_hmem__HF__iff,axiom,
% 0.31/0.46 ! [X3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HF @ A4 ) )
% 0.31/0.46 = ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.31/0.46 & ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % hmem_HF_iff
% 0.31/0.46 thf(fact_247_finite__hfset,axiom,
% 0.31/0.46 ! [A: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_hfset
% 0.31/0.46 thf(fact_248_finite__has__minimal2,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( member1367349282lle_hf @ A @ A4 )
% 0.31/0.46 => ? [X: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ X @ A4 )
% 0.31/0.46 & ( ord_le976219883lle_hf @ X @ A )
% 0.31/0.46 & ! [Xa: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ Xa @ A4 )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ Xa @ X )
% 0.31/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal2
% 0.31/0.46 thf(fact_249_finite__has__minimal2,axiom,
% 0.31/0.46 ! [A4: set_nat,A: nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( member_nat @ A @ A4 )
% 0.31/0.46 => ? [X: nat] :
% 0.31/0.46 ( ( member_nat @ X @ A4 )
% 0.31/0.46 & ( ord_less_eq_nat @ X @ A )
% 0.31/0.46 & ! [Xa: nat] :
% 0.31/0.46 ( ( member_nat @ Xa @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ Xa @ X )
% 0.31/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_minimal2
% 0.31/0.46 thf(fact_250_finite__has__maximal2,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( member1367349282lle_hf @ A @ A4 )
% 0.31/0.46 => ? [X: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ X @ A4 )
% 0.31/0.46 & ( ord_le976219883lle_hf @ A @ X )
% 0.31/0.46 & ! [Xa: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ Xa @ A4 )
% 0.31/0.46 => ( ( ord_le976219883lle_hf @ X @ Xa )
% 0.31/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal2
% 0.31/0.46 thf(fact_251_finite__has__maximal2,axiom,
% 0.31/0.46 ! [A4: set_nat,A: nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( member_nat @ A @ A4 )
% 0.31/0.46 => ? [X: nat] :
% 0.31/0.46 ( ( member_nat @ X @ A4 )
% 0.31/0.46 & ( ord_less_eq_nat @ A @ X )
% 0.31/0.46 & ! [Xa: nat] :
% 0.31/0.46 ( ( member_nat @ Xa @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ X @ Xa )
% 0.31/0.46 => ( X = Xa ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_has_maximal2
% 0.31/0.46 thf(fact_252_finite__subset,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset
% 0.31/0.46 thf(fact_253_finite__subset,axiom,
% 0.31/0.46 ! [A4: set_nat,B4: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset
% 0.31/0.46 thf(fact_254_infinite__super,axiom,
% 0.31/0.46 ! [S: set_HF_Mirabelle_hf,T2: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ S @ T2 )
% 0.31/0.46 => ( ~ ( finite586181922lle_hf @ S )
% 0.31/0.46 => ~ ( finite586181922lle_hf @ T2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_super
% 0.31/0.46 thf(fact_255_infinite__super,axiom,
% 0.31/0.46 ! [S: set_nat,T2: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.31/0.46 => ( ~ ( finite_finite_nat @ S )
% 0.31/0.46 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_super
% 0.31/0.46 thf(fact_256_rev__finite__subset,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rev_finite_subset
% 0.31/0.46 thf(fact_257_rev__finite__subset,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % rev_finite_subset
% 0.31/0.46 thf(fact_258_finite__psubset__induct,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ! [A7: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A7 )
% 0.31/0.46 => ( ! [B7: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1344122901lle_hf @ B7 @ A7 )
% 0.31/0.46 => ( P @ B7 ) )
% 0.31/0.46 => ( P @ A7 ) ) )
% 0.31/0.46 => ( P @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_psubset_induct
% 0.31/0.46 thf(fact_259_finite__psubset__induct,axiom,
% 0.31/0.46 ! [A4: set_nat,P: set_nat > $o] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ! [A7: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A7 )
% 0.31/0.46 => ( ! [B7: set_nat] :
% 0.31/0.46 ( ( ord_less_set_nat @ B7 @ A7 )
% 0.31/0.46 => ( P @ B7 ) )
% 0.31/0.46 => ( P @ A7 ) ) )
% 0.31/0.46 => ( P @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_psubset_induct
% 0.31/0.46 thf(fact_260_card__psubset,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.31/0.46 => ( ord_le1344122901lle_hf @ A4 @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_psubset
% 0.31/0.46 thf(fact_261_card__psubset,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.31/0.46 => ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_psubset
% 0.31/0.46 thf(fact_262_inj__on__HF,axiom,
% 0.31/0.46 inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% 0.31/0.46
% 0.31/0.46 % inj_on_HF
% 0.31/0.46 thf(fact_263_order_Oordering__axioms,axiom,
% 0.31/0.46 orderi1737556723lle_hf @ ord_le976219883lle_hf @ ord_le1310584031lle_hf ).
% 0.31/0.46
% 0.31/0.46 % order.ordering_axioms
% 0.31/0.46 thf(fact_264_order_Oordering__axioms,axiom,
% 0.31/0.46 ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% 0.31/0.46
% 0.31/0.46 % order.ordering_axioms
% 0.31/0.46 thf(fact_265_card__mono,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_mono
% 0.31/0.46 thf(fact_266_card__mono,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_mono
% 0.31/0.46 thf(fact_267_card__seteq,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ B4 ) @ ( finite1213132899lle_hf @ A4 ) )
% 0.31/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_seteq
% 0.31/0.46 thf(fact_268_card__seteq,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
% 0.31/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_seteq
% 0.31/0.46 thf(fact_269_card__subset__eq,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ( ( finite1213132899lle_hf @ A4 )
% 0.31/0.46 = ( finite1213132899lle_hf @ B4 ) )
% 0.31/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_subset_eq
% 0.31/0.46 thf(fact_270_card__subset__eq,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ( ( finite_card_nat @ A4 )
% 0.31/0.46 = ( finite_card_nat @ B4 ) )
% 0.31/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_subset_eq
% 0.31/0.46 thf(fact_271_infinite__arbitrarily__large,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,N2: nat] :
% 0.31/0.46 ( ~ ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ? [B8: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B8 )
% 0.31/0.46 & ( ( finite1213132899lle_hf @ B8 )
% 0.31/0.46 = N2 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B8 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_arbitrarily_large
% 0.31/0.46 thf(fact_272_infinite__arbitrarily__large,axiom,
% 0.31/0.46 ! [A4: set_nat,N2: nat] :
% 0.31/0.46 ( ~ ( finite_finite_nat @ A4 )
% 0.31/0.46 => ? [B8: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B8 )
% 0.31/0.46 & ( ( finite_card_nat @ B8 )
% 0.31/0.46 = N2 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_arbitrarily_large
% 0.31/0.46 thf(fact_273_finite__if__finite__subsets__card__bdd,axiom,
% 0.31/0.46 ! [F3: set_HF_Mirabelle_hf,C3: nat] :
% 0.31/0.46 ( ! [G2: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ G2 @ F3 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ G2 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G2 ) @ C3 ) ) )
% 0.31/0.46 => ( ( finite586181922lle_hf @ F3 )
% 0.31/0.46 & ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F3 ) @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_if_finite_subsets_card_bdd
% 0.31/0.46 thf(fact_274_finite__if__finite__subsets__card__bdd,axiom,
% 0.31/0.46 ! [F3: set_nat,C3: nat] :
% 0.31/0.46 ( ! [G2: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ G2 @ F3 )
% 0.31/0.46 => ( ( finite_finite_nat @ G2 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
% 0.31/0.46 => ( ( finite_finite_nat @ F3 )
% 0.31/0.46 & ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_if_finite_subsets_card_bdd
% 0.31/0.46 thf(fact_275_psubset__card__mono,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le1344122901lle_hf @ A4 @ B4 )
% 0.31/0.46 => ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_card_mono
% 0.31/0.46 thf(fact_276_psubset__card__mono,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_set_nat @ A4 @ B4 )
% 0.31/0.46 => ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_card_mono
% 0.31/0.46 thf(fact_277_obtain__subset__with__card__n,axiom,
% 0.31/0.46 ! [N2: nat,S: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N2 @ ( finite1213132899lle_hf @ S ) )
% 0.31/0.46 => ~ ! [T3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ T3 @ S )
% 0.31/0.46 => ( ( ( finite1213132899lle_hf @ T3 )
% 0.31/0.46 = N2 )
% 0.31/0.46 => ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % obtain_subset_with_card_n
% 0.31/0.46 thf(fact_278_obtain__subset__with__card__n,axiom,
% 0.31/0.46 ! [N2: nat,S: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
% 0.31/0.46 => ~ ! [T3: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ T3 @ S )
% 0.31/0.46 => ( ( ( finite_card_nat @ T3 )
% 0.31/0.46 = N2 )
% 0.31/0.46 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % obtain_subset_with_card_n
% 0.31/0.46 thf(fact_279_linorder__inj__onI,axiom,
% 0.31/0.46 ! [A4: set_se933006839lle_hf,F: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le1344122901lle_hf @ X @ Y4 )
% 0.31/0.46 => ( ( member1490636632lle_hf @ X @ A4 )
% 0.31/0.46 => ( ( member1490636632lle_hf @ Y4 @ A4 )
% 0.31/0.46 => ( ( F @ X )
% 0.31/0.46 != ( F @ Y4 ) ) ) ) )
% 0.31/0.46 => ( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1490636632lle_hf @ X @ A4 )
% 0.31/0.46 => ( ( member1490636632lle_hf @ Y4 @ A4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ X @ Y4 )
% 0.31/0.46 | ( ord_le432112161lle_hf @ Y4 @ X ) ) ) )
% 0.31/0.46 => ( inj_on811196232lle_hf @ F @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_inj_onI
% 0.31/0.46 thf(fact_280_card__le__if__inj__on__rel,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ! [A5: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ A5 @ A4 )
% 0.31/0.46 => ? [B9: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ B9 @ B4 )
% 0.31/0.46 & ( R2 @ A5 @ B9 ) ) )
% 0.31/0.46 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.31/0.46 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.31/0.46 => ( ( member1367349282lle_hf @ B5 @ B4 )
% 0.31/0.46 => ( ( R2 @ A1 @ B5 )
% 0.31/0.46 => ( ( R2 @ A22 @ B5 )
% 0.31/0.46 => ( A1 = A22 ) ) ) ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_if_inj_on_rel
% 0.31/0.46 thf(fact_281_card__le__if__inj__on__rel,axiom,
% 0.31/0.46 ! [B4: set_nat,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > nat > $o] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ! [A5: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( member1367349282lle_hf @ A5 @ A4 )
% 0.31/0.46 => ? [B9: nat] :
% 0.31/0.46 ( ( member_nat @ B9 @ B4 )
% 0.31/0.46 & ( R2 @ A5 @ B9 ) ) )
% 0.31/0.46 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: nat] :
% 0.31/0.46 ( ( member1367349282lle_hf @ A1 @ A4 )
% 0.31/0.46 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 0.31/0.46 => ( ( member_nat @ B5 @ B4 )
% 0.31/0.46 => ( ( R2 @ A1 @ B5 )
% 0.31/0.46 => ( ( R2 @ A22 @ B5 )
% 0.31/0.46 => ( A1 = A22 ) ) ) ) ) )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_if_inj_on_rel
% 0.31/0.46 thf(fact_282_finite__nat__set__iff__bounded__le,axiom,
% 0.31/0.46 ( finite_finite_nat
% 0.31/0.46 = ( ^ [N3: set_nat] :
% 0.31/0.46 ? [M: nat] :
% 0.31/0.46 ! [X2: nat] :
% 0.31/0.46 ( ( member_nat @ X2 @ N3 )
% 0.31/0.46 => ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_nat_set_iff_bounded_le
% 0.31/0.46 thf(fact_283_bounded__nat__set__is__finite,axiom,
% 0.31/0.46 ! [N4: set_nat,N2: nat] :
% 0.31/0.46 ( ! [X: nat] :
% 0.31/0.46 ( ( member_nat @ X @ N4 )
% 0.31/0.46 => ( ord_less_nat @ X @ N2 ) )
% 0.31/0.46 => ( finite_finite_nat @ N4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % bounded_nat_set_is_finite
% 0.31/0.46 thf(fact_284_finite__nat__set__iff__bounded,axiom,
% 0.31/0.46 ( finite_finite_nat
% 0.31/0.46 = ( ^ [N3: set_nat] :
% 0.31/0.46 ? [M: nat] :
% 0.31/0.46 ! [X2: nat] :
% 0.31/0.46 ( ( member_nat @ X2 @ N3 )
% 0.31/0.46 => ( ord_less_nat @ X2 @ M ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_nat_set_iff_bounded
% 0.31/0.46 thf(fact_285_card__le__inj,axiom,
% 0.31/0.46 ! [A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite1450550360lle_hf @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.31/0.46 => ? [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F4 @ A4 ) @ B4 )
% 0.31/0.46 & ( inj_on811196232lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_inj
% 0.31/0.46 thf(fact_286_card__le__inj,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.31/0.46 => ? [F4: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F4 @ A4 ) @ B4 )
% 0.31/0.46 & ( inj_on755450110lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_inj
% 0.31/0.46 thf(fact_287_card__le__inj,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.31/0.46 => ? [F4: hF_Mirabelle_hf > nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F4 @ A4 ) @ B4 )
% 0.31/0.46 & ( inj_on1874279374hf_nat @ F4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_inj
% 0.31/0.46 thf(fact_288_card__le__inj,axiom,
% 0.31/0.46 ! [A4: set_nat,B4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 0.31/0.46 => ? [F4: nat > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F4 @ A4 ) @ B4 )
% 0.31/0.46 & ( inj_on1988990670lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_inj
% 0.31/0.46 thf(fact_289_card__le__inj,axiom,
% 0.31/0.46 ! [A4: set_nat,B4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 0.31/0.46 => ? [F4: nat > nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A4 ) @ B4 )
% 0.31/0.46 & ( inj_on_nat_nat @ F4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_le_inj
% 0.31/0.46 thf(fact_290_card__inj__on__le,axiom,
% 0.31/0.46 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( inj_on811196232lle_hf @ F @ A4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F @ A4 ) @ B4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_inj_on_le
% 0.31/0.46 thf(fact_291_finite__imageI,axiom,
% 0.31/0.46 ! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ F3 )
% 0.31/0.46 => ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageI
% 0.31/0.46 thf(fact_292_finite__imageI,axiom,
% 0.31/0.46 ! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
% 0.31/0.46 ( ( finite586181922lle_hf @ F3 )
% 0.31/0.46 => ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageI
% 0.31/0.46 thf(fact_293_finite__imageI,axiom,
% 0.31/0.46 ! [F3: set_nat,H: nat > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite_finite_nat @ F3 )
% 0.31/0.46 => ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageI
% 0.31/0.46 thf(fact_294_finite__imageI,axiom,
% 0.31/0.46 ! [F3: set_nat,H: nat > nat] :
% 0.31/0.46 ( ( finite_finite_nat @ F3 )
% 0.31/0.46 => ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageI
% 0.31/0.46 thf(fact_295_all__finite__subset__image,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( ! [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) ) )
% 0.31/0.46 => ( P @ B3 ) ) )
% 0.31/0.46 = ( ! [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ A4 ) )
% 0.31/0.46 => ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_finite_subset_image
% 0.31/0.46 thf(fact_296_all__finite__subset__image,axiom,
% 0.31/0.46 ! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( ! [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) ) )
% 0.31/0.46 => ( P @ B3 ) ) )
% 0.31/0.46 = ( ! [B3: set_nat] :
% 0.31/0.46 ( ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ A4 ) )
% 0.31/0.46 => ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_finite_subset_image
% 0.31/0.46 thf(fact_297_all__finite__subset__image,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.31/0.46 ( ( ! [B3: set_nat] :
% 0.31/0.46 ( ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) ) )
% 0.31/0.46 => ( P @ B3 ) ) )
% 0.31/0.46 = ( ! [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ A4 ) )
% 0.31/0.46 => ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_finite_subset_image
% 0.31/0.46 thf(fact_298_all__finite__subset__image,axiom,
% 0.31/0.46 ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.31/0.46 ( ( ! [B3: set_nat] :
% 0.31/0.46 ( ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) ) )
% 0.31/0.46 => ( P @ B3 ) ) )
% 0.31/0.46 = ( ! [B3: set_nat] :
% 0.31/0.46 ( ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ A4 ) )
% 0.31/0.46 => ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_finite_subset_image
% 0.31/0.46 thf(fact_299_ex__finite__subset__image,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( ? [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 & ( P @ B3 ) ) )
% 0.31/0.46 = ( ? [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ A4 )
% 0.31/0.46 & ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_finite_subset_image
% 0.31/0.46 thf(fact_300_ex__finite__subset__image,axiom,
% 0.31/0.46 ! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( ? [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) )
% 0.31/0.46 & ( P @ B3 ) ) )
% 0.31/0.46 = ( ? [B3: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ A4 )
% 0.31/0.46 & ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_finite_subset_image
% 0.31/0.46 thf(fact_301_ex__finite__subset__image,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 0.31/0.46 ( ( ? [B3: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) )
% 0.31/0.46 & ( P @ B3 ) ) )
% 0.31/0.46 = ( ? [B3: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B3 )
% 0.31/0.46 & ( ord_le432112161lle_hf @ B3 @ A4 )
% 0.31/0.46 & ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_finite_subset_image
% 0.31/0.46 thf(fact_302_ex__finite__subset__image,axiom,
% 0.31/0.46 ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
% 0.31/0.46 ( ( ? [B3: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 & ( P @ B3 ) ) )
% 0.31/0.46 = ( ? [B3: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B3 )
% 0.31/0.46 & ( ord_less_eq_set_nat @ B3 @ A4 )
% 0.31/0.46 & ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_finite_subset_image
% 0.31/0.46 thf(fact_303_finite__subset__image,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 => ? [C4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.31/0.46 & ( finite586181922lle_hf @ C4 )
% 0.31/0.46 & ( B4
% 0.31/0.46 = ( image_1743964010lle_hf @ F @ C4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset_image
% 0.31/0.46 thf(fact_304_finite__subset__image,axiom,
% 0.31/0.46 ! [B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.31/0.46 ( ( finite586181922lle_hf @ B4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
% 0.31/0.46 => ? [C4: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.31/0.46 & ( finite_finite_nat @ C4 )
% 0.31/0.46 & ( B4
% 0.31/0.46 = ( image_246164834lle_hf @ F @ C4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset_image
% 0.31/0.46 thf(fact_305_finite__subset__image,axiom,
% 0.31/0.46 ! [B4: set_nat,F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
% 0.31/0.46 => ? [C4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 0.31/0.46 & ( finite586181922lle_hf @ C4 )
% 0.31/0.46 & ( B4
% 0.31/0.46 = ( image_131453538hf_nat @ F @ C4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset_image
% 0.31/0.46 thf(fact_306_finite__subset__image,axiom,
% 0.31/0.46 ! [B4: set_nat,F: nat > nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 => ? [C4: set_nat] :
% 0.31/0.46 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 0.31/0.46 & ( finite_finite_nat @ C4 )
% 0.31/0.46 & ( B4
% 0.31/0.46 = ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_subset_image
% 0.31/0.46 thf(fact_307_finite__surj,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj
% 0.31/0.46 thf(fact_308_finite__surj,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,B4: set_nat,F: hF_Mirabelle_hf > nat] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
% 0.31/0.46 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj
% 0.31/0.46 thf(fact_309_finite__surj,axiom,
% 0.31/0.46 ! [A4: set_nat,B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj
% 0.31/0.46 thf(fact_310_finite__surj,axiom,
% 0.31/0.46 ! [A4: set_nat,B4: set_nat,F: nat > nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 => ( finite_finite_nat @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj
% 0.31/0.46 thf(fact_311_finite__image__iff,axiom,
% 0.31/0.46 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.31/0.46 ( ( inj_on811196232lle_hf @ F @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 0.31/0.46 = ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_image_iff
% 0.31/0.46 thf(fact_312_finite__image__iff,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( inj_on755450110lle_hf @ F @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_image_iff
% 0.31/0.46 thf(fact_313_finite__image__iff,axiom,
% 0.31/0.46 ! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.31/0.46 ( ( inj_on1988990670lle_hf @ F @ A4 )
% 0.31/0.46 => ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
% 0.31/0.46 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_image_iff
% 0.31/0.46 thf(fact_314_finite__image__iff,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( inj_on1874279374hf_nat @ F @ A4 )
% 0.31/0.46 => ( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
% 0.31/0.46 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_image_iff
% 0.31/0.46 thf(fact_315_finite__image__iff,axiom,
% 0.31/0.46 ! [F: nat > nat,A4: set_nat] :
% 0.31/0.46 ( ( inj_on_nat_nat @ F @ A4 )
% 0.31/0.46 => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 = ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_image_iff
% 0.31/0.46 thf(fact_316_finite__imageD,axiom,
% 0.31/0.46 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( ( inj_on811196232lle_hf @ F @ A4 )
% 0.31/0.46 => ( finite1450550360lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageD
% 0.31/0.46 thf(fact_317_finite__imageD,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( ( inj_on755450110lle_hf @ F @ A4 )
% 0.31/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageD
% 0.31/0.46 thf(fact_318_finite__imageD,axiom,
% 0.31/0.46 ! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 0.31/0.46 ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( ( inj_on1988990670lle_hf @ F @ A4 )
% 0.31/0.46 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageD
% 0.31/0.46 thf(fact_319_finite__imageD,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
% 0.31/0.46 => ( ( inj_on1874279374hf_nat @ F @ A4 )
% 0.31/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageD
% 0.31/0.46 thf(fact_320_finite__imageD,axiom,
% 0.31/0.46 ! [F: nat > nat,A4: set_nat] :
% 0.31/0.46 ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 => ( ( inj_on_nat_nat @ F @ A4 )
% 0.31/0.46 => ( finite_finite_nat @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_imageD
% 0.31/0.46 thf(fact_321_card__image,axiom,
% 0.31/0.46 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 0.31/0.46 ( ( inj_on811196232lle_hf @ F @ A4 )
% 0.31/0.46 => ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 0.31/0.46 = ( finite90088345lle_hf @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % card_image
% 0.31/0.46 thf(fact_322_finite__surj__inj,axiom,
% 0.31/0.46 ! [A4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( finite586181922lle_hf @ A4 )
% 0.31/0.46 => ( ( ord_le432112161lle_hf @ A4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 0.31/0.46 => ( inj_on755450110lle_hf @ F @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj_inj
% 0.31/0.46 thf(fact_323_finite__surj__inj,axiom,
% 0.31/0.46 ! [A4: set_nat,F: nat > nat] :
% 0.31/0.46 ( ( finite_finite_nat @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
% 0.31/0.46 => ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_surj_inj
% 0.31/0.46 thf(fact_324_inj__on__finite,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 0.31/0.46 ( ( inj_on1874279374hf_nat @ F @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F @ A4 ) @ B4 )
% 0.31/0.46 => ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( finite586181922lle_hf @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inj_on_finite
% 0.31/0.46 thf(fact_325_inj__on__finite,axiom,
% 0.31/0.46 ! [F: nat > nat,A4: set_nat,B4: set_nat] :
% 0.31/0.46 ( ( inj_on_nat_nat @ F @ A4 )
% 0.31/0.46 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 )
% 0.31/0.46 => ( ( finite_finite_nat @ B4 )
% 0.31/0.46 => ( finite_finite_nat @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inj_on_finite
% 0.31/0.46 thf(fact_326_finite__greaterThanLessThan,axiom,
% 0.31/0.46 ! [L: nat,U4: nat] : ( finite_finite_nat @ ( set_or1544565540an_nat @ L @ U4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % finite_greaterThanLessThan
% 0.31/0.46 thf(fact_327_bot__nat__0_Onot__eq__extremum,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( A != zero_zero_nat )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.not_eq_extremum
% 0.31/0.46 thf(fact_328_less__nat__zero__code,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % less_nat_zero_code
% 0.31/0.46 thf(fact_329_neq0__conv,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( N2 != zero_zero_nat )
% 0.31/0.46 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq0_conv
% 0.31/0.46 thf(fact_330_le0,axiom,
% 0.31/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.31/0.46
% 0.31/0.46 % le0
% 0.31/0.46 thf(fact_331_bot__nat__0_Oextremum,axiom,
% 0.31/0.46 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum
% 0.31/0.46 thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
% 0.31/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.31/0.46
% 0.31/0.46 % less_eq_nat.simps(1)
% 0.31/0.46 thf(fact_333_le__0__eq,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.31/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_0_eq
% 0.31/0.46 thf(fact_334_bot__nat__0_Oextremum__unique,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.31/0.46 = ( A = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum_unique
% 0.31/0.46 thf(fact_335_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.31/0.46 => ( A = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum_uniqueI
% 0.31/0.46 thf(fact_336_ex__least__nat__le,axiom,
% 0.31/0.46 ! [P: nat > $o,N2: nat] :
% 0.31/0.46 ( ( P @ N2 )
% 0.31/0.46 => ( ~ ( P @ zero_zero_nat )
% 0.31/0.46 => ? [K: nat] :
% 0.31/0.46 ( ( ord_less_eq_nat @ K @ N2 )
% 0.31/0.46 & ! [I: nat] :
% 0.31/0.46 ( ( ord_less_nat @ I @ K )
% 0.31/0.46 => ~ ( P @ I ) )
% 0.31/0.46 & ( P @ K ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_least_nat_le
% 0.31/0.46 thf(fact_337_bot__nat__0_Oextremum__strict,axiom,
% 0.31/0.46 ! [A: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nat_0.extremum_strict
% 0.31/0.46 thf(fact_338_infinite__descent0,axiom,
% 0.31/0.46 ! [P: nat > $o,N2: nat] :
% 0.31/0.46 ( ( P @ zero_zero_nat )
% 0.31/0.46 => ( ! [N5: nat] :
% 0.31/0.46 ( ( ord_less_nat @ zero_zero_nat @ N5 )
% 0.31/0.46 => ( ~ ( P @ N5 )
% 0.31/0.46 => ? [M2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M2 @ N5 )
% 0.31/0.46 & ~ ( P @ M2 ) ) ) )
% 0.31/0.46 => ( P @ N2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % infinite_descent0
% 0.31/0.46 thf(fact_339_gr__implies__not0,axiom,
% 0.31/0.46 ! [M3: nat,N2: nat] :
% 0.31/0.46 ( ( ord_less_nat @ M3 @ N2 )
% 0.31/0.46 => ( N2 != zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr_implies_not0
% 0.31/0.46 thf(fact_340_less__zeroE,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % less_zeroE
% 0.31/0.46 thf(fact_341_not__less0,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.31/0.46
% 0.31/0.46 % not_less0
% 0.31/0.46 thf(fact_342_not__gr0,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.31/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_gr0
% 0.31/0.46 thf(fact_343_gr0I,axiom,
% 0.31/0.46 ! [N2: nat] :
% 0.31/0.46 ( ( N2 != zero_zero_nat )
% 0.31/0.46 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % gr0I
% 0.31/0.46 thf(fact_344_RepFun__0,axiom,
% 0.31/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/0.46 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.31/0.46 = zero_z189798548lle_hf ) ).
% 0.31/0.46
% 0.31/0.46 % RepFun_0
% 0.31/0.46 thf(fact_345_HUnion__hempty,axiom,
% 0.31/0.46 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.31/0.46 = zero_z189798548lle_hf ) ).
% 0.31/0.46
% 0.31/0.46 % HUnion_hempty
% 0.31/0.46 thf(fact_346_Replace__0,axiom,
% 0.31/0.46 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/0.46 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 0.31/0.46 = zero_z189798548lle_hf ) ).
% 0.31/0.46
% 0.31/0.46 % Replace_0
% 0.31/0.46
% 0.31/0.46 % Conjectures (2)
% 0.31/0.46 thf(conj_0,hypothesis,
% 0.31/0.46 ! [X6: hF_Mirabelle_hf] :
% 0.31/0.46 ( ( hF_Mirabelle_hmem @ X6 @ a )
% 0.31/0.46 => ( hF_Mirabelle_hmem @ X6 @ b ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(conj_1,conjecture,
% 0.31/0.46 ord_le976219883lle_hf @ a @ b ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.84/1.53 ---- Embedded in THF ---
% 2.84/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.84/1.53 %%% Generated on Tue Feb 24 08:25:51 EST 2026
% 2.84/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.84/1.53 %%% Logic specification used:
% 2.84/1.53 %%% thf(spec, logic, $$dhol).
% 2.84/1.53
% 2.84/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2DTF10317.p
% See solution above
% 2.84/1.53 ------------------------
% 2.84/1.58 ---- Cleaned THF ---
% 2.84/1.58 thf(ty_n_t__Set__Oset_It__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_J,type,
% 2.84/1.58 set_se933006839lle_hf: $tType ).
% 2.84/1.58
% 2.84/1.58 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 set_HF_Mirabelle_hf: $tType ).
% 2.84/1.58
% 2.84/1.58 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 hF_Mirabelle_hf: $tType ).
% 2.84/1.58
% 2.84/1.58 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.84/1.58 set_nat: $tType ).
% 2.84/1.58
% 2.84/1.58 thf(ty_n_t__Nat__Onat,type,
% 2.84/1.58 nat: $tType ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ocard_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 finite1213132899lle_hf: set_HF_Mirabelle_hf > nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 2.84/1.58 finite_card_nat: set_nat > nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 finite90088345lle_hf: set_se933006839lle_hf > nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ofinite_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 finite586181922lle_hf: set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 2.84/1.58 finite_finite_nat: set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 finite1450550360lle_hf: set_se933006839lle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 inj_on755450110lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Fun_Oinj__on_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 2.84/1.58 inj_on1874279374hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 inj_on1988990670lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.84/1.58 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 inj_on811196232lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.84/1.58 zero_zero_nat: nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 2.84/1.58 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 2.84/1.58 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 2.84/1.58 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 2.84/1.58 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 2.84/1.58 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 2.84/1.58 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 2.84/1.58 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 2.84/1.58 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.84/1.58 ord_less_nat: nat > nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.84/1.58 ord_less_set_nat: set_nat > set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 ord_le2017230260lle_hf: ( $o > hF_Mirabelle_hf ) > ( $o > hF_Mirabelle_hf ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 2.84/1.58 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.84/1.58 ord_less_eq_nat: nat > nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.84/1.58 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 order_1640852850lle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 2.84/1.58 order_Greatest_nat: ( nat > $o ) > nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 order_690706045lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 2.84/1.58 order_557194383hf_nat: ( hF_Mirabelle_hf > nat ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 order_671905679lle_hf: ( nat > hF_Mirabelle_hf ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.84/1.58 order_1631207636at_nat: ( nat > nat ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oordering_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 orderi1737556723lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
% 2.84/1.58 ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_OCollect_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 collec1758573718lle_hf: ( set_HF_Mirabelle_hf > $o ) > set_se933006839lle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 image_1743964010lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_Oimage_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
% 2.84/1.58 image_131453538hf_nat: ( hF_Mirabelle_hf > nat ) > set_HF_Mirabelle_hf > set_nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 image_246164834lle_hf: ( nat > hF_Mirabelle_hf ) > set_nat > set_HF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.84/1.58 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set_Oimage_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 image_899003828lle_hf: ( set_HF_Mirabelle_hf > hF_Mirabelle_hf ) > set_se933006839lle_hf > set_HF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 2.84/1.58 set_or1544565540an_nat: nat > nat > set_nat ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 2.84/1.58 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_member_001t__Nat__Onat,type,
% 2.84/1.58 member_nat: nat > set_nat > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_c_member_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 2.84/1.58 member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o ).
% 2.84/1.58
% 2.84/1.58 thf(sy_v_A,type,
% 2.84/1.58 a: hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(sy_v_B,type,
% 2.84/1.58 b: hF_Mirabelle_hf ).
% 2.84/1.58
% 2.84/1.58 thf(fact_0_hf__equalityI,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ! [X: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X @ A )
% 2.84/1.58 = ( hF_Mirabelle_hmem @ X @ B ) )
% 2.84/1.58 => ( A = B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_1_hf__ext,axiom,
% 2.84/1.58 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [X2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 2.84/1.58 = ( hF_Mirabelle_hmem @ X2 @ B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_2_replacement,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U @ X3 )
% 2.84/1.58 => ( ( R @ U @ V )
% 2.84/1.58 => ( ( R @ U @ V2 )
% 2.84/1.58 => ( V2 = V ) ) ) )
% 2.84/1.58 => ? [Z2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [V3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 2.84/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U2 @ X3 )
% 2.84/1.58 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_3_binary__union,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.84/1.58 = ( ( hF_Mirabelle_hmem @ U3 @ X3 )
% 2.84/1.58 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_4_union__of__set,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf] :
% 2.84/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.84/1.58 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ Y3 @ X3 )
% 2.84/1.58 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_5_comprehension,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.84/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [U3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.84/1.58 = ( ( hF_Mirabelle_hmem @ U3 @ X3 )
% 2.84/1.58 & ( P @ U3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_6_less__eq__hf__def,axiom,
% 2.84/1.58 ( ord_le976219883lle_hf
% 2.84/1.58 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.84/1.58 ! [X2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 2.84/1.58 => ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_7_replacement__fun,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.58 ? [Z2: hF_Mirabelle_hf] :
% 2.84/1.58 ! [V3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 2.84/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U2 @ X3 )
% 2.84/1.58 & ( V3
% 2.84/1.58 = ( F @ U2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_8_order__refl,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X3 @ X3 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_9_order__refl,axiom,
% 2.84/1.58 ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_10_PrimReplace__iff,axiom,
% 2.84/1.58 ! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U @ A4 )
% 2.84/1.58 => ( ( R @ U @ V )
% 2.84/1.58 => ( ( R @ U @ V2 )
% 2.84/1.58 => ( V2 = V ) ) ) )
% 2.84/1.58 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A4 @ R ) )
% 2.84/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.84/1.58 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_11_HCollect__iff,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mir818139703ollect @ P @ A4 ) )
% 2.84/1.58 = ( ( P @ X3 )
% 2.84/1.58 & ( hF_Mirabelle_hmem @ X3 @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_12_HUnion__iff,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HUnion @ A4 ) )
% 2.84/1.58 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 2.84/1.58 & ( hF_Mirabelle_hmem @ X3 @ Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_13_Replace__iff,axiom,
% 2.84/1.58 ! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A4 @ R ) )
% 2.84/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.84/1.58 & ( R @ U2 @ V4 )
% 2.84/1.58 & ! [Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( R @ U2 @ Y3 )
% 2.84/1.58 => ( Y3 = V4 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_14_order__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_15_order__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_16_order__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_17_order__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_18_order__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_19_order__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_20_order__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_21_order__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_22_verit__la__disequality,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( A = B )
% 2.84/1.58 | ~ ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_23_Replace__cong,axiom,
% 2.84/1.58 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( A4 = B4 )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.84/1.58 => ( ( P @ X @ Y4 )
% 2.84/1.58 = ( Q @ X @ Y4 ) ) )
% 2.84/1.58 => ( ( hF_Mirabelle_Replace @ A4 @ P )
% 2.84/1.58 = ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_24_dual__order_Oantisym,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ B @ A )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( A = B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_25_dual__order_Oantisym,axiom,
% 2.84/1.58 ! [B: nat,A: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.84/1.58 => ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( A = B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_26_dual__order_Oeq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ B2 @ A2 )
% 2.84/1.58 & ( ord_le976219883lle_hf @ A2 @ B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_27_dual__order_Oeq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [A2: nat,B2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.84/1.58 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_28_dual__order_Otrans,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ B @ A )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ C @ B )
% 2.84/1.58 => ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_29_dual__order_Otrans,axiom,
% 2.84/1.58 ! [B: nat,A: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.84/1.58 => ( ( ord_less_eq_nat @ C @ B )
% 2.84/1.58 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_30_linorder__wlog,axiom,
% 2.84/1.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.84/1.58 ( ! [A5: nat,B5: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A5 @ B5 )
% 2.84/1.58 => ( P @ A5 @ B5 ) )
% 2.84/1.58 => ( ! [A5: nat,B5: nat] :
% 2.84/1.58 ( ( P @ B5 @ A5 )
% 2.84/1.58 => ( P @ A5 @ B5 ) )
% 2.84/1.58 => ( P @ A @ B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_31_dual__order_Orefl,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_32_dual__order_Orefl,axiom,
% 2.84/1.58 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_33_order__trans,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_34_order__trans,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,Z3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_35_order__class_Oorder_Oantisym,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ A )
% 2.84/1.58 => ( A = B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_36_order__class_Oorder_Oantisym,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ A )
% 2.84/1.58 => ( A = B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_37_ord__le__eq__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( B = C )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_38_ord__le__eq__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( B = C )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_39_ord__eq__le__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A = B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_40_ord__eq__le__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( A = B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_41_order__class_Oorder_Oeq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A2 @ B2 )
% 2.84/1.58 & ( ord_le976219883lle_hf @ B2 @ A2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_42_order__class_Oorder_Oeq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [A2: nat,B2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.84/1.58 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_43_antisym__conv,axiom,
% 2.84/1.58 ! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_44_antisym__conv,axiom,
% 2.84/1.58 ! [Y2: nat,X3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_45_le__cases3,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,Z3: nat] :
% 2.84/1.58 ( ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
% 2.84/1.58 => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
% 2.84/1.58 => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
% 2.84/1.58 => ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
% 2.84/1.58 => ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
% 2.84/1.58 => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_46_order_Otrans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_47_order_Otrans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_48_le__cases,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_49_eq__refl,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( X3 = Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_50_eq__refl,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( X3 = Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_51_linear,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_52_antisym,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 2.84/1.58 => ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_53_antisym,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ Y2 @ X3 )
% 2.84/1.58 => ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_54_eq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 2.84/1.58 & ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_55_eq__iff,axiom,
% 2.84/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.84/1.58 = ( ^ [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.84/1.58 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_56_ord__le__eq__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_57_ord__le__eq__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_58_ord__le__eq__subst,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_59_ord__le__eq__subst,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_60_ord__eq__le__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_61_ord__eq__le__subst,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_62_ord__eq__le__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_63_ord__eq__le__subst,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_64_Greatest__equality,axiom,
% 2.84/1.58 ! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ X3 )
% 2.84/1.58 => ( ! [Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ Y4 @ X3 ) )
% 2.84/1.58 => ( ( order_1640852850lle_hf @ P )
% 2.84/1.58 = X3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_65_Greatest__equality,axiom,
% 2.84/1.58 ! [P: nat > $o,X3: nat] :
% 2.84/1.58 ( ( P @ X3 )
% 2.84/1.58 => ( ! [Y4: nat] :
% 2.84/1.58 ( ( P @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ Y4 @ X3 ) )
% 2.84/1.58 => ( ( order_Greatest_nat @ P )
% 2.84/1.58 = X3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_66_GreatestI2__order,axiom,
% 2.84/1.58 ! [P: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( P @ X3 )
% 2.84/1.58 => ( ! [Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ Y4 @ X3 ) )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ X )
% 2.84/1.58 => ( ! [Y5: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ Y5 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ Y5 @ X ) )
% 2.84/1.58 => ( Q @ X ) ) )
% 2.84/1.58 => ( Q @ ( order_1640852850lle_hf @ P ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_67_GreatestI2__order,axiom,
% 2.84/1.58 ! [P: nat > $o,X3: nat,Q: nat > $o] :
% 2.84/1.58 ( ( P @ X3 )
% 2.84/1.58 => ( ! [Y4: nat] :
% 2.84/1.58 ( ( P @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ Y4 @ X3 ) )
% 2.84/1.58 => ( ! [X: nat] :
% 2.84/1.58 ( ( P @ X )
% 2.84/1.58 => ( ! [Y5: nat] :
% 2.84/1.58 ( ( P @ Y5 )
% 2.84/1.58 => ( ord_less_eq_nat @ Y5 @ X ) )
% 2.84/1.58 => ( Q @ X ) ) )
% 2.84/1.58 => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_68_le__rel__bool__arg__iff,axiom,
% 2.84/1.58 ( ord_le2017230260lle_hf
% 2.84/1.58 = ( ^ [X4: $o > hF_Mirabelle_hf,Y6: $o > hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 2.84/1.58 & ( ord_le976219883lle_hf @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_69_le__rel__bool__arg__iff,axiom,
% 2.84/1.58 ( ord_less_eq_o_nat
% 2.84/1.58 = ( ^ [X4: $o > nat,Y6: $o > nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
% 2.84/1.58 & ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_70_hmem__def,axiom,
% 2.84/1.58 ( hF_Mirabelle_hmem
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_71_mem__Collect__eq,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 2.84/1.58 = ( P @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_72_mem__Collect__eq,axiom,
% 2.84/1.58 ! [A: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( member1490636632lle_hf @ A @ ( collec1758573718lle_hf @ P ) )
% 2.84/1.58 = ( P @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_73_Collect__mem__eq,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( collec2046588256lle_hf
% 2.84/1.58 @ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A4 ) )
% 2.84/1.58 = A4 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_74_Collect__mem__eq,axiom,
% 2.84/1.58 ! [A4: set_se933006839lle_hf] :
% 2.84/1.58 ( ( collec1758573718lle_hf
% 2.84/1.58 @ ^ [X2: set_HF_Mirabelle_hf] : ( member1490636632lle_hf @ X2 @ A4 ) )
% 2.84/1.58 = A4 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_75_Collect__cong,axiom,
% 2.84/1.58 ! [P: set_HF_Mirabelle_hf > $o,Q: set_HF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ! [X: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( P @ X )
% 2.84/1.58 = ( Q @ X ) )
% 2.84/1.58 => ( ( collec1758573718lle_hf @ P )
% 2.84/1.58 = ( collec1758573718lle_hf @ Q ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_76_RepFun__iff,axiom,
% 2.84/1.58 ! [V4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
% 2.84/1.58 = ( ? [U2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 2.84/1.58 & ( V4
% 2.84/1.58 = ( F @ U2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_77_less__hf__def,axiom,
% 2.84/1.58 ( ord_le1310584031lle_hf
% 2.84/1.58 = ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A3 @ B3 )
% 2.84/1.58 & ( A3 != B3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_78_antimono__def,axiom,
% 2.84/1.58 ( order_690706045lle_hf
% 2.84/1.58 = ( ^ [F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.58 ! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_79_antimono__def,axiom,
% 2.84/1.58 ( order_557194383hf_nat
% 2.84/1.58 = ( ^ [F2: hF_Mirabelle_hf > nat] :
% 2.84/1.58 ! [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_80_antimono__def,axiom,
% 2.84/1.58 ( order_671905679lle_hf
% 2.84/1.58 = ( ^ [F2: nat > hF_Mirabelle_hf] :
% 2.84/1.58 ! [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_81_antimono__def,axiom,
% 2.84/1.58 ( order_1631207636at_nat
% 2.84/1.58 = ( ^ [F2: nat > nat] :
% 2.84/1.58 ! [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_82_antimonoI,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.58 ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
% 2.84/1.58 => ( order_690706045lle_hf @ F ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_83_antimonoI,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > nat] :
% 2.84/1.58 ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
% 2.84/1.58 => ( order_557194383hf_nat @ F ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_84_antimonoI,axiom,
% 2.84/1.58 ! [F: nat > hF_Mirabelle_hf] :
% 2.84/1.58 ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y4 ) @ ( F @ X ) ) )
% 2.84/1.58 => ( order_671905679lle_hf @ F ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_85_antimonoI,axiom,
% 2.84/1.58 ! [F: nat > nat] :
% 2.84/1.58 ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X ) ) )
% 2.84/1.58 => ( order_1631207636at_nat @ F ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_86_antimonoE,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( order_690706045lle_hf @ F )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_87_antimonoE,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( order_557194383hf_nat @ F )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_88_antimonoE,axiom,
% 2.84/1.58 ! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
% 2.84/1.58 ( ( order_671905679lle_hf @ F )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_89_antimonoE,axiom,
% 2.84/1.58 ! [F: nat > nat,X3: nat,Y2: nat] :
% 2.84/1.58 ( ( order_1631207636at_nat @ F )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_90_antimonoD,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( order_690706045lle_hf @ F )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_91_antimonoD,axiom,
% 2.84/1.58 ! [F: hF_Mirabelle_hf > nat,X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( order_557194383hf_nat @ F )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_92_antimonoD,axiom,
% 2.84/1.58 ! [F: nat > hF_Mirabelle_hf,X3: nat,Y2: nat] :
% 2.84/1.58 ( ( order_671905679lle_hf @ F )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_93_antimonoD,axiom,
% 2.84/1.58 ! [F: nat > nat,X3: nat,Y2: nat] :
% 2.84/1.58 ( ( order_1631207636at_nat @ F )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_94_verit__comp__simplify1_I1_J,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf] :
% 2.84/1.58 ~ ( ord_le1310584031lle_hf @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_95_verit__comp__simplify1_I1_J,axiom,
% 2.84/1.58 ! [A: nat] :
% 2.84/1.58 ~ ( ord_less_nat @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_96_ord__eq__less__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_97_ord__eq__less__subst,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_98_ord__eq__less__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_99_ord__eq__less__subst,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( A
% 2.84/1.58 = ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_100_ord__less__eq__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_101_ord__less__eq__subst,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_102_ord__less__eq__subst,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_103_ord__less__eq__subst,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ( F @ B )
% 2.84/1.58 = C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_104_order__less__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_105_order__less__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_106_order__less__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_107_order__less__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_108_order__less__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_109_order__less__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_110_order__less__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_111_order__less__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_112_gt__ex,axiom,
% 2.84/1.58 ! [X3: nat] :
% 2.84/1.58 ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_113_neqE,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( X3 != Y2 )
% 2.84/1.58 => ( ~ ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_114_neq__iff,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( X3 != Y2 )
% 2.84/1.58 = ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 | ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_115_order_Oasym,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_116_order_Oasym,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_117_less__imp__neq,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( X3 != Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_118_less__imp__neq,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( X3 != Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_119_less__asym,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_120_less__asym,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_121_less__asym_H,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_122_less__asym_H,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_123_less__trans,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_124_less__trans,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,Z3: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_nat @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_125_less__linear,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 | ( X3 = Y2 )
% 2.84/1.58 | ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_126_less__irrefl,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf] :
% 2.84/1.58 ~ ( ord_le1310584031lle_hf @ X3 @ X3 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_127_less__irrefl,axiom,
% 2.84/1.58 ! [X3: nat] :
% 2.84/1.58 ~ ( ord_less_nat @ X3 @ X3 ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_128_ord__eq__less__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A = B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_129_ord__eq__less__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( A = B )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_130_ord__less__eq__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( B = C )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_131_ord__less__eq__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( B = C )
% 2.84/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_132_dual__order_Oasym,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B @ A )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ A @ B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_133_dual__order_Oasym,axiom,
% 2.84/1.58 ! [B: nat,A: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B @ A )
% 2.84/1.58 => ~ ( ord_less_nat @ A @ B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_134_less__imp__not__eq,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( X3 != Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_135_less__imp__not__eq,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( X3 != Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_136_less__not__sym,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_137_less__not__sym,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_138_less__induct,axiom,
% 2.84/1.58 ! [P: nat > $o,A: nat] :
% 2.84/1.58 ( ! [X: nat] :
% 2.84/1.58 ( ! [Y5: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Y5 @ X )
% 2.84/1.58 => ( P @ Y5 ) )
% 2.84/1.58 => ( P @ X ) )
% 2.84/1.58 => ( P @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_139_antisym__conv3,axiom,
% 2.84/1.58 ! [Y2: nat,X3: nat] :
% 2.84/1.58 ( ~ ( ord_less_nat @ Y2 @ X3 )
% 2.84/1.58 => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_140_less__imp__not__eq2,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( Y2 != X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_141_less__imp__not__eq2,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( Y2 != X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_142_less__imp__triv,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,P: $o] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ Y2 @ X3 )
% 2.84/1.58 => P ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_143_less__imp__triv,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,P: $o] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_nat @ Y2 @ X3 )
% 2.84/1.58 => P ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_144_linorder__cases,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( X3 != Y2 )
% 2.84/1.58 => ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_145_dual__order_Oirrefl,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf] :
% 2.84/1.58 ~ ( ord_le1310584031lle_hf @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_146_dual__order_Oirrefl,axiom,
% 2.84/1.58 ! [A: nat] :
% 2.84/1.58 ~ ( ord_less_nat @ A @ A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_147_order_Ostrict__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_148_order_Ostrict__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_149_less__imp__not__less,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_150_less__imp__not__less,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_151_exists__least__iff,axiom,
% 2.84/1.58 ( ( ^ [P2: nat > $o] :
% 2.84/1.58 ? [X5: nat] : ( P2 @ X5 ) )
% 2.84/1.58 = ( ^ [P3: nat > $o] :
% 2.84/1.58 ? [N: nat] :
% 2.84/1.58 ( ( P3 @ N )
% 2.84/1.58 & ! [M: nat] :
% 2.84/1.58 ( ( ord_less_nat @ M @ N )
% 2.84/1.58 => ~ ( P3 @ M ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_152_linorder__less__wlog,axiom,
% 2.84/1.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.84/1.58 ( ! [A5: nat,B5: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A5 @ B5 )
% 2.84/1.58 => ( P @ A5 @ B5 ) )
% 2.84/1.58 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 2.84/1.58 => ( ! [A5: nat,B5: nat] :
% 2.84/1.58 ( ( P @ B5 @ A5 )
% 2.84/1.58 => ( P @ A5 @ B5 ) )
% 2.84/1.58 => ( P @ A @ B ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_153_dual__order_Ostrict__trans,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B @ A )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ C @ B )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_154_dual__order_Ostrict__trans,axiom,
% 2.84/1.58 ! [B: nat,A: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B @ A )
% 2.84/1.58 => ( ( ord_less_nat @ C @ B )
% 2.84/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_155_not__less__iff__gr__or__eq,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 2.84/1.58 = ( ( ord_less_nat @ Y2 @ X3 )
% 2.84/1.58 | ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_156_order_Ostrict__implies__not__eq,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( A != B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_157_order_Ostrict__implies__not__eq,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( A != B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_158_dual__order_Ostrict__implies__not__eq,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B @ A )
% 2.84/1.58 => ( A != B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_159_dual__order_Ostrict__implies__not__eq,axiom,
% 2.84/1.58 ! [B: nat,A: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B @ A )
% 2.84/1.58 => ( A != B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_160_verit__comp__simplify1_I3_J,axiom,
% 2.84/1.58 ! [B6: nat,A6: nat] :
% 2.84/1.58 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 2.84/1.58 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_161_leD,axiom,
% 2.84/1.58 ! [Y2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ Y2 @ X3 )
% 2.84/1.58 => ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_162_leD,axiom,
% 2.84/1.58 ! [Y2: nat,X3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ Y2 @ X3 )
% 2.84/1.58 => ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_163_leI,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_164_le__less,axiom,
% 2.84/1.58 ( ord_le976219883lle_hf
% 2.84/1.58 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X2 @ Y3 )
% 2.84/1.58 | ( X2 = Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_165_le__less,axiom,
% 2.84/1.58 ( ord_less_eq_nat
% 2.84/1.58 = ( ^ [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X2 @ Y3 )
% 2.84/1.58 | ( X2 = Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_166_less__le,axiom,
% 2.84/1.58 ( ord_le1310584031lle_hf
% 2.84/1.58 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 2.84/1.58 & ( X2 != Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_167_less__le,axiom,
% 2.84/1.58 ( ord_less_nat
% 2.84/1.58 = ( ^ [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.84/1.58 & ( X2 != Y3 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_168_order__le__less__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_169_order__le__less__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_170_order__le__less__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_171_order__le__less__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_172_order__le__less__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_173_order__le__less__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_174_order__le__less__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_175_order__le__less__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_176_order__less__le__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_177_order__less__le__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: hF_Mirabelle_hf > nat,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_178_order__less__le__subst1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_179_order__less__le__subst1,axiom,
% 2.84/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_180_order__less__le__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_181_order__less__le__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_182_order__less__le__subst2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > nat,C: nat] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_183_order__less__le__subst2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.84/1.58 => ( ! [X: nat,Y4: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Y4 )
% 2.84/1.58 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
% 2.84/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_184_not__le,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
% 2.84/1.58 = ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_185_not__less,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 2.84/1.58 = ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_186_le__neq__trans,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( A != B )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_187_le__neq__trans,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( A != B )
% 2.84/1.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_188_antisym__conv1,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_189_antisym__conv1,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ~ ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_190_antisym__conv2,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ~ ( ord_le1310584031lle_hf @ X3 @ Y2 ) )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_191_antisym__conv2,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
% 2.84/1.58 = ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_192_less__imp__le,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ord_le976219883lle_hf @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_193_less__imp__le,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_194_le__less__trans,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_195_le__less__trans,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,Z3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_nat @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_196_less__le__trans,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_197_less__le__trans,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat,Z3: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.84/1.58 => ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_198_le__less__linear,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 | ( ord_less_nat @ Y2 @ X3 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_199_le__imp__less__or__eq,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ X3 @ Y2 )
% 2.84/1.58 | ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_200_le__imp__less__or__eq,axiom,
% 2.84/1.58 ! [X3: nat,Y2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 2.84/1.58 => ( ( ord_less_nat @ X3 @ Y2 )
% 2.84/1.58 | ( X3 = Y2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_201_less__le__not__le,axiom,
% 2.84/1.58 ( ord_le1310584031lle_hf
% 2.84/1.58 = ( ^ [X2: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ X2 @ Y3 )
% 2.84/1.58 & ~ ( ord_le976219883lle_hf @ Y3 @ X2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_202_less__le__not__le,axiom,
% 2.84/1.58 ( ord_less_nat
% 2.84/1.58 = ( ^ [X2: nat,Y3: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.84/1.58 & ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_203_not__le__imp__less,axiom,
% 2.84/1.58 ! [Y2: nat,X3: nat] :
% 2.84/1.58 ( ~ ( ord_less_eq_nat @ Y2 @ X3 )
% 2.84/1.58 => ( ord_less_nat @ X3 @ Y2 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_204_order_Ostrict__trans1,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_205_order_Ostrict__trans1,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_nat @ B @ C )
% 2.84/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_206_order_Ostrict__trans2,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ B @ C )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_207_order_Ostrict__trans2,axiom,
% 2.84/1.58 ! [A: nat,B: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.84/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_208_order_Oorder__iff__strict,axiom,
% 2.84/1.58 ( ord_le976219883lle_hf
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A2 @ B2 )
% 2.84/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_209_order_Oorder__iff__strict,axiom,
% 2.84/1.58 ( ord_less_eq_nat
% 2.84/1.58 = ( ^ [A2: nat,B2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A2 @ B2 )
% 2.84/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_210_order_Ostrict__iff__order,axiom,
% 2.84/1.58 ( ord_le1310584031lle_hf
% 2.84/1.58 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ A2 @ B2 )
% 2.84/1.58 & ( A2 != B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_211_order_Ostrict__iff__order,axiom,
% 2.84/1.58 ( ord_less_nat
% 2.84/1.58 = ( ^ [A2: nat,B2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.84/1.58 & ( A2 != B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_212_dual__order_Ostrict__trans1,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ B @ A )
% 2.84/1.58 => ( ( ord_le1310584031lle_hf @ C @ B )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_213_dual__order_Ostrict__trans1,axiom,
% 2.84/1.58 ! [B: nat,A: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.84/1.58 => ( ( ord_less_nat @ C @ B )
% 2.84/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_214_dual__order_Ostrict__trans2,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B @ A )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ C @ B )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_215_dual__order_Ostrict__trans2,axiom,
% 2.84/1.58 ! [B: nat,A: nat,C: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B @ A )
% 2.84/1.58 => ( ( ord_less_eq_nat @ C @ B )
% 2.84/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_216_order_Ostrict__implies__order,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ A @ B )
% 2.84/1.58 => ( ord_le976219883lle_hf @ A @ B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_217_order_Ostrict__implies__order,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_218_dual__order_Oorder__iff__strict,axiom,
% 2.84/1.58 ( ord_le976219883lle_hf
% 2.84/1.58 = ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B2 @ A2 )
% 2.84/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_219_dual__order_Oorder__iff__strict,axiom,
% 2.84/1.58 ( ord_less_eq_nat
% 2.84/1.58 = ( ^ [B2: nat,A2: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B2 @ A2 )
% 2.84/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_220_dual__order_Ostrict__iff__order,axiom,
% 2.84/1.58 ( ord_le1310584031lle_hf
% 2.84/1.58 = ( ^ [B2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le976219883lle_hf @ B2 @ A2 )
% 2.84/1.58 & ( A2 != B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_221_dual__order_Ostrict__iff__order,axiom,
% 2.84/1.58 ( ord_less_nat
% 2.84/1.58 = ( ^ [B2: nat,A2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.84/1.58 & ( A2 != B2 ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_222_dual__order_Ostrict__implies__order,axiom,
% 2.84/1.58 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1310584031lle_hf @ B @ A )
% 2.84/1.58 => ( ord_le976219883lle_hf @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_223_dual__order_Ostrict__implies__order,axiom,
% 2.84/1.58 ! [B: nat,A: nat] :
% 2.84/1.58 ( ( ord_less_nat @ B @ A )
% 2.84/1.58 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_224_order_Onot__eq__order__implies__strict,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A != B )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ A @ B )
% 2.84/1.58 => ( ord_le1310584031lle_hf @ A @ B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_225_order_Onot__eq__order__implies__strict,axiom,
% 2.84/1.58 ! [A: nat,B: nat] :
% 2.84/1.58 ( ( A != B )
% 2.84/1.58 => ( ( ord_less_eq_nat @ A @ B )
% 2.84/1.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_226_RepFun__cong,axiom,
% 2.84/1.58 ! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.58 ( ( A4 = B4 )
% 2.84/1.58 => ( ! [X: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X @ B4 )
% 2.84/1.58 => ( ( F @ X )
% 2.84/1.58 = ( G @ X ) ) )
% 2.84/1.58 => ( ( hF_Mirabelle_RepFun @ A4 @ F )
% 2.84/1.58 = ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_227_minf_I8_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_228_minf_I6_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_229_pinf_I8_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_230_pinf_I6_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_231_complete__interval,axiom,
% 2.84/1.58 ! [A: nat,B: nat,P: nat > $o] :
% 2.84/1.58 ( ( ord_less_nat @ A @ B )
% 2.84/1.58 => ( ( P @ A )
% 2.84/1.58 => ( ~ ( P @ B )
% 2.84/1.58 => ? [C2: nat] :
% 2.84/1.58 ( ( ord_less_eq_nat @ A @ C2 )
% 2.84/1.58 & ( ord_less_eq_nat @ C2 @ B )
% 2.84/1.58 & ! [X6: nat] :
% 2.84/1.58 ( ( ( ord_less_eq_nat @ A @ X6 )
% 2.84/1.58 & ( ord_less_nat @ X6 @ C2 ) )
% 2.84/1.58 => ( P @ X6 ) )
% 2.84/1.58 & ! [D: nat] :
% 2.84/1.58 ( ! [X: nat] :
% 2.84/1.58 ( ( ( ord_less_eq_nat @ A @ X )
% 2.84/1.58 & ( ord_less_nat @ X @ D ) )
% 2.84/1.58 => ( P @ X ) )
% 2.84/1.58 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_232_HF__hfset,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 2.84/1.58 = A ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_233_minf_I7_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_234_minf_I5_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( ord_less_nat @ X6 @ T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_235_minf_I4_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( X6 != T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_236_minf_I3_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( X6 != T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_237_minf_I2_J,axiom,
% 2.84/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.84/1.58 ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Z4 )
% 2.84/1.58 => ( ( P @ X )
% 2.84/1.58 = ( P4 @ X ) ) )
% 2.84/1.58 => ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Z4 )
% 2.84/1.58 => ( ( Q @ X )
% 2.84/1.58 = ( Q2 @ X ) ) )
% 2.84/1.58 => ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( ( ( P @ X6 )
% 2.84/1.58 | ( Q @ X6 ) )
% 2.84/1.58 = ( ( P4 @ X6 )
% 2.84/1.58 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_238_minf_I1_J,axiom,
% 2.84/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.84/1.58 ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Z4 )
% 2.84/1.58 => ( ( P @ X )
% 2.84/1.58 = ( P4 @ X ) ) )
% 2.84/1.58 => ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X @ Z4 )
% 2.84/1.58 => ( ( Q @ X )
% 2.84/1.58 = ( Q2 @ X ) ) )
% 2.84/1.58 => ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.84/1.58 => ( ( ( P @ X6 )
% 2.84/1.58 & ( Q @ X6 ) )
% 2.84/1.58 = ( ( P4 @ X6 )
% 2.84/1.58 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_239_pinf_I7_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( ord_less_nat @ T @ X6 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_240_pinf_I5_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_241_pinf_I4_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( X6 != T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_242_pinf_I3_J,axiom,
% 2.84/1.58 ! [T: nat] :
% 2.84/1.58 ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( X6 != T ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_243_pinf_I2_J,axiom,
% 2.84/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.84/1.58 ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z4 @ X )
% 2.84/1.58 => ( ( P @ X )
% 2.84/1.58 = ( P4 @ X ) ) )
% 2.84/1.58 => ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z4 @ X )
% 2.84/1.58 => ( ( Q @ X )
% 2.84/1.58 = ( Q2 @ X ) ) )
% 2.84/1.58 => ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( ( ( P @ X6 )
% 2.84/1.58 | ( Q @ X6 ) )
% 2.84/1.58 = ( ( P4 @ X6 )
% 2.84/1.58 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_244_pinf_I1_J,axiom,
% 2.84/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.84/1.58 ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z4 @ X )
% 2.84/1.58 => ( ( P @ X )
% 2.84/1.58 = ( P4 @ X ) ) )
% 2.84/1.58 => ( ? [Z4: nat] :
% 2.84/1.58 ! [X: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z4 @ X )
% 2.84/1.58 => ( ( Q @ X )
% 2.84/1.58 = ( Q2 @ X ) ) )
% 2.84/1.58 => ? [Z2: nat] :
% 2.84/1.58 ! [X6: nat] :
% 2.84/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.84/1.58 => ( ( ( P @ X6 )
% 2.84/1.58 & ( Q @ X6 ) )
% 2.84/1.58 = ( ( P4 @ X6 )
% 2.84/1.58 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_245_hfset__HF,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.58 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A4 ) )
% 2.84/1.58 = A4 ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_246_hmem__HF__iff,axiom,
% 2.84/1.58 ! [X3: hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HF @ A4 ) )
% 2.84/1.58 = ( ( member1367349282lle_hf @ X3 @ A4 )
% 2.84/1.58 & ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_247_finite__hfset,axiom,
% 2.84/1.58 ! [A: hF_Mirabelle_hf] : ( finite586181922lle_hf @ ( hF_Mirabelle_hfset @ A ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_248_finite__has__minimal2,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.58 => ( ( member1367349282lle_hf @ A @ A4 )
% 2.84/1.58 => ? [X: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( member1367349282lle_hf @ X @ A4 )
% 2.84/1.58 & ( ord_le976219883lle_hf @ X @ A )
% 2.84/1.58 & ! [Xa: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( member1367349282lle_hf @ Xa @ A4 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ Xa @ X )
% 2.84/1.58 => ( X = Xa ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_249_finite__has__minimal2,axiom,
% 2.84/1.58 ! [A4: set_nat,A: nat] :
% 2.84/1.58 ( ( finite_finite_nat @ A4 )
% 2.84/1.58 => ( ( member_nat @ A @ A4 )
% 2.84/1.58 => ? [X: nat] :
% 2.84/1.58 ( ( member_nat @ X @ A4 )
% 2.84/1.58 & ( ord_less_eq_nat @ X @ A )
% 2.84/1.58 & ! [Xa: nat] :
% 2.84/1.58 ( ( member_nat @ Xa @ A4 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ Xa @ X )
% 2.84/1.58 => ( X = Xa ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_250_finite__has__maximal2,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.58 => ( ( member1367349282lle_hf @ A @ A4 )
% 2.84/1.58 => ? [X: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( member1367349282lle_hf @ X @ A4 )
% 2.84/1.58 & ( ord_le976219883lle_hf @ A @ X )
% 2.84/1.58 & ! [Xa: hF_Mirabelle_hf] :
% 2.84/1.58 ( ( member1367349282lle_hf @ Xa @ A4 )
% 2.84/1.58 => ( ( ord_le976219883lle_hf @ X @ Xa )
% 2.84/1.58 => ( X = Xa ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_251_finite__has__maximal2,axiom,
% 2.84/1.58 ! [A4: set_nat,A: nat] :
% 2.84/1.58 ( ( finite_finite_nat @ A4 )
% 2.84/1.58 => ( ( member_nat @ A @ A4 )
% 2.84/1.58 => ? [X: nat] :
% 2.84/1.58 ( ( member_nat @ X @ A4 )
% 2.84/1.58 & ( ord_less_eq_nat @ A @ X )
% 2.84/1.58 & ! [Xa: nat] :
% 2.84/1.58 ( ( member_nat @ Xa @ A4 )
% 2.84/1.58 => ( ( ord_less_eq_nat @ X @ Xa )
% 2.84/1.58 => ( X = Xa ) ) ) ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_252_finite__subset,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.58 => ( ( finite586181922lle_hf @ B4 )
% 2.84/1.58 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_253_finite__subset,axiom,
% 2.84/1.58 ! [A4: set_nat,B4: set_nat] :
% 2.84/1.58 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.58 => ( ( finite_finite_nat @ B4 )
% 2.84/1.58 => ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_254_infinite__super,axiom,
% 2.84/1.58 ! [S: set_HF_Mirabelle_hf,T2: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le432112161lle_hf @ S @ T2 )
% 2.84/1.58 => ( ~ ( finite586181922lle_hf @ S )
% 2.84/1.58 => ~ ( finite586181922lle_hf @ T2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_255_infinite__super,axiom,
% 2.84/1.58 ! [S: set_nat,T2: set_nat] :
% 2.84/1.58 ( ( ord_less_eq_set_nat @ S @ T2 )
% 2.84/1.58 => ( ~ ( finite_finite_nat @ S )
% 2.84/1.58 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_256_rev__finite__subset,axiom,
% 2.84/1.58 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.58 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.58 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_257_rev__finite__subset,axiom,
% 2.84/1.58 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.58 ( ( finite_finite_nat @ B4 )
% 2.84/1.58 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.58 => ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_258_finite__psubset__induct,axiom,
% 2.84/1.58 ! [A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.58 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.58 => ( ! [A7: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( finite586181922lle_hf @ A7 )
% 2.84/1.58 => ( ! [B7: set_HF_Mirabelle_hf] :
% 2.84/1.58 ( ( ord_le1344122901lle_hf @ B7 @ A7 )
% 2.84/1.58 => ( P @ B7 ) )
% 2.84/1.58 => ( P @ A7 ) ) )
% 2.84/1.58 => ( P @ A4 ) ) ) ).
% 2.84/1.58
% 2.84/1.58 thf(fact_259_finite__psubset__induct,axiom,
% 2.84/1.58 ! [A4: set_nat,P: set_nat > $o] :
% 2.84/1.58 ( ( finite_finite_nat @ A4 )
% 2.84/1.58 => ( ! [A7: set_nat] :
% 2.84/1.58 ( ( finite_finite_nat @ A7 )
% 2.84/1.58 => ( ! [B7: set_nat] :
% 2.84/1.58 ( ( ord_less_set_nat @ B7 @ A7 )
% 2.84/1.58 => ( P @ B7 ) )
% 2.84/1.58 => ( P @ A7 ) ) )
% 2.84/1.58 => ( P @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_260_card__psubset,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.59 => ( ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 2.84/1.59 => ( ord_le1344122901lle_hf @ A4 @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_261_card__psubset,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.59 => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 2.84/1.59 => ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_262_inj__on__HF,axiom,
% 2.84/1.59 inj_on811196232lle_hf @ hF_Mirabelle_HF @ ( collec1758573718lle_hf @ finite586181922lle_hf ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_263_order_Oordering__axioms,axiom,
% 2.84/1.59 orderi1737556723lle_hf @ ord_le976219883lle_hf @ ord_le1310584031lle_hf ).
% 2.84/1.59
% 2.84/1.59 thf(fact_264_order_Oordering__axioms,axiom,
% 2.84/1.59 ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% 2.84/1.59
% 2.84/1.59 thf(fact_265_card__mono,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_266_card__mono,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_267_card__seteq,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ B4 ) @ ( finite1213132899lle_hf @ A4 ) )
% 2.84/1.59 => ( A4 = B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_268_card__seteq,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
% 2.84/1.59 => ( A4 = B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_269_card__subset__eq,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ A4 @ B4 )
% 2.84/1.59 => ( ( ( finite1213132899lle_hf @ A4 )
% 2.84/1.59 = ( finite1213132899lle_hf @ B4 ) )
% 2.84/1.59 => ( A4 = B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_270_card__subset__eq,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.84/1.59 => ( ( ( finite_card_nat @ A4 )
% 2.84/1.59 = ( finite_card_nat @ B4 ) )
% 2.84/1.59 => ( A4 = B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_271_infinite__arbitrarily__large,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,N2: nat] :
% 2.84/1.59 ( ~ ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ? [B8: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B8 )
% 2.84/1.59 & ( ( finite1213132899lle_hf @ B8 )
% 2.84/1.59 = N2 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B8 @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_272_infinite__arbitrarily__large,axiom,
% 2.84/1.59 ! [A4: set_nat,N2: nat] :
% 2.84/1.59 ( ~ ( finite_finite_nat @ A4 )
% 2.84/1.59 => ? [B8: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B8 )
% 2.84/1.59 & ( ( finite_card_nat @ B8 )
% 2.84/1.59 = N2 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_273_finite__if__finite__subsets__card__bdd,axiom,
% 2.84/1.59 ! [F3: set_HF_Mirabelle_hf,C3: nat] :
% 2.84/1.59 ( ! [G2: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ G2 @ F3 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ G2 )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ G2 ) @ C3 ) ) )
% 2.84/1.59 => ( ( finite586181922lle_hf @ F3 )
% 2.84/1.59 & ( ord_less_eq_nat @ ( finite1213132899lle_hf @ F3 ) @ C3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_274_finite__if__finite__subsets__card__bdd,axiom,
% 2.84/1.59 ! [F3: set_nat,C3: nat] :
% 2.84/1.59 ( ! [G2: set_nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ G2 @ F3 )
% 2.84/1.59 => ( ( finite_finite_nat @ G2 )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
% 2.84/1.59 => ( ( finite_finite_nat @ F3 )
% 2.84/1.59 & ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_275_psubset__card__mono,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le1344122901lle_hf @ A4 @ B4 )
% 2.84/1.59 => ( ord_less_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_276_psubset__card__mono,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_set_nat @ A4 @ B4 )
% 2.84/1.59 => ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_277_obtain__subset__with__card__n,axiom,
% 2.84/1.59 ! [N2: nat,S: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_less_eq_nat @ N2 @ ( finite1213132899lle_hf @ S ) )
% 2.84/1.59 => ~ ! [T3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ T3 @ S )
% 2.84/1.59 => ( ( ( finite1213132899lle_hf @ T3 )
% 2.84/1.59 = N2 )
% 2.84/1.59 => ~ ( finite586181922lle_hf @ T3 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_278_obtain__subset__with__card__n,axiom,
% 2.84/1.59 ! [N2: nat,S: set_nat] :
% 2.84/1.59 ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
% 2.84/1.59 => ~ ! [T3: set_nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ T3 @ S )
% 2.84/1.59 => ( ( ( finite_card_nat @ T3 )
% 2.84/1.59 = N2 )
% 2.84/1.59 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_279_linorder__inj__onI,axiom,
% 2.84/1.59 ! [A4: set_se933006839lle_hf,F: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le1344122901lle_hf @ X @ Y4 )
% 2.84/1.59 => ( ( member1490636632lle_hf @ X @ A4 )
% 2.84/1.59 => ( ( member1490636632lle_hf @ Y4 @ A4 )
% 2.84/1.59 => ( ( F @ X )
% 2.84/1.59 != ( F @ Y4 ) ) ) ) )
% 2.84/1.59 => ( ! [X: set_HF_Mirabelle_hf,Y4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( member1490636632lle_hf @ X @ A4 )
% 2.84/1.59 => ( ( member1490636632lle_hf @ Y4 @ A4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ X @ Y4 )
% 2.84/1.59 | ( ord_le432112161lle_hf @ Y4 @ X ) ) ) )
% 2.84/1.59 => ( inj_on811196232lle_hf @ F @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_280_card__le__if__inj__on__rel,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ! [A5: hF_Mirabelle_hf] :
% 2.84/1.59 ( ( member1367349282lle_hf @ A5 @ A4 )
% 2.84/1.59 => ? [B9: hF_Mirabelle_hf] :
% 2.84/1.59 ( ( member1367349282lle_hf @ B9 @ B4 )
% 2.84/1.59 & ( R2 @ A5 @ B9 ) ) )
% 2.84/1.59 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 2.84/1.59 ( ( member1367349282lle_hf @ A1 @ A4 )
% 2.84/1.59 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 2.84/1.59 => ( ( member1367349282lle_hf @ B5 @ B4 )
% 2.84/1.59 => ( ( R2 @ A1 @ B5 )
% 2.84/1.59 => ( ( R2 @ A22 @ B5 )
% 2.84/1.59 => ( A1 = A22 ) ) ) ) ) )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_281_card__le__if__inj__on__rel,axiom,
% 2.84/1.59 ! [B4: set_nat,A4: set_HF_Mirabelle_hf,R2: hF_Mirabelle_hf > nat > $o] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ! [A5: hF_Mirabelle_hf] :
% 2.84/1.59 ( ( member1367349282lle_hf @ A5 @ A4 )
% 2.84/1.59 => ? [B9: nat] :
% 2.84/1.59 ( ( member_nat @ B9 @ B4 )
% 2.84/1.59 & ( R2 @ A5 @ B9 ) ) )
% 2.84/1.59 => ( ! [A1: hF_Mirabelle_hf,A22: hF_Mirabelle_hf,B5: nat] :
% 2.84/1.59 ( ( member1367349282lle_hf @ A1 @ A4 )
% 2.84/1.59 => ( ( member1367349282lle_hf @ A22 @ A4 )
% 2.84/1.59 => ( ( member_nat @ B5 @ B4 )
% 2.84/1.59 => ( ( R2 @ A1 @ B5 )
% 2.84/1.59 => ( ( R2 @ A22 @ B5 )
% 2.84/1.59 => ( A1 = A22 ) ) ) ) ) )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_282_finite__nat__set__iff__bounded__le,axiom,
% 2.84/1.59 ( finite_finite_nat
% 2.84/1.59 = ( ^ [N3: set_nat] :
% 2.84/1.59 ? [M: nat] :
% 2.84/1.59 ! [X2: nat] :
% 2.84/1.59 ( ( member_nat @ X2 @ N3 )
% 2.84/1.59 => ( ord_less_eq_nat @ X2 @ M ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_283_bounded__nat__set__is__finite,axiom,
% 2.84/1.59 ! [N4: set_nat,N2: nat] :
% 2.84/1.59 ( ! [X: nat] :
% 2.84/1.59 ( ( member_nat @ X @ N4 )
% 2.84/1.59 => ( ord_less_nat @ X @ N2 ) )
% 2.84/1.59 => ( finite_finite_nat @ N4 ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_284_finite__nat__set__iff__bounded,axiom,
% 2.84/1.59 ( finite_finite_nat
% 2.84/1.59 = ( ^ [N3: set_nat] :
% 2.84/1.59 ? [M: nat] :
% 2.84/1.59 ! [X2: nat] :
% 2.84/1.59 ( ( member_nat @ X2 @ N3 )
% 2.84/1.59 => ( ord_less_nat @ X2 @ M ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_285_card__le__inj,axiom,
% 2.84/1.59 ! [A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite1450550360lle_hf @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 2.84/1.59 => ? [F4: set_HF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F4 @ A4 ) @ B4 )
% 2.84/1.59 & ( inj_on811196232lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_286_card__le__inj,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 2.84/1.59 => ? [F4: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ ( image_1743964010lle_hf @ F4 @ A4 ) @ B4 )
% 2.84/1.59 & ( inj_on755450110lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_287_card__le__inj,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 2.84/1.59 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite1213132899lle_hf @ A4 ) @ ( finite_card_nat @ B4 ) )
% 2.84/1.59 => ? [F4: hF_Mirabelle_hf > nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F4 @ A4 ) @ B4 )
% 2.84/1.59 & ( inj_on1874279374hf_nat @ F4 @ A4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_288_card__le__inj,axiom,
% 2.84/1.59 ! [A4: set_nat,B4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite_finite_nat @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1213132899lle_hf @ B4 ) )
% 2.84/1.59 => ? [F4: nat > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ ( image_246164834lle_hf @ F4 @ A4 ) @ B4 )
% 2.84/1.59 & ( inj_on1988990670lle_hf @ F4 @ A4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_289_card__le__inj,axiom,
% 2.84/1.59 ! [A4: set_nat,B4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ A4 )
% 2.84/1.59 => ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
% 2.84/1.59 => ? [F4: nat > nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A4 ) @ B4 )
% 2.84/1.59 & ( inj_on_nat_nat @ F4 @ A4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_290_card__inj__on__le,axiom,
% 2.84/1.59 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf,B4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( inj_on811196232lle_hf @ F @ A4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ ( image_899003828lle_hf @ F @ A4 ) @ B4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ord_less_eq_nat @ ( finite90088345lle_hf @ A4 ) @ ( finite1213132899lle_hf @ B4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_291_finite__imageI,axiom,
% 2.84/1.59 ! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ F3 )
% 2.84/1.59 => ( finite586181922lle_hf @ ( image_1743964010lle_hf @ H @ F3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_292_finite__imageI,axiom,
% 2.84/1.59 ! [F3: set_HF_Mirabelle_hf,H: hF_Mirabelle_hf > nat] :
% 2.84/1.59 ( ( finite586181922lle_hf @ F3 )
% 2.84/1.59 => ( finite_finite_nat @ ( image_131453538hf_nat @ H @ F3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_293_finite__imageI,axiom,
% 2.84/1.59 ! [F3: set_nat,H: nat > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite_finite_nat @ F3 )
% 2.84/1.59 => ( finite586181922lle_hf @ ( image_246164834lle_hf @ H @ F3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_294_finite__imageI,axiom,
% 2.84/1.59 ! [F3: set_nat,H: nat > nat] :
% 2.84/1.59 ( ( finite_finite_nat @ F3 )
% 2.84/1.59 => ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_295_all__finite__subset__image,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( ! [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) ) )
% 2.84/1.59 => ( P @ B3 ) ) )
% 2.84/1.59 = ( ! [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ A4 ) )
% 2.84/1.59 => ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_296_all__finite__subset__image,axiom,
% 2.84/1.59 ! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( ! [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) ) )
% 2.84/1.59 => ( P @ B3 ) ) )
% 2.84/1.59 = ( ! [B3: set_nat] :
% 2.84/1.59 ( ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ A4 ) )
% 2.84/1.59 => ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_297_all__finite__subset__image,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 2.84/1.59 ( ( ! [B3: set_nat] :
% 2.84/1.59 ( ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) ) )
% 2.84/1.59 => ( P @ B3 ) ) )
% 2.84/1.59 = ( ! [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ A4 ) )
% 2.84/1.59 => ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_298_all__finite__subset__image,axiom,
% 2.84/1.59 ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
% 2.84/1.59 ( ( ! [B3: set_nat] :
% 2.84/1.59 ( ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) ) )
% 2.84/1.59 => ( P @ B3 ) ) )
% 2.84/1.59 = ( ! [B3: set_nat] :
% 2.84/1.59 ( ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ A4 ) )
% 2.84/1.59 => ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_299_ex__finite__subset__image,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( ? [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 & ( P @ B3 ) ) )
% 2.84/1.59 = ( ? [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ A4 )
% 2.84/1.59 & ( P @ ( image_1743964010lle_hf @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_300_ex__finite__subset__image,axiom,
% 2.84/1.59 ! [F: nat > hF_Mirabelle_hf,A4: set_nat,P: set_HF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( ? [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ ( image_246164834lle_hf @ F @ A4 ) )
% 2.84/1.59 & ( P @ B3 ) ) )
% 2.84/1.59 = ( ? [B3: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ A4 )
% 2.84/1.59 & ( P @ ( image_246164834lle_hf @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_301_ex__finite__subset__image,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,P: set_nat > $o] :
% 2.84/1.59 ( ( ? [B3: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ ( image_131453538hf_nat @ F @ A4 ) )
% 2.84/1.59 & ( P @ B3 ) ) )
% 2.84/1.59 = ( ? [B3: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B3 )
% 2.84/1.59 & ( ord_le432112161lle_hf @ B3 @ A4 )
% 2.84/1.59 & ( P @ ( image_131453538hf_nat @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_302_ex__finite__subset__image,axiom,
% 2.84/1.59 ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
% 2.84/1.59 ( ( ? [B3: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 & ( P @ B3 ) ) )
% 2.84/1.59 = ( ? [B3: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B3 )
% 2.84/1.59 & ( ord_less_eq_set_nat @ B3 @ A4 )
% 2.84/1.59 & ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_303_finite__subset__image,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 => ? [C4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 2.84/1.59 & ( finite586181922lle_hf @ C4 )
% 2.84/1.59 & ( B4
% 2.84/1.59 = ( image_1743964010lle_hf @ F @ C4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_304_finite__subset__image,axiom,
% 2.84/1.59 ! [B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 2.84/1.59 ( ( finite586181922lle_hf @ B4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
% 2.84/1.59 => ? [C4: set_nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 2.84/1.59 & ( finite_finite_nat @ C4 )
% 2.84/1.59 & ( B4
% 2.84/1.59 = ( image_246164834lle_hf @ F @ C4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_305_finite__subset__image,axiom,
% 2.84/1.59 ! [B4: set_nat,F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
% 2.84/1.59 => ? [C4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( ord_le432112161lle_hf @ C4 @ A4 )
% 2.84/1.59 & ( finite586181922lle_hf @ C4 )
% 2.84/1.59 & ( B4
% 2.84/1.59 = ( image_131453538hf_nat @ F @ C4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_306_finite__subset__image,axiom,
% 2.84/1.59 ! [B4: set_nat,F: nat > nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 => ? [C4: set_nat] :
% 2.84/1.59 ( ( ord_less_eq_set_nat @ C4 @ A4 )
% 2.84/1.59 & ( finite_finite_nat @ C4 )
% 2.84/1.59 & ( B4
% 2.84/1.59 = ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_307_finite__surj,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ B4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_308_finite__surj,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,B4: set_nat,F: hF_Mirabelle_hf > nat] :
% 2.84/1.59 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ B4 @ ( image_131453538hf_nat @ F @ A4 ) )
% 2.84/1.59 => ( finite_finite_nat @ B4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_309_finite__surj,axiom,
% 2.84/1.59 ! [A4: set_nat,B4: set_HF_Mirabelle_hf,F: nat > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite_finite_nat @ A4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ B4 @ ( image_246164834lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( finite586181922lle_hf @ B4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_310_finite__surj,axiom,
% 2.84/1.59 ! [A4: set_nat,B4: set_nat,F: nat > nat] :
% 2.84/1.59 ( ( finite_finite_nat @ A4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 => ( finite_finite_nat @ B4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_311_finite__image__iff,axiom,
% 2.84/1.59 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 2.84/1.59 ( ( inj_on811196232lle_hf @ F @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 2.84/1.59 = ( finite1450550360lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_312_finite__image__iff,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( inj_on755450110lle_hf @ F @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_313_finite__image__iff,axiom,
% 2.84/1.59 ! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 2.84/1.59 ( ( inj_on1988990670lle_hf @ F @ A4 )
% 2.84/1.59 => ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
% 2.84/1.59 = ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_314_finite__image__iff,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( inj_on1874279374hf_nat @ F @ A4 )
% 2.84/1.59 => ( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
% 2.84/1.59 = ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_315_finite__image__iff,axiom,
% 2.84/1.59 ! [F: nat > nat,A4: set_nat] :
% 2.84/1.59 ( ( inj_on_nat_nat @ F @ A4 )
% 2.84/1.59 => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 = ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_316_finite__imageD,axiom,
% 2.84/1.59 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( ( inj_on811196232lle_hf @ F @ A4 )
% 2.84/1.59 => ( finite1450550360lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_317_finite__imageD,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( ( inj_on755450110lle_hf @ F @ A4 )
% 2.84/1.59 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_318_finite__imageD,axiom,
% 2.84/1.59 ! [F: nat > hF_Mirabelle_hf,A4: set_nat] :
% 2.84/1.59 ( ( finite586181922lle_hf @ ( image_246164834lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( ( inj_on1988990670lle_hf @ F @ A4 )
% 2.84/1.59 => ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_319_finite__imageD,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite_finite_nat @ ( image_131453538hf_nat @ F @ A4 ) )
% 2.84/1.59 => ( ( inj_on1874279374hf_nat @ F @ A4 )
% 2.84/1.59 => ( finite586181922lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_320_finite__imageD,axiom,
% 2.84/1.59 ! [F: nat > nat,A4: set_nat] :
% 2.84/1.59 ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 => ( ( inj_on_nat_nat @ F @ A4 )
% 2.84/1.59 => ( finite_finite_nat @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_321_card__image,axiom,
% 2.84/1.59 ! [F: set_HF_Mirabelle_hf > hF_Mirabelle_hf,A4: set_se933006839lle_hf] :
% 2.84/1.59 ( ( inj_on811196232lle_hf @ F @ A4 )
% 2.84/1.59 => ( ( finite1213132899lle_hf @ ( image_899003828lle_hf @ F @ A4 ) )
% 2.84/1.59 = ( finite90088345lle_hf @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_322_finite__surj__inj,axiom,
% 2.84/1.59 ! [A4: set_HF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( finite586181922lle_hf @ A4 )
% 2.84/1.59 => ( ( ord_le432112161lle_hf @ A4 @ ( image_1743964010lle_hf @ F @ A4 ) )
% 2.84/1.59 => ( inj_on755450110lle_hf @ F @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_323_finite__surj__inj,axiom,
% 2.84/1.59 ! [A4: set_nat,F: nat > nat] :
% 2.84/1.59 ( ( finite_finite_nat @ A4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
% 2.84/1.59 => ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_324_inj__on__finite,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > nat,A4: set_HF_Mirabelle_hf,B4: set_nat] :
% 2.84/1.59 ( ( inj_on1874279374hf_nat @ F @ A4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ ( image_131453538hf_nat @ F @ A4 ) @ B4 )
% 2.84/1.59 => ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( finite586181922lle_hf @ A4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_325_inj__on__finite,axiom,
% 2.84/1.59 ! [F: nat > nat,A4: set_nat,B4: set_nat] :
% 2.84/1.59 ( ( inj_on_nat_nat @ F @ A4 )
% 2.84/1.59 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 )
% 2.84/1.59 => ( ( finite_finite_nat @ B4 )
% 2.84/1.59 => ( finite_finite_nat @ A4 ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_326_finite__greaterThanLessThan,axiom,
% 2.84/1.59 ! [L: nat,U4: nat] : ( finite_finite_nat @ ( set_or1544565540an_nat @ L @ U4 ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_327_bot__nat__0_Onot__eq__extremum,axiom,
% 2.84/1.59 ! [A: nat] :
% 2.84/1.59 ( ( A != zero_zero_nat )
% 2.84/1.59 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_328_less__nat__zero__code,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_329_neq0__conv,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ( ( N2 != zero_zero_nat )
% 2.84/1.59 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_330_le0,axiom,
% 2.84/1.59 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_331_bot__nat__0_Oextremum,axiom,
% 2.84/1.59 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
% 2.84/1.59 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_333_le__0__eq,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 2.84/1.59 = ( N2 = zero_zero_nat ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_334_bot__nat__0_Oextremum__unique,axiom,
% 2.84/1.59 ! [A: nat] :
% 2.84/1.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.84/1.59 = ( A = zero_zero_nat ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_335_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.84/1.59 ! [A: nat] :
% 2.84/1.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.84/1.59 => ( A = zero_zero_nat ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_336_ex__least__nat__le,axiom,
% 2.84/1.59 ! [P: nat > $o,N2: nat] :
% 2.84/1.59 ( ( P @ N2 )
% 2.84/1.59 => ( ~ ( P @ zero_zero_nat )
% 2.84/1.59 => ? [K: nat] :
% 2.84/1.59 ( ( ord_less_eq_nat @ K @ N2 )
% 2.84/1.59 & ! [I: nat] :
% 2.84/1.59 ( ( ord_less_nat @ I @ K )
% 2.84/1.59 => ~ ( P @ I ) )
% 2.84/1.59 & ( P @ K ) ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_337_bot__nat__0_Oextremum__strict,axiom,
% 2.84/1.59 ! [A: nat] :
% 2.84/1.59 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_338_infinite__descent0,axiom,
% 2.84/1.59 ! [P: nat > $o,N2: nat] :
% 2.84/1.59 ( ( P @ zero_zero_nat )
% 2.84/1.59 => ( ! [N5: nat] :
% 2.84/1.59 ( ( ord_less_nat @ zero_zero_nat @ N5 )
% 2.84/1.59 => ( ~ ( P @ N5 )
% 2.84/1.59 => ? [M2: nat] :
% 2.84/1.59 ( ( ord_less_nat @ M2 @ N5 )
% 2.84/1.59 & ~ ( P @ M2 ) ) ) )
% 2.84/1.59 => ( P @ N2 ) ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_339_gr__implies__not0,axiom,
% 2.84/1.59 ! [M3: nat,N2: nat] :
% 2.84/1.59 ( ( ord_less_nat @ M3 @ N2 )
% 2.84/1.59 => ( N2 != zero_zero_nat ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_340_less__zeroE,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_341_not__less0,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_342_not__gr0,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 2.84/1.59 = ( N2 = zero_zero_nat ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_343_gr0I,axiom,
% 2.84/1.59 ! [N2: nat] :
% 2.84/1.59 ( ( N2 != zero_zero_nat )
% 2.84/1.59 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_344_RepFun__0,axiom,
% 2.84/1.59 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.84/1.59 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 2.84/1.59 = zero_z189798548lle_hf ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_345_HUnion__hempty,axiom,
% 2.84/1.59 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 2.84/1.59 = zero_z189798548lle_hf ) ).
% 2.84/1.59
% 2.84/1.59 thf(fact_346_Replace__0,axiom,
% 2.84/1.59 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.84/1.59 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 2.84/1.59 = zero_z189798548lle_hf ) ).
% 2.84/1.59
% 2.84/1.59 thf(conj_0,hypothesis,
% 2.84/1.59 ! [X6: hF_Mirabelle_hf] :
% 2.84/1.59 ( ( hF_Mirabelle_hmem @ X6 @ a )
% 2.84/1.59 => ( hF_Mirabelle_hmem @ X6 @ b ) ) ).
% 2.84/1.59
% 2.84/1.59 thf(conj_1,conjecture,
% 2.84/1.59 ord_le976219883lle_hf @ a @ b ).
% 2.84/1.59 ------------------------
% 2.94/1.62 % (10420)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_10317 for (2999ds/27Mi)
% 2.94/1.62 % (10423)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_10317 for (2999ds/275Mi)
% 2.94/1.62 % (10419)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_10317 for (2999ds/4Mi)
% 2.94/1.63 % (10424)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/18Mi)
% 2.94/1.63 % (10418)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/183Mi)
% 2.94/1.63 % (10421)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/2Mi)
% 2.94/1.63 % (10422)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_10317 for (2999ds/2Mi)
% 2.94/1.63 % (10419)Instruction limit reached!
% 2.94/1.63 % (10419)------------------------------
% 2.94/1.63 % (10419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (10419)Termination reason: Unknown
% 2.94/1.63 % (10419)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (10419)Memory used [KB]: 1407
% 2.94/1.63 % (10419)Time elapsed: 0.004 s
% 2.94/1.63 % (10419)Instructions burned: 6 (million)
% 2.94/1.63 % (10419)------------------------------
% 2.94/1.63 % (10419)------------------------------
% 2.94/1.63 % (10420)Instruction limit reached!
% 2.94/1.63 % (10420)------------------------------
% 2.94/1.63 % (10420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (10420)Termination reason: Unknown
% 2.94/1.63 % (10420)Termination phase: Property scanning
% 2.94/1.63
% 2.94/1.63 % (10420)Memory used [KB]: 2046
% 2.94/1.63 % (10420)Time elapsed: 0.011 s
% 2.94/1.63 % (10420)Instructions burned: 29 (million)
% 2.94/1.63 % (10420)------------------------------
% 2.94/1.63 % (10420)------------------------------
% 2.94/1.63 % (10421)Instruction limit reached!
% 2.94/1.63 % (10421)------------------------------
% 2.94/1.63 % (10421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (10421)Termination reason: Unknown
% 2.94/1.63 % (10421)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (10421)Memory used [KB]: 1407
% 2.94/1.63 % (10422)Instruction limit reached!
% 2.94/1.63 % (10422)------------------------------
% 2.94/1.63 % (10422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (10422)Termination reason: Unknown
% 2.94/1.63 % (10422)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (10422)Memory used [KB]: 1407
% 2.94/1.63 % (10422)Time elapsed: 0.004 s
% 2.94/1.63 % (10422)Instructions burned: 3 (million)
% 2.94/1.63 % (10422)------------------------------
% 2.94/1.63 % (10422)------------------------------
% 2.94/1.63 % (10421)Time elapsed: 0.004 s
% 2.94/1.63 % (10421)Instructions burned: 3 (million)
% 2.94/1.63 % (10421)------------------------------
% 2.94/1.63 % (10421)------------------------------
% 2.94/1.63 % (10424)Instruction limit reached!
% 2.94/1.63 % (10424)------------------------------
% 2.94/1.63 % (10424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (10424)Termination reason: Unknown
% 2.94/1.63 % (10424)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (10424)Memory used [KB]: 1791
% 2.94/1.63 % (10424)Time elapsed: 0.009 s
% 2.94/1.63 % (10424)Instructions burned: 20 (million)
% 2.94/1.63 % (10424)------------------------------
% 2.94/1.63 % (10424)------------------------------
% 2.94/1.64 % (10426)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_10317 for (2999ds/37Mi)
% 2.94/1.64 % (10425)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/3Mi)
% 2.94/1.64 % (10429)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_10317 for (2999ds/1041Mi)
% 2.94/1.64 % (10425)Instruction limit reached!
% 2.94/1.64 % (10425)------------------------------
% 2.94/1.64 % (10425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (10425)Termination reason: Unknown
% 2.94/1.64 % (10425)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (10425)Memory used [KB]: 1407
% 2.94/1.64 % (10425)Time elapsed: 0.003 s
% 2.94/1.64 % (10425)Instructions burned: 4 (million)
% 2.94/1.64 % (10425)------------------------------
% 2.94/1.64 % (10425)------------------------------
% 2.94/1.65 % (10427)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_10317 for (2999ds/15Mi)
% 2.94/1.65 % (10428)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_10317 for (2999ds/3Mi)
% 2.94/1.65 % (10426)Instruction limit reached!
% 2.94/1.65 % (10426)------------------------------
% 2.94/1.65 % (10426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (10426)Termination reason: Unknown
% 2.94/1.65 % (10426)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (10426)Memory used [KB]: 2174
% 2.94/1.65 % (10426)Time elapsed: 0.011 s
% 2.94/1.65 % (10426)Instructions burned: 37 (million)
% 2.94/1.65 % (10426)------------------------------
% 2.94/1.65 % (10426)------------------------------
% 2.94/1.65 % (10428)Instruction limit reached!
% 2.94/1.65 % (10428)------------------------------
% 2.94/1.65 % (10428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (10428)Termination reason: Unknown
% 2.94/1.65 % (10428)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (10428)Memory used [KB]: 1407
% 2.94/1.65 % (10428)Time elapsed: 0.004 s
% 2.94/1.65 % (10428)Instructions burned: 3 (million)
% 2.94/1.65 % (10428)------------------------------
% 2.94/1.65 % (10428)------------------------------
% 2.94/1.65 % (10427)Instruction limit reached!
% 2.94/1.65 % (10427)------------------------------
% 2.94/1.65 % (10427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (10427)Termination reason: Unknown
% 2.94/1.65 % (10427)Termination phase: shuffling
% 2.94/1.65
% 2.94/1.65 % (10427)Memory used [KB]: 1663
% 2.94/1.65 % (10427)Time elapsed: 0.011 s
% 2.94/1.65 % (10427)Instructions burned: 16 (million)
% 2.94/1.65 % (10427)------------------------------
% 2.94/1.65 % (10427)------------------------------
% 2.94/1.66 % (10431)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/16Mi)
% 2.94/1.66 % (10430)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/7Mi)
% 2.94/1.66 % (10431)Instruction limit reached!
% 2.94/1.66 % (10431)------------------------------
% 2.94/1.66 % (10431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (10431)Termination reason: Unknown
% 2.94/1.66 % (10431)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (10431)Memory used [KB]: 1663
% 2.94/1.66 % (10431)Time elapsed: 0.006 s
% 2.94/1.66 % (10431)Instructions burned: 16 (million)
% 2.94/1.66 % (10431)------------------------------
% 2.94/1.66 % (10431)------------------------------
% 2.94/1.66 % (10430)Instruction limit reached!
% 2.94/1.66 % (10430)------------------------------
% 2.94/1.66 % (10430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (10430)Termination reason: Unknown
% 2.94/1.66 % (10430)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (10430)Memory used [KB]: 1535
% 2.94/1.66 % (10430)Time elapsed: 0.005 s
% 2.94/1.66 % (10430)Instructions burned: 9 (million)
% 2.94/1.66 % (10430)------------------------------
% 2.94/1.66 % (10430)------------------------------
% 2.94/1.66 % (10432)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_10317 for (2999ds/3Mi)
% 2.94/1.67 % (10432)Instruction limit reached!
% 2.94/1.67 % (10432)------------------------------
% 2.94/1.67 % (10432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (10432)Termination reason: Unknown
% 2.94/1.67 % (10432)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (10432)Memory used [KB]: 1407
% 2.94/1.67 % (10432)Time elapsed: 0.004 s
% 2.94/1.67 % (10432)Instructions burned: 3 (million)
% 2.94/1.67 % (10432)------------------------------
% 2.94/1.67 % (10432)------------------------------
% 2.94/1.67 % (10433)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_10317 for (2999ds/3Mi)
% 2.94/1.67 % (10433)Instruction limit reached!
% 2.94/1.67 % (10433)------------------------------
% 2.94/1.67 % (10433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (10433)Termination reason: Unknown
% 2.94/1.67 % (10433)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (10433)Memory used [KB]: 1407
% 2.94/1.67 % (10433)Time elapsed: 0.003 s
% 2.94/1.67 % (10433)Instructions burned: 4 (million)
% 2.94/1.67 % (10433)------------------------------
% 2.94/1.67 % (10433)------------------------------
% 2.94/1.67 % (10434)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/7Mi)
% 2.94/1.67 % (10434)Instruction limit reached!
% 2.94/1.67 % (10434)------------------------------
% 2.94/1.67 % (10434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (10434)Termination reason: Unknown
% 2.94/1.67 % (10434)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (10434)Memory used [KB]: 1535
% 2.94/1.67 % (10434)Time elapsed: 0.004 s
% 2.94/1.67 % (10434)Instructions burned: 9 (million)
% 2.94/1.67 % (10434)------------------------------
% 2.94/1.67 % (10434)------------------------------
% 2.94/1.68 % (10437)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/18Mi)
% 2.94/1.68 % (10435)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/3Mi)
% 2.94/1.68 % (10436)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/4Mi)
% 2.94/1.68 % (10435)Instruction limit reached!
% 2.94/1.68 % (10435)------------------------------
% 2.94/1.68 % (10435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (10435)Termination reason: Unknown
% 2.94/1.68 % (10435)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (10435)Memory used [KB]: 1407
% 2.94/1.68 % (10435)Time elapsed: 0.003 s
% 2.94/1.68 % (10435)Instructions burned: 4 (million)
% 2.94/1.68 % (10435)------------------------------
% 2.94/1.68 % (10435)------------------------------
% 2.94/1.68 % (10436)Instruction limit reached!
% 2.94/1.68 % (10436)------------------------------
% 2.94/1.68 % (10436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (10436)Termination reason: Unknown
% 2.94/1.68 % (10436)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (10436)Memory used [KB]: 1407
% 2.94/1.68 % (10436)Time elapsed: 0.003 s
% 2.94/1.68 % (10436)Instructions burned: 4 (million)
% 2.94/1.68 % (10436)------------------------------
% 2.94/1.68 % (10436)------------------------------
% 2.94/1.68 % (10438)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/710Mi)
% 2.94/1.68 % (10437)Instruction limit reached!
% 2.94/1.68 % (10437)------------------------------
% 2.94/1.68 % (10437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (10437)Termination reason: Unknown
% 2.94/1.68 % (10437)Termination phase: shuffling
% 2.94/1.69
% 2.94/1.69 % (10437)Memory used [KB]: 1791
% 2.94/1.69 % (10437)Time elapsed: 0.008 s
% 2.94/1.69 % (10437)Instructions burned: 18 (million)
% 2.94/1.69 % (10437)------------------------------
% 2.94/1.69 % (10437)------------------------------
% 2.94/1.69 % (10418)Instruction limit reached!
% 2.94/1.69 % (10418)------------------------------
% 2.94/1.69 % (10418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.69 % (10418)Termination reason: Unknown
% 2.94/1.69 % (10418)Termination phase: Saturation
% 2.94/1.69
% 2.94/1.69 % (10418)Memory used [KB]: 7547
% 2.94/1.69 % (10418)Time elapsed: 0.065 s
% 2.94/1.69 % (10418)Instructions burned: 185 (million)
% 2.94/1.69 % (10418)------------------------------
% 2.94/1.69 % (10418)------------------------------
% 2.94/1.69 % (10440)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_10317 for (2999ds/902Mi)
% 2.94/1.70 % (10441)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_10317 for (2999ds/21Mi)
% 2.94/1.70 % (10423)Instruction limit reached!
% 2.94/1.70 % (10423)------------------------------
% 2.94/1.70 % (10423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (10423)Termination reason: Unknown
% 2.94/1.70 % (10423)Termination phase: Saturation
% 2.94/1.70
% 2.94/1.70 % (10423)Memory used [KB]: 7164
% 2.94/1.70 % (10423)Time elapsed: 0.078 s
% 2.94/1.70 % (10423)Instructions burned: 278 (million)
% 2.94/1.70 % (10423)------------------------------
% 2.94/1.70 % (10423)------------------------------
% 2.94/1.70 % (10439)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10317 for (2999ds/6Mi)
% 2.94/1.70 % (10442)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/5Mi)
% 2.94/1.70 % (10439)Instruction limit reached!
% 2.94/1.70 % (10439)------------------------------
% 2.94/1.70 % (10439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (10439)Termination reason: Unknown
% 2.94/1.70 % (10439)Termination phase: shuffling
% 2.94/1.70
% 2.94/1.70 % (10439)Memory used [KB]: 1535
% 2.94/1.70 % (10439)Time elapsed: 0.004 s
% 2.94/1.70 % (10439)Instructions burned: 6 (million)
% 2.94/1.70 % (10439)------------------------------
% 2.94/1.70 % (10439)------------------------------
% 2.94/1.70 % (10442)Instruction limit reached!
% 2.94/1.70 % (10442)------------------------------
% 2.94/1.70 % (10442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (10442)Termination reason: Unknown
% 2.94/1.70 % (10442)Termination phase: shuffling
% 2.94/1.70
% 2.94/1.70 % (10442)Memory used [KB]: 1535
% 2.94/1.70 % (10442)Time elapsed: 0.004 s
% 2.94/1.70 % (10442)Instructions burned: 7 (million)
% 2.94/1.70 % (10442)------------------------------
% 2.94/1.70 % (10442)------------------------------
% 2.94/1.70 % (10441)Instruction limit reached!
% 2.94/1.70 % (10441)------------------------------
% 2.94/1.70 % (10441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (10441)Termination reason: Unknown
% 2.94/1.70 % (10441)Termination phase: shuffling
% 2.94/1.70
% 2.94/1.70 % (10441)Memory used [KB]: 1791
% 2.94/1.70 % (10441)Time elapsed: 0.009 s
% 2.94/1.70 % (10441)Instructions burned: 21 (million)
% 2.94/1.70 % (10441)------------------------------
% 2.94/1.70 % (10441)------------------------------
% 2.94/1.71 % (10443)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_10317 for (2998ds/6Mi)
% 2.94/1.71 % (10443)Instruction limit reached!
% 2.94/1.71 % (10443)------------------------------
% 2.94/1.71 % (10443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.71 % (10443)Termination reason: Unknown
% 2.94/1.71 % (10443)Termination phase: shuffling
% 2.94/1.71
% 2.94/1.71 % (10443)Memory used [KB]: 1535
% 2.94/1.71 % (10443)Time elapsed: 0.003 s
% 2.94/1.71 % (10443)Instructions burned: 6 (million)
% 2.94/1.71 % (10443)------------------------------
% 2.94/1.71 % (10443)------------------------------
% 2.94/1.71 % (10445)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_10317 for (2998ds/779Mi)
% 2.94/1.71 % (10446)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_10317 for (2998ds/19Mi)
% 2.94/1.72 % (10444)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/377Mi)
% 2.94/1.72 % (10447)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/879Mi)
% 2.94/1.72 % (10446)Instruction limit reached!
% 2.94/1.72 % (10446)------------------------------
% 2.94/1.72 % (10446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.72 % (10446)Termination reason: Unknown
% 2.94/1.72 % (10446)Termination phase: shuffling
% 2.94/1.72
% 2.94/1.72 % (10446)Memory used [KB]: 1791
% 2.94/1.72 % (10446)Time elapsed: 0.008 s
% 2.94/1.72 % (10446)Instructions burned: 21 (million)
% 2.94/1.72 % (10446)------------------------------
% 2.94/1.72 % (10446)------------------------------
% 2.94/1.73 % (10448)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/17Mi)
% 2.94/1.74 % (10448)Instruction limit reached!
% 2.94/1.74 % (10448)------------------------------
% 2.94/1.74 % (10448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.74 % (10448)Termination reason: Unknown
% 2.94/1.74 % (10448)Termination phase: shuffling
% 2.94/1.74
% 2.94/1.74 % (10448)Memory used [KB]: 1791
% 2.94/1.74 % (10448)Time elapsed: 0.007 s
% 2.94/1.74 % (10448)Instructions burned: 17 (million)
% 2.94/1.74 % (10448)------------------------------
% 2.94/1.74 % (10448)------------------------------
% 2.94/1.75 % (10449)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/3Mi)
% 2.94/1.75 % (10449)Instruction limit reached!
% 2.94/1.75 % (10449)------------------------------
% 2.94/1.75 % (10449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.75 % (10449)Termination reason: Unknown
% 2.94/1.75 % (10449)Termination phase: shuffling
% 2.94/1.75
% 2.94/1.75 % (10449)Memory used [KB]: 1407
% 2.94/1.75 % (10449)Time elapsed: 0.003 s
% 2.94/1.75 % (10449)Instructions burned: 4 (million)
% 2.94/1.75 % (10449)------------------------------
% 2.94/1.75 % (10449)------------------------------
% 2.94/1.76 % (10450)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/30Mi)
% 2.94/1.77 % (10450)Instruction limit reached!
% 2.94/1.77 % (10450)------------------------------
% 2.94/1.77 % (10450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.77 % (10450)Termination reason: Unknown
% 2.94/1.77 % (10450)Termination phase: Property scanning
% 2.94/1.77
% 2.94/1.77 % (10450)Memory used [KB]: 2046
% 2.94/1.77 % (10450)Time elapsed: 0.012 s
% 2.94/1.77 % (10450)Instructions burned: 31 (million)
% 2.94/1.77 % (10450)------------------------------
% 2.94/1.77 % (10450)------------------------------
% 2.94/1.78 % (10451)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_10317 for (2998ds/127Mi)
% 2.94/1.78 % (10447)First to succeed.
% 2.94/1.78 % (10447)Refutation found. Thanks to Tanya!
% 2.94/1.78 % SZS status Theorem for DTF2THF_10317
% 2.94/1.78 % SZS output start Proof for DTF2THF_10317
% 2.94/1.78 thf(type_def_5, type, set_HF_Mirabelle_hf: $tType).
% 2.94/1.78 thf(type_def_6, type, nat: $tType).
% 2.94/1.78 thf(type_def_8, type, set_nat: $tType).
% 2.94/1.78 thf(type_def_9, type, set_se933006839lle_hf: $tType).
% 2.94/1.78 thf(type_def_10, type, hF_Mirabelle_hf: $tType).
% 2.94/1.78 thf(func_def_0, type, set_se933006839lle_hf: $tType).
% 2.94/1.78 thf(func_def_1, type, set_HF_Mirabelle_hf: $tType).
% 2.94/1.78 thf(func_def_2, type, hF_Mirabelle_hf: $tType).
% 2.94/1.78 thf(func_def_3, type, set_nat: $tType).
% 2.94/1.78 thf(func_def_4, type, nat: $tType).
% 2.94/1.78 thf(func_def_5, type, finite1213132899lle_hf: set_HF_Mirabelle_hf > nat).
% 2.94/1.78 thf(func_def_6, type, finite_card_nat: set_nat > nat).
% 2.94/1.78 thf(func_def_7, type, finite90088345lle_hf: set_se933006839lle_hf > nat).
% 2.94/1.78 thf(func_def_8, type, finite586181922lle_hf: set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_9, type, finite_finite_nat: set_nat > $o).
% 2.94/1.78 thf(func_def_10, type, finite1450550360lle_hf: set_se933006839lle_hf > $o).
% 2.94/1.78 thf(func_def_11, type, inj_on755450110lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf) > set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_12, type, inj_on1874279374hf_nat: (hF_Mirabelle_hf > nat) > set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_13, type, inj_on1988990670lle_hf: (nat > hF_Mirabelle_hf) > set_nat > $o).
% 2.94/1.78 thf(func_def_14, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 2.94/1.78 thf(func_def_15, type, inj_on811196232lle_hf: (set_HF_Mirabelle_hf > hF_Mirabelle_hf) > set_se933006839lle_hf > $o).
% 2.94/1.78 thf(func_def_16, type, zero_z189798548lle_hf: hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_17, type, zero_zero_nat: nat).
% 2.94/1.78 thf(func_def_18, type, hF_Mir818139703ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_19, type, hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_20, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_21, type, hF_Mir1248913145eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_22, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_23, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_24, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_25, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_26, type, ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_27, type, ord_less_nat: nat > nat > $o).
% 2.94/1.78 thf(func_def_28, type, ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_29, type, ord_less_set_nat: set_nat > set_nat > $o).
% 2.94/1.78 thf(func_def_30, type, ord_le2017230260lle_hf: ($o > hF_Mirabelle_hf) > ($o > hF_Mirabelle_hf) > $o).
% 2.94/1.78 thf(func_def_31, type, ord_less_eq_o_nat: ($o > nat) > ($o > nat) > $o).
% 2.94/1.78 thf(func_def_32, type, ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_33, type, ord_less_eq_nat: nat > nat > $o).
% 2.94/1.78 thf(func_def_34, type, ord_le432112161lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_35, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 2.94/1.78 thf(func_def_36, type, order_1640852850lle_hf: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_37, type, order_Greatest_nat: (nat > $o) > nat).
% 2.94/1.78 thf(func_def_38, type, order_690706045lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf) > $o).
% 2.94/1.78 thf(func_def_39, type, order_557194383hf_nat: (hF_Mirabelle_hf > nat) > $o).
% 2.94/1.78 thf(func_def_40, type, order_671905679lle_hf: (nat > hF_Mirabelle_hf) > $o).
% 2.94/1.78 thf(func_def_41, type, order_1631207636at_nat: (nat > nat) > $o).
% 2.94/1.78 thf(func_def_42, type, orderi1737556723lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > $o).
% 2.94/1.78 thf(func_def_43, type, ordering_nat: (nat > nat > $o) > (nat > nat > $o) > $o).
% 2.94/1.78 thf(func_def_44, type, collec2046588256lle_hf: (hF_Mirabelle_hf > $o) > set_HF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_45, type, collec1758573718lle_hf: (set_HF_Mirabelle_hf > $o) > set_se933006839lle_hf).
% 2.94/1.78 thf(func_def_46, type, image_1743964010lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf) > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_47, type, image_131453538hf_nat: (hF_Mirabelle_hf > nat) > set_HF_Mirabelle_hf > set_nat).
% 2.94/1.78 thf(func_def_48, type, image_246164834lle_hf: (nat > hF_Mirabelle_hf) > set_nat > set_HF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_49, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 2.94/1.78 thf(func_def_50, type, image_899003828lle_hf: (set_HF_Mirabelle_hf > hF_Mirabelle_hf) > set_se933006839lle_hf > set_HF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_51, type, set_or1544565540an_nat: nat > nat > set_nat).
% 2.94/1.78 thf(func_def_52, type, member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 2.94/1.78 thf(func_def_53, type, member_nat: nat > set_nat > $o).
% 2.94/1.78 thf(func_def_54, type, member1490636632lle_hf: set_HF_Mirabelle_hf > set_se933006839lle_hf > $o).
% 2.94/1.78 thf(func_def_55, type, a: hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_56, type, b: hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_71, type, sK0: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_72, type, sK1: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_73, type, sK2: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_74, type, sK3: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_75, type, sK4: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_76, type, sK5: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_77, type, sK6: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_78, type, sK7: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_79, type, sK8: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_80, type, sK9: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_81, type, sK10: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_82, type, sK11: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_83, type, sK12: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_84, type, sK13: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_85, type, sK14: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_86, type, sK15: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_87, type, sK16: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_88, type, sK17: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_89, type, sK18: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_90, type, sK19: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_91, type, sK20: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_92, type, sK21: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_93, type, sK22: (hF_Mirabelle_hf > hF_Mirabelle_hf) > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_94, type, sK23: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_95, type, sK24: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_96, type, sK25: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_97, type, sK26: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_98, type, sK27: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_99, type, sK28: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_100, type, sK29: (nat > hF_Mirabelle_hf) > nat).
% 2.94/1.78 thf(func_def_101, type, sK30: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_102, type, sK31: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_103, type, sK32: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_104, type, sK33: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_105, type, sK34: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_106, type, sK35: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_107, type, sK36: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_108, type, sK37: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_109, type, sK38: (hF_Mirabelle_hf > nat) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_110, type, sK39: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_111, type, sK40: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_112, type, sK41: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_113, type, sK42: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_115, type, sK44: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_116, type, sK45: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_117, type, sK46: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_118, type, sK47: hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_119, type, sK48: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_120, type, sK49: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_121, type, sK50: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_122, type, sK51: nat > (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_123, type, sK52: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_124, type, sK53: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_125, type, sK54: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_126, type, sK55: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_127, type, sK56: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_128, type, sK57: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_129, type, sK58: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_130, type, sK59: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_131, type, sK60: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_132, type, sK61: (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_133, type, sK62: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_134, type, sK63: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_135, type, sK64: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_136, type, sK65: hF_Mirabelle_hf > hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_137, type, sK66: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_138, type, sK67: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_139, type, sK68: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_140, type, sK69: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_141, type, sK70: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_142, type, sK71: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_143, type, sK72: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_144, type, sK73: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_145, type, sK74: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_146, type, sK75: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_147, type, sK76: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_148, type, sK77: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_149, type, sK78: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_150, type, sK79: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_151, type, sK80: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_152, type, sK81: (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_153, type, sK82: nat > (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_154, type, sK83: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_155, type, sK84: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_156, type, sK85: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_157, type, sK86: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_158, type, sK87: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_159, type, sK88: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_160, type, sK89: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_161, type, sK90: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_162, type, sK91: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_163, type, sK92: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_164, type, sK93: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_165, type, sK94: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_166, type, sK95: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_167, type, sK96: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_168, type, sK97: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_169, type, sK98: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_170, type, sK99: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_171, type, sK100: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_172, type, sK101: (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_173, type, sK102: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_174, type, sK103: (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_175, type, sK104: nat > (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_176, type, sK105: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_177, type, sK106: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_178, type, sK107: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_179, type, sK108: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_180, type, sK109: hF_Mirabelle_hf > hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_181, type, sK110: nat > (nat > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_182, type, sK111: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_183, type, sK112: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_184, type, sK113: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_185, type, sK114: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_186, type, sK115: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(func_def_187, type, sK116: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.94/1.78 thf(f1920,plain,(
% 2.94/1.78 $false),
% 2.94/1.78 inference(trivial_inequality_removal,[status(thm)],[f1919])).
% 2.94/1.78 thf(f1919,plain,(
% 2.94/1.78 ($true = $false)),
% 2.94/1.78 inference(forward_demodulation,[status(thm)],[f1918,f1327])).
% 2.94/1.78 thf(f1327,plain,(
% 2.94/1.78 ($false = (member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ b)))),
% 2.94/1.78 inference(binary_proxy_clausification,[status(thm)],[f1326])).
% 2.94/1.78 thf(f1326,plain,(
% 2.94/1.78 ($false = ((member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ a)) => (member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ b))))),
% 2.94/1.78 inference(beta_eta_normalization,[status(thm)],[f1325])).
% 2.94/1.78 thf(f1325,plain,(
% 2.94/1.78 ($false = ((^[Y0 : hF_Mirabelle_hf]: ((member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ a)) => (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ b)))) @ sK47))),
% 2.94/1.78 inference(sigma_clausification,[status(thm)],[f1324])).
% 2.94/1.78 thf(f1324,plain,(
% 2.94/1.78 ($true != (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ a)) => (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ b))))))),
% 2.94/1.78 inference(beta_eta_normalization,[status(thm)],[f1243])).
% 2.94/1.78 thf(f1243,plain,(
% 2.94/1.78 ($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: (((^[Y3 : hF_Mirabelle_hf]: ((^[Y4 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y3 @ (hF_Mirabelle_hfset @ Y4))))) @ Y2 @ Y0) => ((^[Y3 : hF_Mirabelle_hf]: ((^[Y4 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y3 @ (hF_Mirabelle_hfset @ Y4))))) @ Y2 @ Y1))))))) @ a @ b))),
% 2.94/1.78 inference(definition_unfolding,[status(thm)],[f1179,f1203])).
% 2.94/1.78 thf(f1203,plain,(
% 2.94/1.78 (ord_le976219883lle_hf = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: (((^[Y3 : hF_Mirabelle_hf]: ((^[Y4 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y3 @ (hF_Mirabelle_hfset @ Y4))))) @ Y2 @ Y0) => ((^[Y3 : hF_Mirabelle_hf]: ((^[Y4 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y3 @ (hF_Mirabelle_hfset @ Y4))))) @ Y2 @ Y1))))))))),
% 2.94/1.78 inference(definition_unfolding,[status(thm)],[f1159,f1164,f1164])).
% 2.94/1.78 thf(f1164,plain,(
% 2.94/1.78 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 2.94/1.78 inference(cnf_transformation,[status(thm)],[f624])).
% 2.94/1.78 thf(f624,plain,(
% 2.94/1.78 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 2.94/1.78 inference(fool_elimination,[status(thm)],[f623])).
% 2.94/1.78 thf(f623,plain,(
% 2.94/1.78 ((^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ X1))) = hF_Mirabelle_hmem)),
% 2.94/1.78 inference(rectify,[status(thm)],[f71])).
% 2.94/1.78 thf(f71,axiom,(
% 2.94/1.78 ((^[X5 : hF_Mirabelle_hf, X6 : hF_Mirabelle_hf] : (member1367349282lle_hf @ X5 @ (hF_Mirabelle_hfset @ X6))) = hF_Mirabelle_hmem)),
% 2.94/1.78 file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',fact_70_hmem__def)).
% 2.94/1.78 thf(f1159,plain,(
% 2.94/1.78 (ord_le976219883lle_hf = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ Y2 @ Y0) => (hF_Mirabelle_hmem @ Y2 @ Y1))))))))),
% 2.94/1.78 inference(cnf_transformation,[status(thm)],[f738])).
% 2.94/1.78 thf(f738,plain,(
% 2.94/1.78 (ord_le976219883lle_hf = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ Y2 @ Y0) => (hF_Mirabelle_hmem @ Y2 @ Y1))))))))),
% 2.94/1.78 inference(fool_elimination,[status(thm)],[f737])).
% 2.94/1.78 thf(f737,plain,(
% 2.94/1.78 (ord_le976219883lle_hf = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (! [X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X2 @ X0) => (hF_Mirabelle_hmem @ X2 @ X1)))))),
% 2.94/1.78 inference(rectify,[status(thm)],[f7])).
% 2.94/1.78 thf(f7,axiom,(
% 2.94/1.78 (ord_le976219883lle_hf = (^[X20 : hF_Mirabelle_hf, X21 : hF_Mirabelle_hf] : (! [X7 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X7 @ X20) => (hF_Mirabelle_hmem @ X7 @ X21)))))),
% 2.94/1.78 file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',fact_6_less__eq__hf__def)).
% 2.94/1.78 thf(f1179,plain,(
% 2.94/1.78 ((ord_le976219883lle_hf @ a @ b) != $true)),
% 2.94/1.78 inference(cnf_transformation,[status(thm)],[f1032])).
% 2.94/1.78 thf(f1032,plain,(
% 2.94/1.78 ((ord_le976219883lle_hf @ a @ b) != $true)),
% 2.94/1.78 inference(flattening,[status(thm)],[f670])).
% 2.94/1.78 thf(f670,plain,(
% 2.94/1.78 ~((ord_le976219883lle_hf @ a @ b) = $true)),
% 2.94/1.78 inference(fool_elimination,[status(thm)],[f669])).
% 2.94/1.78 thf(f669,plain,(
% 2.94/1.78 ~(ord_le976219883lle_hf @ a @ b)),
% 2.94/1.78 inference(rectify,[status(thm)],[f350])).
% 2.94/1.78 thf(f350,negated_conjecture,(
% 2.94/1.78 ~(ord_le976219883lle_hf @ a @ b)),
% 2.94/1.78 inference(negated_conjecture,[status(cth)],[f349])).
% 2.94/1.78 thf(f349,conjecture,(
% 2.94/1.78 (ord_le976219883lle_hf @ a @ b)),
% 2.94/1.78 file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',conj_1)).
% 2.94/1.78 thf(f1918,plain,(
% 2.94/1.78 ($true = (member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ b)))),
% 2.94/1.78 inference(trivial_inequality_removal,[status(thm)],[f1917])).
% 2.94/1.78 thf(f1917,plain,(
% 2.94/1.78 ($true != $true) | ($true = (member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ b)))),
% 2.94/1.78 inference(superposition,[status(thm)],[f1773,f1328])).
% 2.94/1.78 thf(f1328,plain,(
% 2.94/1.78 ($true = (member1367349282lle_hf @ sK47 @ (hF_Mirabelle_hfset @ a)))),
% 2.94/1.78 inference(binary_proxy_clausification,[status(thm)],[f1326])).
% 2.94/1.78 thf(f1773,plain,(
% 2.94/1.78 ( ! [X0 : hF_Mirabelle_hf] : (($true != (member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ a))) | ((member1367349282lle_hf @ X0 @ (hF_Mirabelle_hfset @ b)) = $true)) )),
% 2.94/1.78 inference(beta_eta_normalization,[status(thm)],[f1241])).
% 2.94/1.78 thf(f1241,plain,(
% 2.94/1.78 ( ! [X0 : hF_Mirabelle_hf] : (($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X0 @ a)) | ($true = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member1367349282lle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X0 @ b))) )),
% 2.94/1.78 inference(definition_unfolding,[status(thm)],[f1177,f1164,f1164])).
% 2.94/1.78 thf(f1177,plain,(
% 2.94/1.78 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | ($true = (hF_Mirabelle_hmem @ X0 @ b))) )),
% 2.94/1.78 inference(cnf_transformation,[status(thm)],[f1066])).
% 2.94/1.78 thf(f1066,plain,(
% 2.94/1.78 ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | ($true = (hF_Mirabelle_hmem @ X0 @ b)))),
% 2.94/1.78 inference(ennf_transformation,[status(thm)],[f867])).
% 2.94/1.78 thf(f867,plain,(
% 2.94/1.78 ! [X0 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X0 @ a)) => ($true = (hF_Mirabelle_hmem @ X0 @ b)))),
% 2.94/1.78 inference(fool_elimination,[status(thm)],[f866])).
% 2.94/1.78 thf(f866,plain,(
% 2.94/1.78 ! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ a) => (hF_Mirabelle_hmem @ X0 @ b))),
% 2.94/1.78 inference(rectify,[status(thm)],[f348])).
% 2.94/1.78 thf(f348,axiom,(
% 2.94/1.78 ! [X46 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X46 @ a) => (hF_Mirabelle_hmem @ X46 @ b))),
% 2.94/1.78 file('/export/starexec/sandbox/tmp/tmp.6AqA4vdshl/DTF2THF_10317.p',conj_0)).
% 2.94/1.78 % SZS output end Proof for DTF2THF_10317
% 2.94/1.78 % (10447)------------------------------
% 2.94/1.78 % (10447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.78 % (10447)Termination reason: Refutation
% 2.94/1.78
% 2.94/1.78 % (10447)Memory used [KB]: 7419
% 2.94/1.78 % (10447)Time elapsed: 0.065 s
% 2.94/1.78 % (10447)Instructions burned: 202 (million)
% 2.94/1.78 % (10447)------------------------------
% 2.94/1.78 % (10447)------------------------------
% 2.94/1.78 % (10417)Success in time 0.192 s
%------------------------------------------------------------------------------