%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP149^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n016.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 08:55:56 AM UTC 2026
% Result : Theorem 2.72s 1.71s
% Output : Refutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 8
% Syntax : Number of formulae : 45 ( 7 unt; 0 typ; 2 def)
% Number of atoms : 287 ( 83 equ; 0 cnn)
% Maximal formula atoms : 6 ( 6 avg)
% Number of connectives : 706 ( 70 ~; 43 |; 24 &; 552 @)
% ( 2 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 102 usr; 18 con; 0-5 aty)
% Number of variables : 46 ( 0 ^; 34 !; 12 ?; 46 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
set_real: $tType ).
thf(type_def_8,type,
a: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
set_set_real: $tType ).
thf(type_def_11,type,
set_set_a: $tType ).
thf(type_def_12,type,
bounde7994401a_real: $tType ).
thf(func_def_0,type,
bounde7994401a_real: $tType ).
thf(func_def_1,type,
set_set_real: $tType ).
thf(func_def_2,type,
set_set_a: $tType ).
thf(func_def_3,type,
set_real: $tType ).
thf(func_def_4,type,
set_a: $tType ).
thf(func_def_5,type,
real: $tType ).
thf(func_def_6,type,
a: $tType ).
thf(func_def_7,type,
elemen1140313242l_real: real > real > set_real ).
thf(func_def_8,type,
elemen49976720ball_a: a > real > set_a ).
thf(func_def_9,type,
auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
thf(func_def_10,type,
minus_minus_real: real > real > real ).
thf(func_def_11,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_12,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_13,type,
minus_minus_a: a > a > a ).
thf(func_def_14,type,
uminus_uminus_a_a: ( a > a ) > a > a ).
thf(func_def_15,type,
uminus_uminus_real: real > real ).
thf(func_def_16,type,
uminus773214379t_real: set_real > set_real ).
thf(func_def_17,type,
uminus_uminus_set_a: set_a > set_a ).
thf(func_def_18,type,
uminus_uminus_a: a > a ).
thf(func_def_19,type,
zero_zero_real: real ).
thf(func_def_20,type,
zero_zero_a: a ).
thf(func_def_21,type,
initia826609931terval: set_real > $o ).
thf(func_def_22,type,
inner_4346926r_real: real > real > real ).
thf(func_def_23,type,
inner_1173012732nner_a: a > a > real ).
thf(func_def_24,type,
auto_l630715367iant_a: ( a > a ) > set_a > set_a > $o ).
thf(func_def_25,type,
line_c1152468841ment_a: a > a > set_a ).
thf(func_def_26,type,
line_open_segment_a: a > a > set_a ).
thf(func_def_27,type,
oDE_au1996717075pped_a: ( a > a ) > set_a > a > set_a > $o ).
thf(func_def_28,type,
ord_less_real: real > real > $o ).
thf(func_def_29,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_30,type,
ord_le861030508t_real: set_set_real > set_set_real > $o ).
thf(func_def_31,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(func_def_32,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_33,type,
ord_less_a: a > a > $o ).
thf(func_def_34,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_35,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_36,type,
ord_le1733505144t_real: set_set_real > set_set_real > $o ).
thf(func_def_37,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_38,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_39,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_40,type,
top_top_set_real: set_real ).
thf(func_def_41,type,
top_top_set_a: set_a ).
thf(func_def_42,type,
period720806154rbit_a: ( a > a ) > set_a > a > $o ).
thf(func_def_43,type,
period1305449585riod_a: ( a > a ) > set_a > a > real ).
thf(func_def_44,type,
period138238489rbit_a: ( a > a ) > set_a > a > $o ).
thf(func_def_45,type,
poinca659159244_rot_a: a > a ).
thf(func_def_46,type,
poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
thf(func_def_47,type,
reacha1191408304tion_a: ( a > a ) > set_a > ( a > real ) > ( a > bounde7994401a_real ) > set_a > $o ).
thf(func_def_48,type,
real_V1934908667t_real: real > real > real ).
thf(func_def_49,type,
real_V1514887919dist_a: a > a > real ).
thf(func_def_50,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_51,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(func_def_52,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_53,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_54,type,
set_or656347191t_real: real > real > set_real ).
thf(func_def_55,type,
set_or1619725421t_real: set_real > set_real > set_set_real ).
thf(func_def_56,type,
set_or2094724627_set_a: set_a > set_a > set_set_a ).
thf(func_def_57,type,
set_or411607219Most_a: a > a > set_a ).
thf(func_def_58,type,
member_real: real > set_real > $o ).
thf(func_def_59,type,
member_set_real: set_real > set_set_real > $o ).
thf(func_def_60,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_61,type,
member_a: a > set_a > $o ).
thf(func_def_62,type,
x: set_a ).
thf(func_def_63,type,
a2: a ).
thf(func_def_64,type,
b: a ).
thf(func_def_65,type,
d: real ).
thf(func_def_66,type,
f: a > a ).
thf(func_def_67,type,
s: a ).
thf(func_def_69,type,
x2: a ).
thf(func_def_70,type,
z: a ).
thf(func_def_84,type,
sP0: real > ( real > real > $o ) > real > $o ).
thf(func_def_85,type,
sP1: a > a > $o ).
thf(func_def_86,type,
sK2: a ).
thf(func_def_87,type,
sK3: a > a ).
thf(func_def_88,type,
sK4: a > a ).
thf(func_def_89,type,
sK5: a > real ).
thf(func_def_90,type,
sK6: a > real ).
thf(func_def_91,type,
sK7: real > ( real > real > $o ) > real > real ).
thf(func_def_92,type,
sK8: real > real > ( real > real > $o ) > real > real ).
thf(func_def_93,type,
sK9: real > real > ( real > real > $o ) > real > real ).
thf(func_def_94,type,
sK10: ( real > real > $o ) > real ).
thf(func_def_95,type,
sK11: ( real > real > $o ) > real ).
thf(func_def_96,type,
sK12: ( real > real > $o ) > real ).
thf(func_def_97,type,
sK13: set_real > set_real > real ).
thf(func_def_98,type,
sK14: real > real ).
thf(func_def_99,type,
sK15: set_a > set_a > a ).
thf(func_def_100,type,
sK16: a > real ).
thf(func_def_101,type,
sK17: real > real > a ).
thf(func_def_102,type,
sK18: ( real > $o ) > ( real > $o ) > real ).
thf(func_def_103,type,
sK19: ( a > $o ) > ( a > $o ) > a ).
thf(func_def_104,type,
sK20: a > a ).
thf(func_def_105,type,
sK21: a ).
thf(func_def_106,type,
sK22: real ).
thf(func_def_107,type,
sK23: a > real ).
thf(func_def_108,type,
sK24: set_a > a ).
thf(func_def_109,type,
sK25: ( a > $o ) > ( a > $o ) > a ).
thf(func_def_110,type,
sK26: set_real > set_real > real ).
thf(func_def_111,type,
sK27: a > a > a ).
thf(func_def_112,type,
sK28: real > real > a ).
thf(func_def_113,type,
sK29: real > real > a ).
thf(func_def_114,type,
sK30: ( real > $o ) > ( real > $o ) > real ).
thf(func_def_115,type,
sK31: a > real ).
thf(func_def_116,type,
sK32: a > real ).
thf(func_def_117,type,
sK33: set_a > a ).
thf(func_def_118,type,
sK34: real > a > a ).
thf(func_def_119,type,
sK35: set_a > set_a > a ).
thf(func_def_121,type,
ph37:
!>[X0: $tType] : X0 ).
thf(f2177,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1949,f2136,f2175]) ).
thf(f2175,plain,
( ~ spl36_1
| ~ spl36_2 ),
inference(avatar_split_clause,[status(thm)],[f2174,f1947,f1943]) ).
thf(f1943,definition,
( spl36_1
<=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl36_1])],[avatar_definition]) ).
thf(f1947,definition,
( spl36_2
<=> ! [X0: real] :
( ( $true
!= ( member_a @ ( sK17 @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ X0 ) @ ( elemen49976720ball_a @ x2 @ d ) ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl36_2])],[avatar_definition]) ).
thf(f2174,plain,
( ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ~ spl36_2 ),
inference(trivial_inequality_removal,[status(thm)],[f2173]) ).
thf(f2173,plain,
( ( $true != $true )
| ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ~ spl36_2 ),
inference(forward_demodulation,[status(thm)],[f2165,f1638]) ).
thf(f1638,plain,
( ( ord_less_real @ zero_zero_real @ d )
= $true ),
inference(cnf_transformation,[status(thm)],[f630]) ).
thf(f630,plain,
( ( ord_less_real @ zero_zero_real @ d )
= $true ),
inference(fool_elimination,[status(thm)],[f629]) ).
thf(f629,plain,
ord_less_real @ zero_zero_real @ d,
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
ord_less_real @ zero_zero_real @ d,
file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_0_d_I1_J) ).
thf(f2165,plain,
( ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ d )
!= $true )
| ~ spl36_2 ),
inference(trivial_inequality_removal,[status(thm)],[f2164]) ).
thf(f2164,plain,
( ( $true != $true )
| ( ( ord_less_real @ zero_zero_real @ d )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ~ spl36_2 ),
inference(duplicate_literal_removal,[status(thm)],[f2159]) ).
thf(f2159,plain,
( ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ d )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ d )
!= $true )
| ( $true != $true )
| ~ spl36_2 ),
inference(superposition,[status(thm)],[f1948,f1508]) ).
thf(f1508,plain,
! [X0: real,X1: real] :
( ( ( member_a @ ( sK17 @ X1 @ X0 ) @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true )
| ( ( ord_less_real @ zero_zero_real @ X1 )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1248]) ).
thf(f1248,plain,
! [X0: real] :
( ! [X1: real] :
( ( ( $true
!= ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ ( sK17 @ X1 @ X0 ) ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) )
& ( ( member_a @ ( sK17 @ X1 @ X0 ) @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true ) )
| ( ( ord_less_real @ zero_zero_real @ X1 )
!= $true ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK17])],[f1005,f1247]) ).
thf(f1247,plain,
! [X0: real,X1: real] :
( ? [X2: a] :
( ( ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
& ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true ) )
=> ( ( $true
!= ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ ( sK17 @ X1 @ X0 ) ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) )
& ( ( member_a @ ( sK17 @ X1 @ X0 ) @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1005,plain,
! [X0: real] :
( ! [X1: real] :
( ? [X2: a] :
( ( ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
& ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true ) )
| ( ( ord_less_real @ zero_zero_real @ X1 )
!= $true ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f763]) ).
thf(f763,plain,
~ ? [X0: real] :
( ? [X1: real] :
( ! [X2: a] :
( ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ X0 ) )
= $true )
=> ( ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) )
& ( ( ord_less_real @ zero_zero_real @ X1 )
= $true ) )
& ( ( ord_less_real @ zero_zero_real @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f762]) ).
thf(f762,plain,
~ ? [X0: real] :
( ? [X1: real] :
( ( ord_less_real @ zero_zero_real @ X1 )
& ! [X2: a] :
( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ X0 ) )
=> ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
& ( ord_less_real @ zero_zero_real @ X0 ) ),
inference(rectify,[status(thm)],[f354]) ).
thf(f354,negated_conjecture,
~ ? [X35: real] :
( ? [X56: real] :
( ( ord_less_real @ zero_zero_real @ X56 )
& ! [X26: a] :
( ( member_a @ X26 @ ( elemen49976720ball_a @ x2 @ X35 ) )
=> ( ord_less_eq_real @ X56 @ ( inner_1173012732nner_a @ ( f @ X26 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
& ( ord_less_real @ zero_zero_real @ X35 ) ),
inference(negated_conjecture,[status(cth)],[f353]) ).
thf(f353,conjecture,
? [X35: real] :
( ? [X56: real] :
( ( ord_less_real @ zero_zero_real @ X56 )
& ! [X26: a] :
( ( member_a @ X26 @ ( elemen49976720ball_a @ x2 @ X35 ) )
=> ( ord_less_eq_real @ X56 @ ( inner_1173012732nner_a @ ( f @ X26 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
& ( ord_less_real @ zero_zero_real @ X35 ) ),
file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',conj_0) ).
thf(f1948,plain,
( ! [X0: real] :
( ( $true
!= ( member_a @ ( sK17 @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ X0 ) @ ( elemen49976720ball_a @ x2 @ d ) ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) )
| ~ spl36_2 ),
inference(avatar_component_clause,[status(thm)],[f1947]) ).
thf(f2136,plain,
spl36_1,
inference(avatar_contradiction_clause,[status(thm)],[f2135]) ).
thf(f2135,plain,
( $false
| spl36_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2128]) ).
thf(f2128,plain,
( ( $true != $true )
| spl36_1 ),
inference(superposition,[status(thm)],[f1992,f1569]) ).
thf(f1569,plain,
( ( member_a @ sK21 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1271]) ).
thf(f1271,plain,
( ( ( member_a @ sK21 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
& ! [X1: a] :
( ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK21])],[f1193,f1270]) ).
thf(f1270,plain,
( ? [X0: a] :
( ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
& ! [X1: a] :
( ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ) )
=> ( ( ( member_a @ sK21 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
& ! [X1: a] :
( ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1193,plain,
? [X0: a] :
( ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
& ! [X1: a] :
( ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f735]) ).
thf(f735,plain,
~ ! [X0: a] :
( ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
=> ~ ! [X1: a] :
( ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
=> ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f734]) ).
thf(f734,plain,
~ ! [X0: a] :
( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ~ ! [X1: a] :
( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ),
inference(rectify,[status(thm)],[f12]) ).
thf(f12,axiom,
~ ! [X3: a] :
( ( member_a @ X3 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ X3 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062) ).
thf(f1992,plain,
( ( ( member_a @ sK21 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| spl36_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1980]) ).
thf(f1980,plain,
( ( $true != $true )
| ( ( member_a @ sK21 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true )
| spl36_1 ),
inference(superposition,[status(thm)],[f1945,f1670]) ).
thf(f1670,plain,
! [X0: a] :
( ( $true
= ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) )
| ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1177]) ).
thf(f1177,plain,
! [X0: a] :
( ( ( zero_zero_a
!= ( f @ X0 ) )
& ( ( member_a @ X0 @ x )
= $true )
& ( $true
= ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
| ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f530]) ).
thf(f530,plain,
! [X0: a] :
( ( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
= $true )
=> ( ( zero_zero_a
!= ( f @ X0 ) )
& ( ( member_a @ X0 @ x )
= $true )
& ( $true
= ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f529]) ).
thf(f529,plain,
! [X0: a] :
( ( member_a @ X0 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ( member_a @ X0 @ x )
& ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X0 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
& ( zero_zero_a
!= ( f @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,axiom,
! [X1: a] :
( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ( member_a @ X1 @ x )
& ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
& ( zero_zero_a
!= ( f @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_8_d_I2_J) ).
thf(f1945,plain,
( ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| spl36_1 ),
inference(avatar_component_clause,[status(thm)],[f1943]) ).
thf(f1949,plain,
( ~ spl36_1
| spl36_2 ),
inference(avatar_split_clause,[status(thm)],[f1941,f1947,f1943]) ).
thf(f1941,plain,
! [X0: real] :
( ( $true
!= ( member_a @ ( sK17 @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ X0 ) @ ( elemen49976720ball_a @ x2 @ d ) ) )
| ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1929]) ).
thf(f1929,plain,
! [X0: real] :
( ( $true
!= ( member_a @ ( sK17 @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ X0 ) @ ( elemen49976720ball_a @ x2 @ d ) ) )
| ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
!= $true )
| ( $true != $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f1509,f1568]) ).
thf(f1568,plain,
! [X1: a] :
( ( ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ sK21 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X1 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
= $true )
| ( ( member_a @ X1 @ ( elemen49976720ball_a @ x2 @ d ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1271]) ).
thf(f1509,plain,
! [X0: real,X1: real] :
( ( $true
!= ( ord_less_eq_real @ X1 @ ( inner_1173012732nner_a @ ( f @ ( sK17 @ X1 @ X0 ) ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) )
| ( ( ord_less_real @ zero_zero_real @ X1 )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1248]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP149^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n016.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:28:21 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP149^1 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer Poincare_Bendixson problem prob_755__19581512_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : Poincare_Bendixson/prob_755__19581512_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v9.1.0, 0.50 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 424 ( 195 unt; 71 typ; 0 def)
% 0.28/0.46 % Number of atoms : 935 ( 305 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 9 ( 2 avg)
% 0.28/0.46 % Number of connectives : 2844 ( 77 ~; 18 |; 53 &;2418 @)
% 0.28/0.46 % ( 0 <=>; 278 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 21 ( 5 avg)
% 0.28/0.46 % Number of types : 8 ( 7 usr)
% 0.28/0.46 % Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 65 ( 64 usr; 12 con; 0-5 aty)
% 0.28/0.46 % Number of variables : 756 ( 58 ^; 688 !; 10 ?; 756 :)
% 0.28/0.46 % SPC : TH0_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:44:24.334
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (7)
% 0.28/0.46 thf(ty_n_t__Bounded____Linear____Function__Oblinfun_Itf__a_Mt__Real__Oreal_J,type,
% 0.28/0.46 bounde7994401a_real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.28/0.46 set_set_real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.28/0.46 set_set_a: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 set_real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 set_a: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.28/0.46 real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_n_tf__a,type,
% 0.28/0.46 a: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (64)
% 0.28/0.46 thf(sy_c_Elementary__Metric__Spaces_Ocball_001t__Real__Oreal,type,
% 0.28/0.46 elemen1140313242l_real: real > real > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Elementary__Metric__Spaces_Ocball_001tf__a,type,
% 0.28/0.46 elemen49976720ball_a: a > real > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
% 0.28/0.46 auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.28/0.46 minus_minus_real: real > real > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 minus_minus_set_real: set_real > set_real > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 minus_minus_set_a: set_a > set_a > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
% 0.28/0.46 minus_minus_a: a > a > a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
% 0.28/0.46 uminus_uminus_a_a: ( a > a ) > a > a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.28/0.46 uminus_uminus_real: real > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 uminus773214379t_real: set_real > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 uminus_uminus_set_a: set_a > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
% 0.28/0.46 uminus_uminus_a: a > a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.28/0.46 zero_zero_real: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 0.28/0.46 zero_zero_a: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.28/0.46 initia826609931terval: set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001t__Real__Oreal,type,
% 0.28/0.46 inner_4346926r_real: real > real > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001tf__a,type,
% 0.28/0.46 inner_1173012732nner_a: a > a > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant_001tf__a,type,
% 0.28/0.46 auto_l630715367iant_a: ( a > a ) > set_a > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Line__Segment_Oclosed__segment_001tf__a,type,
% 0.28/0.46 line_c1152468841ment_a: a > a > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
% 0.28/0.46 line_open_segment_a: a > a > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped_001tf__a,type,
% 0.28/0.46 oDE_au1996717075pped_a: ( a > a ) > set_a > a > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.28/0.46 ord_less_real: real > real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 ord_less_set_real: set_real > set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.28/0.46 ord_le861030508t_real: set_set_real > set_set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.28/0.46 ord_less_set_set_a: set_set_a > set_set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 ord_less_set_a: set_a > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 0.28/0.46 ord_less_a: a > a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.28/0.46 ord_less_eq_real: real > real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 ord_less_eq_set_real: set_real > set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.28/0.46 ord_le1733505144t_real: set_set_real > set_set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.28/0.46 ord_le318720350_set_a: set_set_a > set_set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.28/0.46 ord_less_eq_a: a > a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 top_top_set_real: set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 top_top_set_a: set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
% 0.28/0.46 period720806154rbit_a: ( a > a ) > set_a > a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
% 0.28/0.46 period1305449585riod_a: ( a > a ) > set_a > a > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
% 0.28/0.46 period138238489rbit_a: ( a > a ) > set_a > a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Orot_001tf__a,type,
% 0.28/0.46 poinca659159244_rot_a: a > a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
% 0.28/0.46 poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection_001tf__a,type,
% 0.28/0.46 reacha1191408304tion_a: ( a > a ) > set_a > ( a > real ) > ( a > bounde7994401a_real ) > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.28/0.46 real_V1934908667t_real: real > real > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001tf__a,type,
% 0.28/0.46 real_V1514887919dist_a: a > a > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.28/0.46 collect_real: ( real > $o ) > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 collect_set_real: ( set_real > $o ) > set_set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 collect_set_a: ( set_a > $o ) > set_set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.28/0.46 collect_a: ( a > $o ) > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 0.28/0.46 set_or656347191t_real: real > real > set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 set_or1619725421t_real: set_real > set_real > set_set_real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 set_or2094724627_set_a: set_a > set_a > set_set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
% 0.28/0.46 set_or411607219Most_a: a > a > set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member_001t__Real__Oreal,type,
% 0.28/0.46 member_real: real > set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.28/0.46 member_set_real: set_real > set_set_real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.28/0.46 member_set_a: set_a > set_set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member_001tf__a,type,
% 0.28/0.46 member_a: a > set_a > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_X,type,
% 0.28/0.46 x: set_a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_a,type,
% 0.28/0.46 a2: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_b,type,
% 0.28/0.46 b: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_d____,type,
% 0.28/0.46 d: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_f,type,
% 0.28/0.46 f: a > a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_s____,type,
% 0.28/0.46 s: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_thesis,type,
% 0.28/0.46 thesis: $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_x____,type,
% 0.28/0.46 x2: a ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_z,type,
% 0.28/0.46 z: a ).
% 0.28/0.46
% 0.28/0.46 % Relevant facts (352)
% 0.28/0.46 thf(fact_0_d_I1_J,axiom,
% 0.28/0.46 ord_less_real @ zero_zero_real @ d ).
% 0.28/0.46
% 0.28/0.46 % d(1)
% 0.28/0.46 thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062,axiom,
% 0.28/0.46 member_a @ s @ ( elemen49976720ball_a @ x2 @ d ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>s \<in> cball x d\<close>
% 0.28/0.46 thf(fact_2_seg_I1_J,axiom,
% 0.28/0.46 a2 != b ).
% 0.28/0.46
% 0.28/0.46 % seg(1)
% 0.28/0.46 thf(fact_3_nrm__dot,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % nrm_dot
% 0.28/0.46 thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062,axiom,
% 0.28/0.46 member_a @ x2 @ x ).
% 0.28/0.46
% 0.28/0.46 % \<open>x \<in> X\<close>
% 0.28/0.46 thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.28/0.46 ! [X2: a] :
% 0.28/0.46 ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ s ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<close>
% 0.28/0.46 thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.28/0.46 ( ( f @ x2 )
% 0.28/0.46 != zero_zero_a ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>f x \<noteq> (0::'a)\<close>
% 0.28/0.46 thf(fact_7_direction_I2_J,axiom,
% 0.28/0.46 ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % direction(2)
% 0.28/0.46 thf(fact_8_d_I2_J,axiom,
% 0.28/0.46 ! [Y: a] :
% 0.28/0.46 ( ( member_a @ Y @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
% 0.28/0.46 & ( member_a @ Y @ x )
% 0.28/0.46 & ( ( f @ Y )
% 0.28/0.46 != zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % d(2)
% 0.28/0.46 thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.28/0.46 ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ x2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>0 < f x \<bullet> rot (a - b)\<close>
% 0.28/0.46 thf(fact_10_dot__ortho,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ X @ ( poinca659159244_rot_a @ X ) )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % dot_ortho
% 0.28/0.46 thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.28/0.46 ~ ! [S: a] :
% 0.28/0.46 ( ( member_a @ S @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.28/0.46 => ~ ! [X2: a] :
% 0.28/0.46 ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ S ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>thesis. (\<And>s. \<lbrakk>s \<in> cball x d; \<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.28/0.46 thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062,axiom,
% 0.28/0.46 member_a @ x2 @ ( line_c1152468841ment_a @ a2 @ b ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>x \<in> {a--b}\<close>
% 0.28/0.46 thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.28/0.46 ~ ! [D: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ D )
% 0.28/0.46 => ~ ! [Y2: a] :
% 0.28/0.46 ( ( member_a @ Y2 @ ( elemen49976720ball_a @ x2 @ D ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
% 0.28/0.46 & ( member_a @ Y2 @ x )
% 0.28/0.46 & ( ( f @ Y2 )
% 0.28/0.46 != zero_zero_a ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>thesis. (\<And>d. \<lbrakk>0 < d; \<And>y. y \<in> cball x d \<Longrightarrow> 0 < f y \<bullet> rot (a - b) \<and> y \<in> X \<and> f y \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.28/0.46 thf(fact_14_transversal,axiom,
% 0.28/0.46 poinca522724647ment_a @ f @ x @ a2 @ b ).
% 0.28/0.46
% 0.28/0.46 % transversal
% 0.28/0.46 thf(fact_15_inner__gt__zero__iff,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ ( inner_4346926r_real @ X @ X ) )
% 0.28/0.46 = ( X != zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_gt_zero_iff
% 0.28/0.46 thf(fact_16_inner__gt__zero__iff,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) )
% 0.28/0.46 = ( X != zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_gt_zero_iff
% 0.28/0.46 thf(fact_17_centre__in__cball,axiom,
% 0.28/0.46 ! [X: real,E: real] :
% 0.28/0.46 ( ( member_real @ X @ ( elemen1140313242l_real @ X @ E ) )
% 0.28/0.46 = ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % centre_in_cball
% 0.28/0.46 thf(fact_18_centre__in__cball,axiom,
% 0.28/0.46 ! [X: a,E: real] :
% 0.28/0.46 ( ( member_a @ X @ ( elemen49976720ball_a @ X @ E ) )
% 0.28/0.46 = ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % centre_in_cball
% 0.28/0.46 thf(fact_19_all__zero__iff,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( ! [U: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ X @ U )
% 0.28/0.46 = zero_zero_real ) )
% 0.28/0.46 = ( X = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % all_zero_iff
% 0.28/0.46 thf(fact_20_all__zero__iff,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( ! [U: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ X @ U )
% 0.28/0.46 = zero_zero_real ) )
% 0.28/0.46 = ( X = zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % all_zero_iff
% 0.28/0.46 thf(fact_21_inner__zero__left,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ zero_zero_real @ X )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % inner_zero_left
% 0.28/0.46 thf(fact_22_inner__zero__left,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ zero_zero_a @ X )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % inner_zero_left
% 0.28/0.46 thf(fact_23_inner__zero__right,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ X @ zero_zero_real )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % inner_zero_right
% 0.28/0.46 thf(fact_24_inner__zero__right,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ X @ zero_zero_a )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % inner_zero_right
% 0.28/0.46 thf(fact_25_inner__eq__zero__iff,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( ( inner_4346926r_real @ X @ X )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 = ( X = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_eq_zero_iff
% 0.28/0.46 thf(fact_26_inner__eq__zero__iff,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( ( inner_1173012732nner_a @ X @ X )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 = ( X = zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_eq_zero_iff
% 0.28/0.46 thf(fact_27_diff__gt__0__iff__gt,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
% 0.28/0.46 = ( ord_less_a @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_gt_0_iff_gt
% 0.28/0.46 thf(fact_28_diff__gt__0__iff__gt,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.28/0.46 = ( ord_less_real @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_gt_0_iff_gt
% 0.28/0.46 thf(fact_29_diff__ge__0__iff__ge,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
% 0.28/0.46 = ( ord_less_eq_a @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_ge_0_iff_ge
% 0.28/0.46 thf(fact_30_diff__ge__0__iff__ge,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.28/0.46 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_ge_0_iff_ge
% 0.28/0.46 thf(fact_31_rot__0,axiom,
% 0.28/0.46 ( ( poinca659159244_rot_a @ zero_zero_a )
% 0.28/0.46 = zero_zero_a ) ).
% 0.28/0.46
% 0.28/0.46 % rot_0
% 0.28/0.46 thf(fact_32_rot__eq__0__iff,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( ( poinca659159244_rot_a @ X )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 = ( X = zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % rot_eq_0_iff
% 0.28/0.46 thf(fact_33_seg_I3_J,axiom,
% 0.28/0.46 ! [Z: a] :
% 0.28/0.46 ( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % seg(3)
% 0.28/0.46 thf(fact_34_direction_I1_J,axiom,
% 0.28/0.46 member_a @ z @ ( line_c1152468841ment_a @ a2 @ b ) ).
% 0.28/0.46
% 0.28/0.46 % direction(1)
% 0.28/0.46 thf(fact_35_transversal__segment__reverse,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_reverse
% 0.28/0.46 thf(fact_36_transversal__segment__commute,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 = ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_commute
% 0.28/0.46 thf(fact_37_in__segment__inner__rot2,axiom,
% 0.28/0.46 ! [X: a,A: a,B: a,Y: a] :
% 0.28/0.46 ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.28/0.46 => ( ( member_a @ Y @ ( line_c1152468841ment_a @ A @ B ) )
% 0.28/0.46 => ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
% 0.28/0.46 = zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_segment_inner_rot2
% 0.28/0.46 thf(fact_38_in__segment__inner__rot,axiom,
% 0.28/0.46 ! [X: a,A: a,B: a] :
% 0.28/0.46 ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.28/0.46 => ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
% 0.28/0.46 = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_segment_inner_rot
% 0.28/0.46 thf(fact_39_seg_I2_J,axiom,
% 0.28/0.46 ord_less_eq_set_a @ ( line_c1152468841ment_a @ a2 @ b ) @ x ).
% 0.28/0.46
% 0.28/0.46 % seg(2)
% 0.28/0.46 thf(fact_40_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( minus_minus_a @ A @ A )
% 0.28/0.46 = zero_zero_a ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.28/0.46 thf(fact_41_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( minus_minus_real @ A @ A )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.28/0.46 thf(fact_42_diff__zero,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( minus_minus_a @ A @ zero_zero_a )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % diff_zero
% 0.28/0.46 thf(fact_43_diff__zero,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % diff_zero
% 0.28/0.46 thf(fact_44_diff__0__right,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( minus_minus_a @ A @ zero_zero_a )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0_right
% 0.28/0.46 thf(fact_45_diff__0__right,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0_right
% 0.28/0.46 thf(fact_46_diff__self,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( minus_minus_a @ A @ A )
% 0.28/0.46 = zero_zero_a ) ).
% 0.28/0.46
% 0.28/0.46 % diff_self
% 0.28/0.46 thf(fact_47_diff__self,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( minus_minus_real @ A @ A )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % diff_self
% 0.28/0.46 thf(fact_48_subset__cball,axiom,
% 0.28/0.46 ! [D2: real,E: real,X: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ D2 @ E )
% 0.28/0.46 => ( ord_less_eq_set_real @ ( elemen1140313242l_real @ X @ D2 ) @ ( elemen1140313242l_real @ X @ E ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_cball
% 0.28/0.46 thf(fact_49_subset__cball,axiom,
% 0.28/0.46 ! [D2: real,E: real,X: a] :
% 0.28/0.46 ( ( ord_less_eq_real @ D2 @ E )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( elemen49976720ball_a @ X @ D2 ) @ ( elemen49976720ball_a @ X @ E ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_cball
% 0.28/0.46 thf(fact_50_transversal__segment__def,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ A @ B )
% 0.28/0.46 = ( ( A != B )
% 0.28/0.46 & ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ A @ B ) @ x )
% 0.28/0.46 & ! [X3: a] :
% 0.28/0.46 ( ( member_a @ X3 @ ( line_c1152468841ment_a @ A @ B ) )
% 0.28/0.46 => ( ( inner_1173012732nner_a @ ( f @ X3 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
% 0.28/0.46 != zero_zero_real ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_def
% 0.28/0.46 thf(fact_51_transversal__segmentE,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( ( X != Y )
% 0.28/0.46 => ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
% 0.28/0.46 => ~ ! [Z2: a] :
% 0.28/0.46 ( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) )
% 0.28/0.46 => ~ ( ( X != Y )
% 0.28/0.46 => ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
% 0.28/0.46 => ~ ! [Z2: a] :
% 0.28/0.46 ( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segmentE
% 0.28/0.46 thf(fact_52_transversal__segment__posD_I2_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_posD(2)
% 0.28/0.46 thf(fact_53_transversal__segment__negD_I2_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_negD(2)
% 0.28/0.46 thf(fact_54_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: set_real,P: set_real > $o] :
% 0.28/0.46 ( ( member_set_real @ A @ ( collect_set_real @ P ) )
% 0.28/0.46 = ( P @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_55_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: set_a,P: set_a > $o] :
% 0.28/0.46 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 0.28/0.46 = ( P @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_56_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: a,P: a > $o] :
% 0.28/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.28/0.46 = ( P @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_57_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: real,P: real > $o] :
% 0.28/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.28/0.46 = ( P @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_58_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A2: set_set_real] :
% 0.28/0.46 ( ( collect_set_real
% 0.28/0.46 @ ^ [X3: set_real] : ( member_set_real @ X3 @ A2 ) )
% 0.28/0.46 = A2 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_59_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A2: set_set_a] :
% 0.28/0.46 ( ( collect_set_a
% 0.28/0.46 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 0.28/0.46 = A2 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_60_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A2: set_a] :
% 0.28/0.46 ( ( collect_a
% 0.28/0.46 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.28/0.46 = A2 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_61_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A2: set_real] :
% 0.28/0.46 ( ( collect_real
% 0.28/0.46 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.28/0.46 = A2 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_62_transversal__segment__sign__less,axiom,
% 0.28/0.46 ! [W: a,Y: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ W @ Y )
% 0.28/0.46 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real )
% 0.28/0.46 => ! [X2: a] :
% 0.28/0.46 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ W @ Y ) )
% 0.28/0.46 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_sign_less
% 0.28/0.46 thf(fact_63_transversal__segment__pos,axiom,
% 0.28/0.46 ! [X: a,Y: a,W: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.28/0.46 => ! [X2: a] :
% 0.28/0.46 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_pos
% 0.28/0.46 thf(fact_64_transversal__segment__neg,axiom,
% 0.28/0.46 ! [X: a,Y: a,W: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.28/0.46 => ! [X2: a] :
% 0.28/0.46 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_neg
% 0.28/0.46 thf(fact_65_transversal__segment__posD_I1_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.28/0.46 => ( X != Y ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_posD(1)
% 0.28/0.46 thf(fact_66_transversal__segment__posD_I3_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Za: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_posD(3)
% 0.28/0.46 thf(fact_67_transversal__segment__negD_I1_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.28/0.46 => ( X != Y ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_negD(1)
% 0.28/0.46 thf(fact_68_transversal__segment__negD_I3_J,axiom,
% 0.28/0.46 ! [X: a,Y: a,Za: a,Z: a] :
% 0.28/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.28/0.46 => ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.28/0.46 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.28/0.46 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_negD(3)
% 0.28/0.46 thf(fact_69__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.28/0.46 ! [Z: a] :
% 0.28/0.46 ( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % \<open>\<And>z. z \<in> {a--b} \<Longrightarrow> 0 < f z \<bullet> rot (a - b)\<close>
% 0.28/0.46 thf(fact_70_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.28/0.46 poinca522724647ment_a = poinca522724647ment_a ).
% 0.28/0.46
% 0.28/0.46 % c1_on_open_R2.transversal_segment.cong
% 0.28/0.46 thf(fact_71_zero__reorient,axiom,
% 0.28/0.46 ! [X: real] :
% 0.28/0.46 ( ( zero_zero_real = X )
% 0.28/0.46 = ( X = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_reorient
% 0.28/0.46 thf(fact_72_zero__reorient,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( zero_zero_a = X )
% 0.28/0.46 = ( X = zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_reorient
% 0.28/0.46 thf(fact_73_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.28/0.46 ! [A: a,C: a,B: a] :
% 0.28/0.46 ( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
% 0.28/0.46 = ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.28/0.46 thf(fact_74_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.28/0.46 ! [A: real,C: real,B: real] :
% 0.28/0.46 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.28/0.46 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.28/0.46 thf(fact_75_diff__eq__diff__eq,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a,D2: a] :
% 0.28/0.46 ( ( ( minus_minus_a @ A @ B )
% 0.28/0.46 = ( minus_minus_a @ C @ D2 ) )
% 0.28/0.46 => ( ( A = B )
% 0.28/0.46 = ( C = D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_eq
% 0.28/0.46 thf(fact_76_diff__eq__diff__eq,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real,D2: real] :
% 0.28/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.28/0.46 = ( minus_minus_real @ C @ D2 ) )
% 0.28/0.46 => ( ( A = B )
% 0.28/0.46 = ( C = D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_eq
% 0.28/0.46 thf(fact_77_inner__commute,axiom,
% 0.28/0.46 ( inner_1173012732nner_a
% 0.28/0.46 = ( ^ [X3: a,Y3: a] : ( inner_1173012732nner_a @ Y3 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_commute
% 0.28/0.46 thf(fact_78_eq__iff__diff__eq__0,axiom,
% 0.28/0.46 ( ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 ) )
% 0.28/0.46 = ( ^ [A3: a,B2: a] :
% 0.28/0.46 ( ( minus_minus_a @ A3 @ B2 )
% 0.28/0.46 = zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_iff_diff_eq_0
% 0.28/0.46 thf(fact_79_eq__iff__diff__eq__0,axiom,
% 0.28/0.46 ( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
% 0.28/0.46 = ( ^ [A3: real,B2: real] :
% 0.28/0.46 ( ( minus_minus_real @ A3 @ B2 )
% 0.28/0.46 = zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_iff_diff_eq_0
% 0.28/0.46 thf(fact_80_diff__eq__diff__less__eq,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a,D2: a] :
% 0.28/0.46 ( ( ( minus_minus_a @ A @ B )
% 0.28/0.46 = ( minus_minus_a @ C @ D2 ) )
% 0.28/0.46 => ( ( ord_less_eq_a @ A @ B )
% 0.28/0.46 = ( ord_less_eq_a @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_less_eq
% 0.28/0.46 thf(fact_81_diff__eq__diff__less__eq,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real,D2: real] :
% 0.28/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.28/0.46 = ( minus_minus_real @ C @ D2 ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.28/0.46 = ( ord_less_eq_real @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_less_eq
% 0.28/0.46 thf(fact_82_diff__right__mono,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.28/0.46 => ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_right_mono
% 0.28/0.46 thf(fact_83_diff__right__mono,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.28/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_right_mono
% 0.28/0.46 thf(fact_84_diff__left__mono,axiom,
% 0.28/0.46 ! [B: a,A: a,C: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ B @ A )
% 0.28/0.46 => ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_left_mono
% 0.28/0.46 thf(fact_85_diff__left__mono,axiom,
% 0.28/0.46 ! [B: real,A: real,C: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.28/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_left_mono
% 0.28/0.46 thf(fact_86_diff__mono,axiom,
% 0.28/0.46 ! [A: a,B: a,D2: a,C: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.28/0.46 => ( ( ord_less_eq_a @ D2 @ C )
% 0.28/0.46 => ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_mono
% 0.28/0.46 thf(fact_87_diff__mono,axiom,
% 0.28/0.46 ! [A: real,B: real,D2: real,C: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.28/0.46 => ( ( ord_less_eq_real @ D2 @ C )
% 0.28/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_mono
% 0.28/0.46 thf(fact_88_diff__strict__right__mono,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a] :
% 0.28/0.46 ( ( ord_less_a @ A @ B )
% 0.28/0.46 => ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_right_mono
% 0.28/0.46 thf(fact_89_diff__strict__right__mono,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real] :
% 0.28/0.46 ( ( ord_less_real @ A @ B )
% 0.28/0.46 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_right_mono
% 0.28/0.46 thf(fact_90_diff__strict__left__mono,axiom,
% 0.28/0.46 ! [B: a,A: a,C: a] :
% 0.28/0.46 ( ( ord_less_a @ B @ A )
% 0.28/0.46 => ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_left_mono
% 0.28/0.46 thf(fact_91_diff__strict__left__mono,axiom,
% 0.28/0.46 ! [B: real,A: real,C: real] :
% 0.28/0.46 ( ( ord_less_real @ B @ A )
% 0.28/0.46 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_left_mono
% 0.28/0.46 thf(fact_92_diff__eq__diff__less,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a,D2: a] :
% 0.28/0.46 ( ( ( minus_minus_a @ A @ B )
% 0.28/0.46 = ( minus_minus_a @ C @ D2 ) )
% 0.28/0.46 => ( ( ord_less_a @ A @ B )
% 0.28/0.46 = ( ord_less_a @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_less
% 0.28/0.46 thf(fact_93_diff__eq__diff__less,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real,D2: real] :
% 0.28/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.28/0.46 = ( minus_minus_real @ C @ D2 ) )
% 0.28/0.46 => ( ( ord_less_real @ A @ B )
% 0.28/0.46 = ( ord_less_real @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_eq_diff_less
% 0.28/0.46 thf(fact_94_diff__strict__mono,axiom,
% 0.28/0.46 ! [A: a,B: a,D2: a,C: a] :
% 0.28/0.46 ( ( ord_less_a @ A @ B )
% 0.28/0.46 => ( ( ord_less_a @ D2 @ C )
% 0.28/0.46 => ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_mono
% 0.28/0.46 thf(fact_95_diff__strict__mono,axiom,
% 0.28/0.46 ! [A: real,B: real,D2: real,C: real] :
% 0.28/0.46 ( ( ord_less_real @ A @ B )
% 0.28/0.46 => ( ( ord_less_real @ D2 @ C )
% 0.28/0.46 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_strict_mono
% 0.28/0.46 thf(fact_96_inner__diff__right,axiom,
% 0.28/0.46 ! [X: real,Y: real,Z: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ X @ ( minus_minus_real @ Y @ Z ) )
% 0.28/0.46 = ( minus_minus_real @ ( inner_4346926r_real @ X @ Y ) @ ( inner_4346926r_real @ X @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_diff_right
% 0.28/0.46 thf(fact_97_inner__diff__right,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ X @ ( minus_minus_a @ Y @ Z ) )
% 0.28/0.46 = ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Y ) @ ( inner_1173012732nner_a @ X @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_diff_right
% 0.28/0.46 thf(fact_98_inner__diff__left,axiom,
% 0.28/0.46 ! [X: real,Y: real,Z: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ ( minus_minus_real @ X @ Y ) @ Z )
% 0.28/0.46 = ( minus_minus_real @ ( inner_4346926r_real @ X @ Z ) @ ( inner_4346926r_real @ Y @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_diff_left
% 0.28/0.46 thf(fact_99_inner__diff__left,axiom,
% 0.28/0.46 ! [X: a,Y: a,Z: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ Z )
% 0.28/0.46 = ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Z ) @ ( inner_1173012732nner_a @ Y @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_diff_left
% 0.28/0.46 thf(fact_100_mem__cball__leI,axiom,
% 0.28/0.46 ! [X: real,Y: real,E: real,F: real] :
% 0.28/0.46 ( ( member_real @ X @ ( elemen1140313242l_real @ Y @ E ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ E @ F )
% 0.28/0.46 => ( member_real @ X @ ( elemen1140313242l_real @ Y @ F ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_cball_leI
% 0.28/0.46 thf(fact_101_mem__cball__leI,axiom,
% 0.28/0.46 ! [X: a,Y: a,E: real,F: real] :
% 0.28/0.46 ( ( member_a @ X @ ( elemen49976720ball_a @ Y @ E ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ E @ F )
% 0.28/0.46 => ( member_a @ X @ ( elemen49976720ball_a @ Y @ F ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_cball_leI
% 0.28/0.46 thf(fact_102_le__iff__diff__le__0,axiom,
% 0.28/0.46 ( ord_less_eq_a
% 0.28/0.46 = ( ^ [A3: a,B2: a] : ( ord_less_eq_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_iff_diff_le_0
% 0.28/0.46 thf(fact_103_le__iff__diff__le__0,axiom,
% 0.28/0.46 ( ord_less_eq_real
% 0.28/0.46 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_iff_diff_le_0
% 0.28/0.46 thf(fact_104_less__iff__diff__less__0,axiom,
% 0.28/0.46 ( ord_less_a
% 0.28/0.46 = ( ^ [A3: a,B2: a] : ( ord_less_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_iff_diff_less_0
% 0.28/0.46 thf(fact_105_less__iff__diff__less__0,axiom,
% 0.28/0.46 ( ord_less_real
% 0.28/0.46 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_iff_diff_less_0
% 0.28/0.46 thf(fact_106_inner__ge__zero,axiom,
% 0.28/0.46 ! [X: a] : ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_ge_zero
% 0.28/0.46 thf(fact_107_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( member_a @ X @ x )
% 0.28/0.46 => ( ( ( f @ X )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 => ( ( period1305449585riod_a @ f @ x @ X )
% 0.28/0.46 = zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % fixed_point_imp_closed_orbit_period_zero(2)
% 0.28/0.46 thf(fact_108_that,axiom,
% 0.28/0.46 ! [D2: real,B3: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.28/0.46 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.28/0.46 => ( ! [X4: a,Y5: a] :
% 0.28/0.46 ( ( member_a @ X4 @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X4 @ Y5 ) @ D2 )
% 0.28/0.46 => ( ord_less_eq_real @ B3 @ ( inner_1173012732nner_a @ ( f @ Y5 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
% 0.28/0.46 => thesis ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % that
% 0.28/0.46 thf(fact_109_inner__nonneg__nonneg,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.28/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.28/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( inner_4346926r_real @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_nonneg_nonneg
% 0.28/0.46 thf(fact_110_inner__nonneg__nonneg,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.28/0.46 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.28/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_nonneg_nonneg
% 0.28/0.46 thf(fact_111_in__closed__segment__iff__rot,axiom,
% 0.28/0.46 ! [A: a,B: a,X: a] :
% 0.28/0.46 ( ( A != B )
% 0.28/0.46 => ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.28/0.46 = ( ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 & ( member_real @ ( inner_1173012732nner_a @ X @ ( minus_minus_a @ B @ A ) ) @ ( set_or656347191t_real @ ( inner_1173012732nner_a @ A @ ( minus_minus_a @ B @ A ) ) @ ( inner_1173012732nner_a @ B @ ( minus_minus_a @ B @ A ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_closed_segment_iff_rot
% 0.28/0.46 thf(fact_112_sectionD_I5_J,axiom,
% 0.28/0.46 ! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
% 0.28/0.46 ( ( reacha1191408304tion_a @ f @ x @ S2 @ Ds @ S3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ S3 @ x ) ) ).
% 0.28/0.46
% 0.28/0.46 % sectionD(5)
% 0.28/0.46 thf(fact_113_transversal__segment__exists,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( member_a @ X @ x )
% 0.28/0.46 => ( ( ( f @ X )
% 0.28/0.46 != zero_zero_a )
% 0.28/0.46 => ~ ! [A4: a,B4: a] :
% 0.28/0.46 ( ( member_a @ X @ ( line_open_segment_a @ A4 @ B4 ) )
% 0.28/0.46 => ~ ( poinca522724647ment_a @ f @ x @ A4 @ B4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % transversal_segment_exists
% 0.28/0.46 thf(fact_114_invariant__iff__compl__invariant,axiom,
% 0.28/0.46 ! [M: set_a] :
% 0.28/0.46 ( ( auto_l630715367iant_a @ f @ x @ M )
% 0.28/0.46 = ( auto_l630715367iant_a @ f @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % invariant_iff_compl_invariant
% 0.28/0.46 thf(fact_115_cball__eq__cball__iff,axiom,
% 0.28/0.46 ! [X: a,D2: real,Y: a,E: real] :
% 0.28/0.46 ( ( ( elemen49976720ball_a @ X @ D2 )
% 0.28/0.46 = ( elemen49976720ball_a @ Y @ E ) )
% 0.28/0.46 = ( ( ( ord_less_real @ D2 @ zero_zero_real )
% 0.28/0.46 & ( ord_less_real @ E @ zero_zero_real ) )
% 0.28/0.46 | ( ( X = Y )
% 0.28/0.46 & ( D2 = E ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cball_eq_cball_iff
% 0.28/0.46 thf(fact_116_Bolzano,axiom,
% 0.28/0.46 ! [A: real,B: real,P: real > real > $o] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.28/0.46 => ( ! [A4: real,B4: real,C2: real] :
% 0.28/0.46 ( ( P @ A4 @ B4 )
% 0.28/0.46 => ( ( P @ B4 @ C2 )
% 0.28/0.46 => ( ( ord_less_eq_real @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq_real @ B4 @ C2 )
% 0.28/0.46 => ( P @ A4 @ C2 ) ) ) ) )
% 0.28/0.46 => ( ! [X4: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ X4 )
% 0.28/0.46 => ( ( ord_less_eq_real @ X4 @ B )
% 0.28/0.46 => ? [D3: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.28/0.46 & ! [A4: real,B4: real] :
% 0.28/0.46 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 0.28/0.46 & ( ord_less_eq_real @ X4 @ B4 )
% 0.28/0.46 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
% 0.28/0.46 => ( P @ A4 @ B4 ) ) ) ) )
% 0.28/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Bolzano
% 0.28/0.46 thf(fact_117_mem__cball,axiom,
% 0.28/0.46 ! [Y: real,X: real,E: real] :
% 0.28/0.46 ( ( member_real @ Y @ ( elemen1140313242l_real @ X @ E ) )
% 0.28/0.46 = ( ord_less_eq_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_cball
% 0.28/0.46 thf(fact_118_mem__cball,axiom,
% 0.28/0.46 ! [Y: a,X: a,E: real] :
% 0.28/0.46 ( ( member_a @ Y @ ( elemen49976720ball_a @ X @ E ) )
% 0.28/0.46 = ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_cball
% 0.28/0.46 thf(fact_119_perfect__choose__dist,axiom,
% 0.28/0.46 ! [R: real,X: a] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ R )
% 0.28/0.46 => ? [A4: a] :
% 0.28/0.46 ( ( A4 != X )
% 0.28/0.46 & ( ord_less_real @ ( real_V1514887919dist_a @ A4 @ X ) @ R ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % perfect_choose_dist
% 0.28/0.46 thf(fact_120_interval__inner__leI_I2_J,axiom,
% 0.28/0.46 ! [X: a,A: a,B: a,I: a] :
% 0.28/0.46 ( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
% 0.28/0.46 => ( ( ord_less_eq_a @ zero_zero_a @ I )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ X @ I ) @ ( inner_1173012732nner_a @ B @ I ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % interval_inner_leI(2)
% 0.28/0.46 thf(fact_121_interval__inner__leI_I2_J,axiom,
% 0.28/0.46 ! [X: real,A: real,B: real,I: real] :
% 0.28/0.46 ( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ I )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_4346926r_real @ X @ I ) @ ( inner_4346926r_real @ B @ I ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % interval_inner_leI(2)
% 0.28/0.46 thf(fact_122_interval__inner__leI_I1_J,axiom,
% 0.28/0.46 ! [X: a,A: a,B: a,I: a] :
% 0.28/0.46 ( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
% 0.28/0.46 => ( ( ord_less_eq_a @ zero_zero_a @ I )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ A @ I ) @ ( inner_1173012732nner_a @ X @ I ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % interval_inner_leI(1)
% 0.28/0.46 thf(fact_123_interval__inner__leI_I1_J,axiom,
% 0.28/0.46 ! [X: real,A: real,B: real,I: real] :
% 0.28/0.46 ( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
% 0.28/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ I )
% 0.28/0.46 => ( ord_less_eq_real @ ( inner_4346926r_real @ A @ I ) @ ( inner_4346926r_real @ X @ I ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % interval_inner_leI(1)
% 0.28/0.46 thf(fact_124_eucl__less__le__not__le,axiom,
% 0.28/0.46 ( ord_less_real
% 0.28/0.46 = ( ^ [X3: real,Y3: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.28/0.46 & ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eucl_less_le_not_le
% 0.28/0.46 thf(fact_125_invariant__def,axiom,
% 0.28/0.46 ! [M: set_a] :
% 0.28/0.46 ( ( auto_l630715367iant_a @ f @ x @ M )
% 0.28/0.46 = ( ! [X3: a] :
% 0.28/0.46 ( ( member_a @ X3 @ M )
% 0.28/0.46 => ( oDE_au1996717075pped_a @ f @ x @ X3 @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % invariant_def
% 0.28/0.46 thf(fact_126_periodic__orbit__period_I1_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % periodic_orbit_period(1)
% 0.28/0.46 thf(fact_127_zero__less__dist__iff,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) )
% 0.28/0.46 = ( X != Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_less_dist_iff
% 0.28/0.46 thf(fact_128_dist__le__zero__iff,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real )
% 0.28/0.46 = ( X = Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % dist_le_zero_iff
% 0.28/0.46 thf(fact_129_atLeastatMost__subset__iff,axiom,
% 0.28/0.46 ! [A: set_a,B: set_a,C: set_a,D2: set_a] :
% 0.28/0.46 ( ( ord_le318720350_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
% 0.28/0.46 = ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_set_a @ C @ A )
% 0.28/0.46 & ( ord_less_eq_set_a @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_subset_iff
% 0.28/0.46 thf(fact_130_atLeastatMost__subset__iff,axiom,
% 0.28/0.46 ! [A: set_real,B: set_real,C: set_real,D2: set_real] :
% 0.28/0.46 ( ( ord_le1733505144t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
% 0.28/0.46 = ( ~ ( ord_less_eq_set_real @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_set_real @ C @ A )
% 0.28/0.46 & ( ord_less_eq_set_real @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_subset_iff
% 0.28/0.46 thf(fact_131_atLeastatMost__subset__iff,axiom,
% 0.28/0.46 ! [A: a,B: a,C: a,D2: a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ ( set_or411607219Most_a @ A @ B ) @ ( set_or411607219Most_a @ C @ D2 ) )
% 0.28/0.46 = ( ~ ( ord_less_eq_a @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_a @ C @ A )
% 0.28/0.46 & ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_subset_iff
% 0.28/0.46 thf(fact_132_atLeastatMost__subset__iff,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real,D2: real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
% 0.28/0.46 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_real @ C @ A )
% 0.28/0.46 & ( ord_less_eq_real @ B @ D2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_subset_iff
% 0.28/0.46 thf(fact_133_dist__self,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( real_V1514887919dist_a @ X @ X )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % dist_self
% 0.28/0.46 thf(fact_134_atLeastAtMost__iff,axiom,
% 0.28/0.46 ! [I: a,L: a,U2: a] :
% 0.28/0.46 ( ( member_a @ I @ ( set_or411607219Most_a @ L @ U2 ) )
% 0.28/0.46 = ( ( ord_less_eq_a @ L @ I )
% 0.28/0.46 & ( ord_less_eq_a @ I @ U2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastAtMost_iff
% 0.28/0.46 thf(fact_135_atLeastAtMost__iff,axiom,
% 0.28/0.46 ! [I: set_a,L: set_a,U2: set_a] :
% 0.28/0.46 ( ( member_set_a @ I @ ( set_or2094724627_set_a @ L @ U2 ) )
% 0.28/0.46 = ( ( ord_less_eq_set_a @ L @ I )
% 0.28/0.46 & ( ord_less_eq_set_a @ I @ U2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastAtMost_iff
% 0.28/0.46 thf(fact_136_atLeastAtMost__iff,axiom,
% 0.28/0.46 ! [I: set_real,L: set_real,U2: set_real] :
% 0.28/0.46 ( ( member_set_real @ I @ ( set_or1619725421t_real @ L @ U2 ) )
% 0.28/0.46 = ( ( ord_less_eq_set_real @ L @ I )
% 0.28/0.46 & ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastAtMost_iff
% 0.28/0.46 thf(fact_137_atLeastAtMost__iff,axiom,
% 0.28/0.46 ! [I: real,L: real,U2: real] :
% 0.28/0.46 ( ( member_real @ I @ ( set_or656347191t_real @ L @ U2 ) )
% 0.28/0.46 = ( ( ord_less_eq_real @ L @ I )
% 0.28/0.46 & ( ord_less_eq_real @ I @ U2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastAtMost_iff
% 0.28/0.46 thf(fact_138_Icc__eq__Icc,axiom,
% 0.28/0.46 ! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
% 0.28/0.46 ( ( ( set_or2094724627_set_a @ L @ H )
% 0.28/0.46 = ( set_or2094724627_set_a @ L2 @ H2 ) )
% 0.28/0.46 = ( ( ( L = L2 )
% 0.28/0.46 & ( H = H2 ) )
% 0.28/0.46 | ( ~ ( ord_less_eq_set_a @ L @ H )
% 0.28/0.46 & ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Icc_eq_Icc
% 0.28/0.46 thf(fact_139_Icc__eq__Icc,axiom,
% 0.28/0.46 ! [L: set_real,H: set_real,L2: set_real,H2: set_real] :
% 0.28/0.46 ( ( ( set_or1619725421t_real @ L @ H )
% 0.28/0.46 = ( set_or1619725421t_real @ L2 @ H2 ) )
% 0.28/0.46 = ( ( ( L = L2 )
% 0.28/0.46 & ( H = H2 ) )
% 0.28/0.46 | ( ~ ( ord_less_eq_set_real @ L @ H )
% 0.28/0.46 & ~ ( ord_less_eq_set_real @ L2 @ H2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Icc_eq_Icc
% 0.28/0.46 thf(fact_140_Icc__eq__Icc,axiom,
% 0.28/0.46 ! [L: real,H: real,L2: real,H2: real] :
% 0.28/0.46 ( ( ( set_or656347191t_real @ L @ H )
% 0.28/0.46 = ( set_or656347191t_real @ L2 @ H2 ) )
% 0.28/0.46 = ( ( ( L = L2 )
% 0.28/0.46 & ( H = H2 ) )
% 0.28/0.46 | ( ~ ( ord_less_eq_real @ L @ H )
% 0.28/0.46 & ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Icc_eq_Icc
% 0.28/0.46 thf(fact_141_dist__eq__0__iff,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( ( real_V1514887919dist_a @ X @ Y )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 = ( X = Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % dist_eq_0_iff
% 0.28/0.46 thf(fact_142_dist__commute,axiom,
% 0.28/0.46 ( real_V1514887919dist_a
% 0.28/0.46 = ( ^ [X3: a,Y3: a] : ( real_V1514887919dist_a @ Y3 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dist_commute
% 0.28/0.46 thf(fact_143_dist__commute__lessI,axiom,
% 0.28/0.46 ! [Y: a,X: a,E: real] :
% 0.28/0.46 ( ( ord_less_real @ ( real_V1514887919dist_a @ Y @ X ) @ E )
% 0.28/0.46 => ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % dist_commute_lessI
% 0.28/0.46 thf(fact_144_atLeastatMost__psubset__iff,axiom,
% 0.28/0.46 ! [A: set_a,B: set_a,C: set_a,D2: set_a] :
% 0.28/0.46 ( ( ord_less_set_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
% 0.28/0.46 = ( ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_set_a @ C @ A )
% 0.28/0.46 & ( ord_less_eq_set_a @ B @ D2 )
% 0.28/0.46 & ( ( ord_less_set_a @ C @ A )
% 0.28/0.46 | ( ord_less_set_a @ B @ D2 ) ) ) )
% 0.28/0.46 & ( ord_less_eq_set_a @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_psubset_iff
% 0.28/0.46 thf(fact_145_atLeastatMost__psubset__iff,axiom,
% 0.28/0.46 ! [A: set_real,B: set_real,C: set_real,D2: set_real] :
% 0.28/0.46 ( ( ord_le861030508t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
% 0.28/0.46 = ( ( ~ ( ord_less_eq_set_real @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_set_real @ C @ A )
% 0.28/0.46 & ( ord_less_eq_set_real @ B @ D2 )
% 0.28/0.46 & ( ( ord_less_set_real @ C @ A )
% 0.28/0.46 | ( ord_less_set_real @ B @ D2 ) ) ) )
% 0.28/0.46 & ( ord_less_eq_set_real @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_psubset_iff
% 0.28/0.46 thf(fact_146_atLeastatMost__psubset__iff,axiom,
% 0.28/0.46 ! [A: real,B: real,C: real,D2: real] :
% 0.28/0.46 ( ( ord_less_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
% 0.28/0.46 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.28/0.46 | ( ( ord_less_eq_real @ C @ A )
% 0.28/0.46 & ( ord_less_eq_real @ B @ D2 )
% 0.28/0.46 & ( ( ord_less_real @ C @ A )
% 0.28/0.46 | ( ord_less_real @ B @ D2 ) ) ) )
% 0.28/0.46 & ( ord_less_eq_real @ C @ D2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastatMost_psubset_iff
% 0.28/0.46 thf(fact_147_zero__le__dist,axiom,
% 0.28/0.46 ! [X: a,Y: a] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_le_dist
% 0.28/0.46 thf(fact_148_dist__not__less__zero,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ~ ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % dist_not_less_zero
% 0.28/0.46 thf(fact_149_dist__pos__lt,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( X != Y )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dist_pos_lt
% 0.28/0.46 thf(fact_150_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.periodic_orbit_period(1)
% 0.28/0.46 thf(fact_151_periodic__orbit__def,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.28/0.46 = ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.46 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % periodic_orbit_def
% 0.28/0.46 thf(fact_152_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( member_a @ X @ x )
% 0.28/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 => ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 = zero_zero_real ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(2)
% 0.28/0.46 thf(fact_153_closed__orbit__period__zero__fixed__point,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.46 => ( ( ( period1305449585riod_a @ f @ x @ X )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 => ( ( f @ X )
% 0.28/0.46 = zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % closed_orbit_period_zero_fixed_point
% 0.28/0.46 thf(fact_154_closed__orbit__period__nonneg,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % closed_orbit_period_nonneg
% 0.28/0.46 thf(fact_155_closed__orbit__in__domain,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.46 => ( member_a @ X @ x ) ) ).
% 0.28/0.46
% 0.28/0.46 % closed_orbit_in_domain
% 0.28/0.46 thf(fact_156_add_Oinverse__inverse,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % add.inverse_inverse
% 0.28/0.46 thf(fact_157_add_Oinverse__inverse,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = A ) ).
% 0.28/0.46
% 0.28/0.46 % add.inverse_inverse
% 0.28/0.46 thf(fact_158_neg__equal__iff__equal,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ( uminus_uminus_a @ A )
% 0.28/0.46 = ( uminus_uminus_a @ B ) )
% 0.28/0.46 = ( A = B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_iff_equal
% 0.28/0.46 thf(fact_159_neg__equal__iff__equal,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ( uminus_uminus_real @ A )
% 0.28/0.46 = ( uminus_uminus_real @ B ) )
% 0.28/0.46 = ( A = B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_iff_equal
% 0.28/0.46 thf(fact_160_rev_Oclosed__orbit__eq__rev,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 = ( period720806154rbit_a @ f @ x @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.closed_orbit_eq_rev
% 0.28/0.46 thf(fact_161_rev_Oclosed__orbit__in__domain,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 => ( member_a @ X @ x ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.closed_orbit_in_domain
% 0.28/0.46 thf(fact_162_rev_Oinvariant__eq__rev,axiom,
% 0.28/0.46 ! [M: set_a] :
% 0.28/0.46 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.28/0.46 = ( auto_l630715367iant_a @ f @ x @ M ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.invariant_eq_rev
% 0.28/0.46 thf(fact_163_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( member_a @ X @ x )
% 0.28/0.46 => ( ( ( f @ X )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 => ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % fixed_point_imp_closed_orbit_period_zero(1)
% 0.28/0.46 thf(fact_164_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( member_a @ X @ x )
% 0.28/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(1)
% 0.28/0.46 thf(fact_165_rev_OsectionD_I5_J,axiom,
% 0.28/0.46 ! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
% 0.28/0.46 ( ( reacha1191408304tion_a @ ( uminus_uminus_a_a @ f ) @ x @ S2 @ Ds @ S3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ S3 @ x ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.sectionD(5)
% 0.28/0.46 thf(fact_166_rev_Oinvariant__iff__compl__invariant,axiom,
% 0.28/0.46 ! [M: set_a] :
% 0.28/0.46 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.28/0.46 = ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.invariant_iff_compl_invariant
% 0.28/0.46 thf(fact_167_closed__orbit__periodic,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.46 => ( ( ( f @ X )
% 0.28/0.46 != zero_zero_a )
% 0.28/0.46 => ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % closed_orbit_periodic
% 0.28/0.46 thf(fact_168_rev_Oinvariant__def,axiom,
% 0.28/0.46 ! [M: set_a] :
% 0.28/0.46 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.28/0.46 = ( ! [X3: a] :
% 0.28/0.46 ( ( member_a @ X3 @ M )
% 0.28/0.46 => ( oDE_au1996717075pped_a @ ( uminus_uminus_a_a @ f ) @ x @ X3 @ M ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.invariant_def
% 0.28/0.46 thf(fact_169_rev_Oclosed__orbit__periodic,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.28/0.46 != zero_zero_a )
% 0.28/0.46 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.closed_orbit_periodic
% 0.28/0.46 thf(fact_170_add_Oinverse__neutral,axiom,
% 0.28/0.46 ( ( uminus_uminus_a @ zero_zero_a )
% 0.28/0.46 = zero_zero_a ) ).
% 0.28/0.46
% 0.28/0.46 % add.inverse_neutral
% 0.28/0.46 thf(fact_171_add_Oinverse__neutral,axiom,
% 0.28/0.46 ( ( uminus_uminus_real @ zero_zero_real )
% 0.28/0.46 = zero_zero_real ) ).
% 0.28/0.46
% 0.28/0.46 % add.inverse_neutral
% 0.28/0.46 thf(fact_172_neg__0__equal__iff__equal,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( zero_zero_a
% 0.28/0.46 = ( uminus_uminus_a @ A ) )
% 0.28/0.46 = ( zero_zero_a = A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_equal_iff_equal
% 0.28/0.46 thf(fact_173_neg__0__equal__iff__equal,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( zero_zero_real
% 0.28/0.46 = ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( zero_zero_real = A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_equal_iff_equal
% 0.28/0.46 thf(fact_174_neg__equal__0__iff__equal,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( ( uminus_uminus_a @ A )
% 0.28/0.46 = zero_zero_a )
% 0.28/0.46 = ( A = zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_0_iff_equal
% 0.28/0.46 thf(fact_175_neg__equal__0__iff__equal,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ( uminus_uminus_real @ A )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 = ( A = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_0_iff_equal
% 0.28/0.46 thf(fact_176_equal__neg__zero,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( A
% 0.28/0.46 = ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( A = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % equal_neg_zero
% 0.28/0.46 thf(fact_177_neg__equal__zero,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ( uminus_uminus_real @ A )
% 0.28/0.46 = A )
% 0.28/0.46 = ( A = zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_zero
% 0.28/0.46 thf(fact_178_neg__le__iff__le,axiom,
% 0.28/0.46 ! [B: a,A: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.28/0.46 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_le_iff_le
% 0.28/0.46 thf(fact_179_neg__le__iff__le,axiom,
% 0.28/0.46 ! [B: real,A: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_le_iff_le
% 0.28/0.46 thf(fact_180_neg__less__iff__less,axiom,
% 0.28/0.46 ! [B: a,A: a] :
% 0.28/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.28/0.46 = ( ord_less_a @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_iff_less
% 0.28/0.46 thf(fact_181_neg__less__iff__less,axiom,
% 0.28/0.46 ! [B: real,A: real] :
% 0.28/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_iff_less
% 0.28/0.46 thf(fact_182_minus__diff__eq,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
% 0.28/0.46 = ( minus_minus_a @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_diff_eq
% 0.28/0.46 thf(fact_183_minus__diff__eq,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.28/0.46 = ( minus_minus_real @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_diff_eq
% 0.28/0.46 thf(fact_184_inner__minus__left,axiom,
% 0.28/0.46 ! [X: real,Y: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ ( uminus_uminus_real @ X ) @ Y )
% 0.28/0.46 = ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_minus_left
% 0.28/0.46 thf(fact_185_inner__minus__left,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ ( uminus_uminus_a @ X ) @ Y )
% 0.28/0.46 = ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_minus_left
% 0.28/0.46 thf(fact_186_inner__minus__right,axiom,
% 0.28/0.46 ! [X: real,Y: real] :
% 0.28/0.46 ( ( inner_4346926r_real @ X @ ( uminus_uminus_real @ Y ) )
% 0.28/0.46 = ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_minus_right
% 0.28/0.46 thf(fact_187_inner__minus__right,axiom,
% 0.28/0.46 ! [X: a,Y: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ X @ ( uminus_uminus_a @ Y ) )
% 0.28/0.46 = ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inner_minus_right
% 0.28/0.46 thf(fact_188_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.closed_orbit_period_nonneg
% 0.28/0.46 thf(fact_189_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 => ( ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 = zero_zero_real )
% 0.28/0.46 => ( ( uminus_uminus_a_a @ f @ X )
% 0.28/0.46 = zero_zero_a ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.closed_orbit_period_zero_fixed_point
% 0.28/0.46 thf(fact_190_rev_Operiodic__orbit__def,axiom,
% 0.28/0.46 ! [X: a] :
% 0.28/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 = ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.46 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rev.periodic_orbit_def
% 0.28/0.46 thf(fact_191_neg__0__le__iff__le,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.28/0.46 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_le_iff_le
% 0.28/0.46 thf(fact_192_neg__0__le__iff__le,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_le_iff_le
% 0.28/0.46 thf(fact_193_neg__le__0__iff__le,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.28/0.46 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_le_0_iff_le
% 0.28/0.46 thf(fact_194_neg__le__0__iff__le,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.28/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_le_0_iff_le
% 0.28/0.46 thf(fact_195_less__eq__neg__nonpos,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_eq_neg_nonpos
% 0.28/0.46 thf(fact_196_neg__less__eq__nonneg,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.28/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_eq_nonneg
% 0.28/0.46 thf(fact_197_less__neg__neg,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_neg_neg
% 0.28/0.46 thf(fact_198_neg__less__pos,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.28/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_pos
% 0.28/0.46 thf(fact_199_neg__0__less__iff__less,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.28/0.46 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_less_iff_less
% 0.28/0.46 thf(fact_200_neg__0__less__iff__less,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.28/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_0_less_iff_less
% 0.28/0.46 thf(fact_201_neg__less__0__iff__less,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.28/0.46 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_0_iff_less
% 0.28/0.46 thf(fact_202_neg__less__0__iff__less,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.28/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_0_iff_less
% 0.28/0.46 thf(fact_203_diff__0,axiom,
% 0.28/0.46 ! [A: a] :
% 0.28/0.46 ( ( minus_minus_a @ zero_zero_a @ A )
% 0.28/0.46 = ( uminus_uminus_a @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0
% 0.28/0.46 thf(fact_204_diff__0,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.28/0.46 = ( uminus_uminus_real @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0
% 0.28/0.46 thf(fact_205_le__imp__neg__le,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.28/0.46 => ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_imp_neg_le
% 0.28/0.46 thf(fact_206_le__imp__neg__le,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.28/0.46 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_imp_neg_le
% 0.28/0.46 thf(fact_207_minus__le__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
% 0.28/0.46 = ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_le_iff
% 0.28/0.46 thf(fact_208_minus__le__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.28/0.46 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_le_iff
% 0.28/0.46 thf(fact_209_le__minus__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
% 0.28/0.46 = ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_minus_iff
% 0.28/0.46 thf(fact_210_le__minus__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.28/0.46 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_minus_iff
% 0.28/0.46 thf(fact_211_minus__less__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
% 0.28/0.46 = ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_less_iff
% 0.28/0.46 thf(fact_212_minus__less__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.28/0.46 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_less_iff
% 0.28/0.46 thf(fact_213_less__minus__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
% 0.28/0.46 = ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_minus_iff
% 0.28/0.46 thf(fact_214_less__minus__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.28/0.46 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_minus_iff
% 0.28/0.46 thf(fact_215_minus__diff__commute,axiom,
% 0.28/0.46 ! [B: a,A: a] :
% 0.28/0.46 ( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
% 0.28/0.46 = ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_diff_commute
% 0.28/0.46 thf(fact_216_minus__diff__commute,axiom,
% 0.28/0.46 ! [B: real,A: real] :
% 0.28/0.46 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.28/0.46 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_diff_commute
% 0.28/0.46 thf(fact_217_equation__minus__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( A
% 0.28/0.46 = ( uminus_uminus_a @ B ) )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( uminus_uminus_a @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equation_minus_iff
% 0.28/0.46 thf(fact_218_equation__minus__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( A
% 0.28/0.46 = ( uminus_uminus_real @ B ) )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equation_minus_iff
% 0.28/0.46 thf(fact_219_minus__equation__iff,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ( uminus_uminus_a @ A )
% 0.28/0.46 = B )
% 0.28/0.46 = ( ( uminus_uminus_a @ B )
% 0.28/0.46 = A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_equation_iff
% 0.28/0.46 thf(fact_220_minus__equation__iff,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ( uminus_uminus_real @ A )
% 0.28/0.46 = B )
% 0.28/0.46 = ( ( uminus_uminus_real @ B )
% 0.28/0.46 = A ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_equation_iff
% 0.28/0.46 thf(fact_221_verit__minus__simplify_I3_J,axiom,
% 0.28/0.46 ! [B: a] :
% 0.28/0.46 ( ( minus_minus_a @ zero_zero_a @ B )
% 0.28/0.46 = ( uminus_uminus_a @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_minus_simplify(3)
% 0.28/0.46 thf(fact_222_verit__minus__simplify_I3_J,axiom,
% 0.28/0.46 ! [B: real] :
% 0.28/0.46 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.28/0.46 = ( uminus_uminus_real @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_minus_simplify(3)
% 0.28/0.46 thf(fact_223_psubsetI,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( A2 != B3 )
% 0.28/0.46 => ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetI
% 0.28/0.46 thf(fact_224_psubsetI,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( A2 != B3 )
% 0.28/0.46 => ( ord_less_set_real @ A2 @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetI
% 0.28/0.46 thf(fact_225_uminus__apply,axiom,
% 0.28/0.46 ( uminus_uminus_a_a
% 0.28/0.46 = ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % uminus_apply
% 0.28/0.46 thf(fact_226_compl__le__compl__iff,axiom,
% 0.28/0.46 ! [X: set_a,Y: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
% 0.28/0.46 = ( ord_less_eq_set_a @ Y @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_compl_iff
% 0.28/0.46 thf(fact_227_compl__le__compl__iff,axiom,
% 0.28/0.46 ! [X: set_real,Y: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ ( uminus773214379t_real @ Y ) )
% 0.28/0.46 = ( ord_less_eq_set_real @ Y @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_compl_iff
% 0.28/0.46 thf(fact_228_rot__diff__commute,axiom,
% 0.28/0.46 ! [B: a,A: a] :
% 0.28/0.46 ( ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) )
% 0.28/0.46 = ( uminus_uminus_a @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rot_diff_commute
% 0.28/0.46 thf(fact_229_nrm__reverse,axiom,
% 0.28/0.46 ! [A: a,X: a,Y: a] :
% 0.28/0.46 ( ( inner_1173012732nner_a @ A @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
% 0.28/0.46 = ( inner_1173012732nner_a @ ( uminus_uminus_a @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % nrm_reverse
% 0.28/0.46 thf(fact_230_verit__minus__simplify_I4_J,axiom,
% 0.28/0.46 ! [B: a] :
% 0.28/0.46 ( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
% 0.28/0.46 = B ) ).
% 0.28/0.46
% 0.28/0.46 % verit_minus_simplify(4)
% 0.28/0.46 thf(fact_231_verit__minus__simplify_I4_J,axiom,
% 0.28/0.46 ! [B: real] :
% 0.28/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.28/0.46 = B ) ).
% 0.28/0.46
% 0.28/0.46 % verit_minus_simplify(4)
% 0.28/0.46 thf(fact_232_Compl__subset__Compl__iff,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
% 0.28/0.46 = ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Compl_subset_Compl_iff
% 0.28/0.46 thf(fact_233_Compl__subset__Compl__iff,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ A2 ) @ ( uminus773214379t_real @ B3 ) )
% 0.28/0.46 = ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Compl_subset_Compl_iff
% 0.28/0.46 thf(fact_234_Compl__anti__mono,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Compl_anti_mono
% 0.28/0.46 thf(fact_235_Compl__anti__mono,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ B3 ) @ ( uminus773214379t_real @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Compl_anti_mono
% 0.28/0.46 thf(fact_236_subset__antisym,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.28/0.46 => ( A2 = B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_antisym
% 0.28/0.46 thf(fact_237_subset__antisym,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_real @ B3 @ A2 )
% 0.28/0.46 => ( A2 = B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_antisym
% 0.28/0.46 thf(fact_238_subsetI,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ! [X4: a] :
% 0.28/0.46 ( ( member_a @ X4 @ A2 )
% 0.28/0.46 => ( member_a @ X4 @ B3 ) )
% 0.28/0.46 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % subsetI
% 0.28/0.46 thf(fact_239_subsetI,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ! [X4: real] :
% 0.28/0.46 ( ( member_real @ X4 @ A2 )
% 0.28/0.46 => ( member_real @ X4 @ B3 ) )
% 0.28/0.46 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % subsetI
% 0.28/0.46 thf(fact_240_Diff__idemp,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
% 0.28/0.46 = ( minus_minus_set_a @ A2 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_idemp
% 0.28/0.46 thf(fact_241_Diff__iff,axiom,
% 0.28/0.46 ! [C: real,A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.28/0.46 = ( ( member_real @ C @ A2 )
% 0.28/0.46 & ~ ( member_real @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_iff
% 0.28/0.46 thf(fact_242_Diff__iff,axiom,
% 0.28/0.46 ! [C: a,A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.28/0.46 = ( ( member_a @ C @ A2 )
% 0.28/0.46 & ~ ( member_a @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Diff_iff
% 0.28/0.46 thf(fact_243_DiffI,axiom,
% 0.28/0.46 ! [C: real,A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( member_real @ C @ A2 )
% 0.28/0.46 => ( ~ ( member_real @ C @ B3 )
% 0.28/0.46 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffI
% 0.28/0.46 thf(fact_244_DiffI,axiom,
% 0.28/0.46 ! [C: a,A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( member_a @ C @ A2 )
% 0.28/0.46 => ( ~ ( member_a @ C @ B3 )
% 0.28/0.46 => ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffI
% 0.28/0.46 thf(fact_245_rot__rot,axiom,
% 0.28/0.46 ! [V: a] :
% 0.28/0.46 ( ( poinca659159244_rot_a @ ( poinca659159244_rot_a @ V ) )
% 0.28/0.46 = ( uminus_uminus_a @ V ) ) ).
% 0.28/0.46
% 0.28/0.46 % rot_rot
% 0.28/0.46 thf(fact_246_verit__la__disequality,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 | ~ ( ord_less_eq_real @ A @ B )
% 0.28/0.46 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_la_disequality
% 0.28/0.46 thf(fact_247_verit__comp__simplify1_I1_J,axiom,
% 0.28/0.46 ! [A: real] :
% 0.28/0.46 ~ ( ord_less_real @ A @ A ) ).
% 0.28/0.46
% 0.28/0.46 % verit_comp_simplify1(1)
% 0.28/0.46 thf(fact_248_verit__negate__coefficient_I3_J,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( ( uminus_uminus_a @ A )
% 0.28/0.46 = ( uminus_uminus_a @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_negate_coefficient(3)
% 0.28/0.46 thf(fact_249_verit__negate__coefficient_I3_J,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( ( uminus_uminus_real @ A )
% 0.28/0.46 = ( uminus_uminus_real @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_negate_coefficient(3)
% 0.28/0.46 thf(fact_250_Collect__mono__iff,axiom,
% 0.28/0.46 ! [P: a > $o,Q: a > $o] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.28/0.46 = ( ! [X3: a] :
% 0.28/0.46 ( ( P @ X3 )
% 0.28/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mono_iff
% 0.28/0.46 thf(fact_251_Collect__mono__iff,axiom,
% 0.28/0.46 ! [P: real > $o,Q: real > $o] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.28/0.46 = ( ! [X3: real] :
% 0.28/0.46 ( ( P @ X3 )
% 0.28/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mono_iff
% 0.28/0.46 thf(fact_252_set__eq__subset,axiom,
% 0.28/0.46 ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
% 0.28/0.46 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.28/0.46 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % set_eq_subset
% 0.28/0.46 thf(fact_253_set__eq__subset,axiom,
% 0.28/0.46 ( ( ^ [Y4: set_real,Z3: set_real] : ( Y4 = Z3 ) )
% 0.28/0.46 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.28/0.46 & ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % set_eq_subset
% 0.28/0.46 thf(fact_254_subset__trans,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_trans
% 0.28/0.46 thf(fact_255_subset__trans,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.28/0.46 => ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_trans
% 0.28/0.46 thf(fact_256_Collect__mono,axiom,
% 0.28/0.46 ! [P: a > $o,Q: a > $o] :
% 0.28/0.46 ( ! [X4: a] :
% 0.28/0.46 ( ( P @ X4 )
% 0.28/0.46 => ( Q @ X4 ) )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mono
% 0.28/0.46 thf(fact_257_Collect__mono,axiom,
% 0.28/0.46 ! [P: real > $o,Q: real > $o] :
% 0.28/0.46 ( ! [X4: real] :
% 0.28/0.46 ( ( P @ X4 )
% 0.28/0.46 => ( Q @ X4 ) )
% 0.28/0.46 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mono
% 0.28/0.46 thf(fact_258_subset__refl,axiom,
% 0.28/0.46 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.28/0.46
% 0.28/0.46 % subset_refl
% 0.28/0.46 thf(fact_259_subset__refl,axiom,
% 0.28/0.46 ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% 0.28/0.46
% 0.28/0.46 % subset_refl
% 0.28/0.46 thf(fact_260_subset__iff,axiom,
% 0.28/0.46 ( ord_less_eq_set_a
% 0.28/0.46 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.46 ! [T: a] :
% 0.28/0.46 ( ( member_a @ T @ A5 )
% 0.28/0.46 => ( member_a @ T @ B5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_iff
% 0.28/0.46 thf(fact_261_subset__iff,axiom,
% 0.28/0.46 ( ord_less_eq_set_real
% 0.28/0.46 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.46 ! [T: real] :
% 0.28/0.46 ( ( member_real @ T @ A5 )
% 0.28/0.46 => ( member_real @ T @ B5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_iff
% 0.28/0.46 thf(fact_262_equalityD2,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityD2
% 0.28/0.46 thf(fact_263_equalityD2,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityD2
% 0.28/0.46 thf(fact_264_equalityD1,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityD1
% 0.28/0.46 thf(fact_265_equalityD1,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityD1
% 0.28/0.46 thf(fact_266_subset__eq,axiom,
% 0.28/0.46 ( ord_less_eq_set_a
% 0.28/0.46 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.46 ! [X3: a] :
% 0.28/0.46 ( ( member_a @ X3 @ A5 )
% 0.28/0.46 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_eq
% 0.28/0.46 thf(fact_267_subset__eq,axiom,
% 0.28/0.46 ( ord_less_eq_set_real
% 0.28/0.46 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.46 ! [X3: real] :
% 0.28/0.46 ( ( member_real @ X3 @ A5 )
% 0.28/0.46 => ( member_real @ X3 @ B5 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_eq
% 0.28/0.46 thf(fact_268_equalityE,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityE
% 0.28/0.46 thf(fact_269_equalityE,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( A2 = B3 )
% 0.28/0.46 => ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ~ ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equalityE
% 0.28/0.46 thf(fact_270_subsetD,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a,C: a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( member_a @ C @ A2 )
% 0.28/0.46 => ( member_a @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subsetD
% 0.28/0.46 thf(fact_271_subsetD,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real,C: real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( member_real @ C @ A2 )
% 0.28/0.46 => ( member_real @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subsetD
% 0.28/0.46 thf(fact_272_in__mono,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a,X: a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( member_a @ X @ A2 )
% 0.28/0.46 => ( member_a @ X @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_mono
% 0.28/0.46 thf(fact_273_in__mono,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real,X: real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( member_real @ X @ A2 )
% 0.28/0.46 => ( member_real @ X @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % in_mono
% 0.28/0.46 thf(fact_274_DiffD2,axiom,
% 0.28/0.46 ! [C: real,A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.28/0.46 => ~ ( member_real @ C @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD2
% 0.28/0.46 thf(fact_275_DiffD2,axiom,
% 0.28/0.46 ! [C: a,A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.28/0.46 => ~ ( member_a @ C @ B3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD2
% 0.28/0.46 thf(fact_276_DiffD1,axiom,
% 0.28/0.46 ! [C: real,A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.28/0.46 => ( member_real @ C @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD1
% 0.28/0.46 thf(fact_277_DiffD1,axiom,
% 0.28/0.46 ! [C: a,A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.28/0.46 => ( member_a @ C @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffD1
% 0.28/0.46 thf(fact_278_DiffE,axiom,
% 0.28/0.46 ! [C: real,A2: set_real,B3: set_real] :
% 0.28/0.46 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.28/0.46 => ~ ( ( member_real @ C @ A2 )
% 0.28/0.46 => ( member_real @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffE
% 0.28/0.46 thf(fact_279_DiffE,axiom,
% 0.28/0.46 ! [C: a,A2: set_a,B3: set_a] :
% 0.28/0.46 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.28/0.46 => ~ ( ( member_a @ C @ A2 )
% 0.28/0.46 => ( member_a @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % DiffE
% 0.28/0.46 thf(fact_280_psubsetD,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a,C: a] :
% 0.28/0.46 ( ( ord_less_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( member_a @ C @ A2 )
% 0.28/0.46 => ( member_a @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetD
% 0.28/0.46 thf(fact_281_psubsetD,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real,C: real] :
% 0.28/0.46 ( ( ord_less_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( member_real @ C @ A2 )
% 0.28/0.46 => ( member_real @ C @ B3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetD
% 0.28/0.46 thf(fact_282_verit__comp__simplify1_I3_J,axiom,
% 0.28/0.46 ! [B6: real,A6: real] :
% 0.28/0.46 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 0.28/0.46 = ( ord_less_real @ A6 @ B6 ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_comp_simplify1(3)
% 0.28/0.46 thf(fact_283_compl__le__swap2,axiom,
% 0.28/0.46 ! [Y: set_a,X: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap2
% 0.28/0.46 thf(fact_284_compl__le__swap2,axiom,
% 0.28/0.46 ! [Y: set_real,X: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ X )
% 0.28/0.46 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap2
% 0.28/0.46 thf(fact_285_compl__le__swap1,axiom,
% 0.28/0.46 ! [Y: set_a,X: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
% 0.28/0.46 => ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap1
% 0.28/0.46 thf(fact_286_compl__le__swap1,axiom,
% 0.28/0.46 ! [Y: set_real,X: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ Y @ ( uminus773214379t_real @ X ) )
% 0.28/0.46 => ( ord_less_eq_set_real @ X @ ( uminus773214379t_real @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap1
% 0.28/0.46 thf(fact_287_compl__mono,axiom,
% 0.28/0.46 ! [X: set_a,Y: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ X @ Y )
% 0.28/0.46 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_mono
% 0.28/0.46 thf(fact_288_compl__mono,axiom,
% 0.28/0.46 ! [X: set_real,Y: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ X @ Y )
% 0.28/0.46 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ ( uminus773214379t_real @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_mono
% 0.28/0.46 thf(fact_289_verit__negate__coefficient_I2_J,axiom,
% 0.28/0.46 ! [A: a,B: a] :
% 0.28/0.46 ( ( ord_less_a @ A @ B )
% 0.28/0.46 => ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_negate_coefficient(2)
% 0.28/0.46 thf(fact_290_verit__negate__coefficient_I2_J,axiom,
% 0.28/0.46 ! [A: real,B: real] :
% 0.28/0.46 ( ( ord_less_real @ A @ B )
% 0.28/0.46 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % verit_negate_coefficient(2)
% 0.28/0.46 thf(fact_291_fun__Compl__def,axiom,
% 0.28/0.46 ( uminus_uminus_a_a
% 0.28/0.46 = ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % fun_Compl_def
% 0.28/0.46 thf(fact_292_double__diff,axiom,
% 0.28/0.46 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.28/0.46 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.28/0.46 => ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
% 0.28/0.46 = A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_diff
% 0.28/0.46 thf(fact_293_double__diff,axiom,
% 0.28/0.46 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.28/0.46 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.46 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.28/0.47 => ( ( minus_minus_set_real @ B3 @ ( minus_minus_set_real @ C3 @ A2 ) )
% 0.28/0.47 = A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % double_diff
% 0.28/0.47 thf(fact_294_Diff__subset,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_subset
% 0.28/0.47 thf(fact_295_Diff__subset,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ A2 ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_subset
% 0.28/0.47 thf(fact_296_Diff__mono,axiom,
% 0.28/0.47 ! [A2: set_a,C3: set_a,D4: set_a,B3: set_a] :
% 0.28/0.47 ( ( ord_less_eq_set_a @ A2 @ C3 )
% 0.28/0.47 => ( ( ord_less_eq_set_a @ D4 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_mono
% 0.28/0.47 thf(fact_297_Diff__mono,axiom,
% 0.28/0.47 ! [A2: set_real,C3: set_real,D4: set_real,B3: set_real] :
% 0.28/0.47 ( ( ord_less_eq_set_real @ A2 @ C3 )
% 0.28/0.47 => ( ( ord_less_eq_set_real @ D4 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ ( minus_minus_set_real @ C3 @ D4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_mono
% 0.28/0.47 thf(fact_298_subset__iff__psubset__eq,axiom,
% 0.28/0.47 ( ord_less_eq_set_a
% 0.28/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.47 ( ( ord_less_set_a @ A5 @ B5 )
% 0.28/0.47 | ( A5 = B5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_iff_psubset_eq
% 0.28/0.47 thf(fact_299_subset__iff__psubset__eq,axiom,
% 0.28/0.47 ( ord_less_eq_set_real
% 0.28/0.47 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.47 ( ( ord_less_set_real @ A5 @ B5 )
% 0.28/0.47 | ( A5 = B5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_iff_psubset_eq
% 0.28/0.47 thf(fact_300_subset__psubset__trans,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.28/0.47 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_set_a @ B3 @ C3 )
% 0.28/0.47 => ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_psubset_trans
% 0.28/0.47 thf(fact_301_subset__psubset__trans,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.28/0.47 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_set_real @ B3 @ C3 )
% 0.28/0.47 => ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_psubset_trans
% 0.28/0.47 thf(fact_302_subset__not__subset__eq,axiom,
% 0.28/0.47 ( ord_less_set_a
% 0.28/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.47 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.28/0.47 & ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_not_subset_eq
% 0.28/0.47 thf(fact_303_subset__not__subset__eq,axiom,
% 0.28/0.47 ( ord_less_set_real
% 0.28/0.47 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.47 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.28/0.47 & ~ ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % subset_not_subset_eq
% 0.28/0.47 thf(fact_304_psubset__subset__trans,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.28/0.47 ( ( ord_less_set_a @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.28/0.47 => ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_subset_trans
% 0.28/0.47 thf(fact_305_psubset__subset__trans,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.28/0.47 ( ( ord_less_set_real @ A2 @ B3 )
% 0.28/0.47 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.28/0.47 => ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_subset_trans
% 0.28/0.47 thf(fact_306_psubset__imp__subset,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a] :
% 0.28/0.47 ( ( ord_less_set_a @ A2 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_imp_subset
% 0.28/0.47 thf(fact_307_psubset__imp__subset,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real] :
% 0.28/0.47 ( ( ord_less_set_real @ A2 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_imp_subset
% 0.28/0.47 thf(fact_308_psubset__eq,axiom,
% 0.28/0.47 ( ord_less_set_a
% 0.28/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.28/0.47 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.28/0.47 & ( A5 != B5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_eq
% 0.28/0.47 thf(fact_309_psubset__eq,axiom,
% 0.28/0.47 ( ord_less_set_real
% 0.28/0.47 = ( ^ [A5: set_real,B5: set_real] :
% 0.28/0.47 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.28/0.47 & ( A5 != B5 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_eq
% 0.28/0.47 thf(fact_310_psubsetE,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a] :
% 0.28/0.47 ( ( ord_less_set_a @ A2 @ B3 )
% 0.28/0.47 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubsetE
% 0.28/0.47 thf(fact_311_psubsetE,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real] :
% 0.28/0.47 ( ( ord_less_set_real @ A2 @ B3 )
% 0.28/0.47 => ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.28/0.47 => ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubsetE
% 0.28/0.47 thf(fact_312_psubset__imp__ex__mem,axiom,
% 0.28/0.47 ! [A2: set_real,B3: set_real] :
% 0.28/0.47 ( ( ord_less_set_real @ A2 @ B3 )
% 0.28/0.47 => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_imp_ex_mem
% 0.28/0.47 thf(fact_313_psubset__imp__ex__mem,axiom,
% 0.28/0.47 ! [A2: set_a,B3: set_a] :
% 0.28/0.47 ( ( ord_less_set_a @ A2 @ B3 )
% 0.28/0.47 => ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % psubset_imp_ex_mem
% 0.28/0.47 thf(fact_314_rev_Oopen__existence__ivl_H,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ~ ! [A4: real] :
% 0.28/0.47 ( ( ord_less_real @ zero_zero_real @ A4 )
% 0.28/0.47 => ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.open_existence_ivl'
% 0.28/0.47 thf(fact_315_open__existence__ivl_H,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ~ ! [A4: real] :
% 0.28/0.47 ( ( ord_less_real @ zero_zero_real @ A4 )
% 0.28/0.47 => ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % open_existence_ivl'
% 0.28/0.47 thf(fact_316_rev_Oopen__existence__ivl0,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ? [A4: real,B4: real] :
% 0.28/0.47 ( ( ord_less_real @ A4 @ zero_zero_real )
% 0.28/0.47 & ( ord_less_real @ zero_zero_real @ B4 )
% 0.28/0.47 & ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.open_existence_ivl0
% 0.28/0.47 thf(fact_317_existence__ivl__zero,axiom,
% 0.28/0.47 ! [X0: a] :
% 0.28/0.47 ( ( member_a @ X0 @ x )
% 0.28/0.47 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % existence_ivl_zero
% 0.28/0.47 thf(fact_318_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.28/0.47 ! [T2: real,T0: real,X0: a] :
% 0.28/0.47 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( member_a @ X0 @ x ) ) ).
% 0.28/0.47
% 0.28/0.47 % general.mem_existence_ivl_iv_defined(2)
% 0.28/0.47 thf(fact_319_Compl__iff,axiom,
% 0.28/0.47 ! [C: a,A2: set_a] :
% 0.28/0.47 ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
% 0.28/0.47 = ( ~ ( member_a @ C @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Compl_iff
% 0.28/0.47 thf(fact_320_Compl__iff,axiom,
% 0.28/0.47 ! [C: real,A2: set_real] :
% 0.28/0.47 ( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
% 0.28/0.47 = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Compl_iff
% 0.28/0.47 thf(fact_321_ComplI,axiom,
% 0.28/0.47 ! [C: a,A2: set_a] :
% 0.28/0.47 ( ~ ( member_a @ C @ A2 )
% 0.28/0.47 => ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ComplI
% 0.28/0.47 thf(fact_322_ComplI,axiom,
% 0.28/0.47 ! [C: real,A2: set_real] :
% 0.28/0.47 ( ~ ( member_real @ C @ A2 )
% 0.28/0.47 => ( member_real @ C @ ( uminus773214379t_real @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ComplI
% 0.28/0.47 thf(fact_323_rev_Oexistence__ivl__zero,axiom,
% 0.28/0.47 ! [X0: a] :
% 0.28/0.47 ( ( member_a @ X0 @ x )
% 0.28/0.47 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.existence_ivl_zero
% 0.28/0.47 thf(fact_324_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.28/0.47 ! [T2: real,T0: real,X0: a] :
% 0.28/0.47 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.28/0.47 => ( member_a @ X0 @ x ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.general.mem_existence_ivl_iv_defined(2)
% 0.28/0.47 thf(fact_325_local_Oivl__subset__existence__ivl,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.ivl_subset_existence_ivl
% 0.28/0.47 thf(fact_326_local_Oivl__subset__existence__ivl_H,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.ivl_subset_existence_ivl'
% 0.28/0.47 thf(fact_327_open__existence__ivl0,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ? [A4: real,B4: real] :
% 0.28/0.47 ( ( ord_less_real @ A4 @ zero_zero_real )
% 0.28/0.47 & ( ord_less_real @ zero_zero_real @ B4 )
% 0.28/0.47 & ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % open_existence_ivl0
% 0.28/0.47 thf(fact_328_local_Orev_Oivl__subset__existence__ivl,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.28/0.47 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.rev.ivl_subset_existence_ivl
% 0.28/0.47 thf(fact_329_local_Orev_Oivl__subset__existence__ivl_H,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.28/0.47 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.rev.ivl_subset_existence_ivl'
% 0.28/0.47 thf(fact_330_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( member_a @ X0 @ x ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.mem_existence_ivl_iv_defined(2)
% 0.28/0.47 thf(fact_331_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.28/0.47 => ( member_a @ X0 @ x ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.rev.mem_existence_ivl_iv_defined(2)
% 0.28/0.47 thf(fact_332_ComplD,axiom,
% 0.28/0.47 ! [C: a,A2: set_a] :
% 0.28/0.47 ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
% 0.28/0.47 => ~ ( member_a @ C @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ComplD
% 0.28/0.47 thf(fact_333_ComplD,axiom,
% 0.28/0.47 ! [C: real,A2: set_real] :
% 0.28/0.47 ( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
% 0.28/0.47 => ~ ( member_real @ C @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ComplD
% 0.28/0.47 thf(fact_334_rev_Omvar_Ointerval__axioms,axiom,
% 0.28/0.47 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.mvar.interval_axioms
% 0.28/0.47 thf(fact_335_mvar_Ointerval__axioms,axiom,
% 0.28/0.47 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mvar.interval_axioms
% 0.28/0.47 thf(fact_336_rev_Oclosed__orbit__global__existence,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.47 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.47 = top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.closed_orbit_global_existence
% 0.28/0.47 thf(fact_337_rev_Ofixpoint__sol_I1_J,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.28/0.47 = zero_zero_a )
% 0.28/0.47 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.28/0.47 = top_top_set_real ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev.fixpoint_sol(1)
% 0.28/0.47 thf(fact_338_interval__axioms,axiom,
% 0.28/0.47 initia826609931terval @ top_top_set_real ).
% 0.28/0.47
% 0.28/0.47 % interval_axioms
% 0.28/0.47 thf(fact_339_UNIV__I,axiom,
% 0.28/0.47 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.28/0.47
% 0.28/0.47 % UNIV_I
% 0.28/0.47 thf(fact_340_UNIV__I,axiom,
% 0.28/0.47 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.28/0.47
% 0.28/0.47 % UNIV_I
% 0.28/0.47 thf(fact_341_local_Omem__existence__ivl__subset,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.mem_existence_ivl_subset
% 0.28/0.47 thf(fact_342_local_Oexistence__ivl__initial__time,axiom,
% 0.28/0.47 ! [X0: a] :
% 0.28/0.47 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.28/0.47 => ( ( member_a @ X0 @ x )
% 0.28/0.47 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.existence_ivl_initial_time
% 0.28/0.47 thf(fact_343_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.28/0.47 ! [T2: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.28/0.47 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.rev.mem_existence_ivl_subset
% 0.28/0.47 thf(fact_344_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.28/0.47 ! [T2: real,T0: real,X0: a] :
% 0.28/0.47 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( member_real @ T0 @ top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % general.mem_existence_ivl_iv_defined(1)
% 0.28/0.47 thf(fact_345_general_Oexistence__ivl__initial__time,axiom,
% 0.28/0.47 ! [T0: real,X0: a] :
% 0.28/0.47 ( ( member_real @ T0 @ top_top_set_real )
% 0.28/0.47 => ( ( member_a @ X0 @ x )
% 0.28/0.47 => ( member_real @ ( minus_minus_real @ T0 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % general.existence_ivl_initial_time
% 0.28/0.47 thf(fact_346_mem__existence__ivl__shift__autonomous2,axiom,
% 0.28/0.47 ! [T2: real,S2: real,X: a] :
% 0.28/0.47 ( ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.28/0.47 => ( ( member_a @ X @ x )
% 0.28/0.47 => ( ! [S: real,T3: real,X4: a] :
% 0.28/0.47 ( ( member_a @ X4 @ x )
% 0.28/0.47 => ( ( f @ X4 )
% 0.28/0.47 = ( f @ X4 ) ) )
% 0.28/0.47 => ( ( top_top_set_real = top_top_set_real )
% 0.28/0.47 => ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_existence_ivl_shift_autonomous2
% 0.28/0.47 thf(fact_347_general_Omem__existence__ivl__subset,axiom,
% 0.28/0.47 ! [T2: real,T0: real,X0: a] :
% 0.28/0.47 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.28/0.47 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % general.mem_existence_ivl_subset
% 0.28/0.47 thf(fact_348_local_Oexistence__ivl__subset,axiom,
% 0.28/0.47 ! [X0: a] : ( ord_less_eq_set_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) @ top_top_set_real ) ).
% 0.28/0.47
% 0.28/0.47 % local.existence_ivl_subset
% 0.28/0.47 thf(fact_349_fixpoint__sol_I1_J,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( member_a @ X @ x )
% 0.28/0.47 => ( ( ( f @ X )
% 0.28/0.47 = zero_zero_a )
% 0.28/0.47 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.28/0.47 = top_top_set_real ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % fixpoint_sol(1)
% 0.28/0.47 thf(fact_350_closed__orbit__global__existence,axiom,
% 0.28/0.47 ! [X: a] :
% 0.28/0.47 ( ( period720806154rbit_a @ f @ x @ X )
% 0.28/0.47 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.28/0.47 = top_top_set_real ) ) ).
% 0.28/0.47
% 0.28/0.47 % closed_orbit_global_existence
% 0.28/0.47 thf(fact_351_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.28/0.47 ! [X0: a] :
% 0.28/0.47 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.28/0.47 => ( ( member_a @ X0 @ x )
% 0.28/0.47 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % local.rev.existence_ivl_initial_time
% 0.28/0.47
% 0.28/0.47 % Conjectures (1)
% 0.28/0.47 thf(conj_0,conjecture,
% 0.28/0.47 ? [D3: real] :
% 0.28/0.47 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.28/0.47 & ? [B7: real] :
% 0.28/0.47 ( ( ord_less_real @ zero_zero_real @ B7 )
% 0.28/0.47 & ! [X4: a] :
% 0.28/0.47 ( ( member_a @ X4 @ ( elemen49976720ball_a @ x2 @ D3 ) )
% 0.28/0.47 => ( ord_less_eq_real @ B7 @ ( inner_1173012732nner_a @ ( f @ X4 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 0.29/1.50 ---- Embedded in THF ---
% 0.29/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.29/1.50 %%% Generated on Tue Feb 24 08:28:22 EST 2026
% 0.29/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.29/1.50 %%% Logic specification used:
% 0.29/1.50 %%% thf(spec, logic, $$dhol).
% 0.29/1.50
% 0.29/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2DTF11562.p
% See solution above
% 0.29/1.50 ------------------------
% 0.29/1.53 ---- Cleaned THF ---
% 0.29/1.53 thf(ty_n_t__Bounded____Linear____Function__Oblinfun_Itf__a_Mt__Real__Oreal_J,type,
% 0.29/1.53 bounde7994401a_real: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/1.53 set_set_real: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/1.53 set_set_a: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 set_real: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 set_a: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_t__Real__Oreal,type,
% 0.29/1.53 real: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(ty_n_tf__a,type,
% 0.29/1.53 a: $tType ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Elementary__Metric__Spaces_Ocball_001t__Real__Oreal,type,
% 0.29/1.53 elemen1140313242l_real: real > real > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Elementary__Metric__Spaces_Ocball_001tf__a,type,
% 0.29/1.53 elemen49976720ball_a: a > real > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
% 0.29/1.53 auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 0.29/1.53 minus_minus_real: real > real > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 minus_minus_set_real: set_real > set_real > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 minus_minus_set_a: set_a > set_a > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
% 0.29/1.53 minus_minus_a: a > a > a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
% 0.29/1.53 uminus_uminus_a_a: ( a > a ) > a > a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.29/1.53 uminus_uminus_real: real > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 uminus773214379t_real: set_real > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 uminus_uminus_set_a: set_a > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
% 0.29/1.53 uminus_uminus_a: a > a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.29/1.53 zero_zero_real: real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 0.29/1.53 zero_zero_a: a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.29/1.53 initia826609931terval: set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001t__Real__Oreal,type,
% 0.29/1.53 inner_4346926r_real: real > real > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001tf__a,type,
% 0.29/1.53 inner_1173012732nner_a: a > a > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant_001tf__a,type,
% 0.29/1.53 auto_l630715367iant_a: ( a > a ) > set_a > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Line__Segment_Oclosed__segment_001tf__a,type,
% 0.29/1.53 line_c1152468841ment_a: a > a > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
% 0.29/1.53 line_open_segment_a: a > a > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped_001tf__a,type,
% 0.29/1.53 oDE_au1996717075pped_a: ( a > a ) > set_a > a > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.29/1.53 ord_less_real: real > real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 ord_less_set_real: set_real > set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/1.53 ord_le861030508t_real: set_set_real > set_set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/1.53 ord_less_set_set_a: set_set_a > set_set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 ord_less_set_a: set_a > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 0.29/1.53 ord_less_a: a > a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.29/1.53 ord_less_eq_real: real > real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 ord_less_eq_set_real: set_real > set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/1.53 ord_le1733505144t_real: set_set_real > set_set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/1.53 ord_le318720350_set_a: set_set_a > set_set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.29/1.53 ord_less_eq_a: a > a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 top_top_set_real: set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 top_top_set_a: set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
% 0.29/1.53 period720806154rbit_a: ( a > a ) > set_a > a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
% 0.29/1.53 period1305449585riod_a: ( a > a ) > set_a > a > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
% 0.29/1.53 period138238489rbit_a: ( a > a ) > set_a > a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Orot_001tf__a,type,
% 0.29/1.53 poinca659159244_rot_a: a > a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
% 0.29/1.53 poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection_001tf__a,type,
% 0.29/1.53 reacha1191408304tion_a: ( a > a ) > set_a > ( a > real ) > ( a > bounde7994401a_real ) > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.29/1.53 real_V1934908667t_real: real > real > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001tf__a,type,
% 0.29/1.53 real_V1514887919dist_a: a > a > real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.29/1.53 collect_real: ( real > $o ) > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 collect_set_real: ( set_real > $o ) > set_set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 collect_set_a: ( set_a > $o ) > set_set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.29/1.53 collect_a: ( a > $o ) > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 0.29/1.53 set_or656347191t_real: real > real > set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 set_or1619725421t_real: set_real > set_real > set_set_real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 set_or2094724627_set_a: set_a > set_a > set_set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
% 0.29/1.53 set_or411607219Most_a: a > a > set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_member_001t__Real__Oreal,type,
% 0.29/1.53 member_real: real > set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/1.53 member_set_real: set_real > set_set_real > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.29/1.53 member_set_a: set_a > set_set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_c_member_001tf__a,type,
% 0.29/1.53 member_a: a > set_a > $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_X,type,
% 0.29/1.53 x: set_a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_a,type,
% 0.29/1.53 a2: a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_b,type,
% 0.29/1.53 b: a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_d____,type,
% 0.29/1.53 d: real ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_f,type,
% 0.29/1.53 f: a > a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_s____,type,
% 0.29/1.53 s: a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_thesis,type,
% 0.29/1.53 thesis: $o ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_x____,type,
% 0.29/1.53 x2: a ).
% 0.29/1.53
% 0.29/1.53 thf(sy_v_z,type,
% 0.29/1.53 z: a ).
% 0.29/1.53
% 0.29/1.53 thf(fact_0_d_I1_J,axiom,
% 0.29/1.53 ord_less_real @ zero_zero_real @ d ).
% 0.29/1.53
% 0.29/1.53 thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062,axiom,
% 0.29/1.53 member_a @ s @ ( elemen49976720ball_a @ x2 @ d ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_2_seg_I1_J,axiom,
% 0.29/1.53 a2 != b ).
% 0.29/1.53
% 0.29/1.53 thf(fact_3_nrm__dot,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062,axiom,
% 0.29/1.53 member_a @ x2 @ x ).
% 0.29/1.53
% 0.29/1.53 thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.29/1.53 ! [X2: a] :
% 0.29/1.53 ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ s ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.29/1.53 ( ( f @ x2 )
% 0.29/1.53 != zero_zero_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_7_direction_I2_J,axiom,
% 0.29/1.53 ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_8_d_I2_J,axiom,
% 0.29/1.53 ! [Y: a] :
% 0.29/1.53 ( ( member_a @ Y @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
% 0.29/1.53 & ( member_a @ Y @ x )
% 0.29/1.53 & ( ( f @ Y )
% 0.29/1.53 != zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.29/1.53 ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ x2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_10_dot__ortho,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ X @ ( poinca659159244_rot_a @ X ) )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/1.53 ~ ! [S: a] :
% 0.29/1.53 ( ( member_a @ S @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.29/1.53 => ~ ! [X2: a] :
% 0.29/1.53 ( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ S ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062,axiom,
% 0.29/1.53 member_a @ x2 @ ( line_c1152468841ment_a @ a2 @ b ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/1.53 ~ ! [D: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ D )
% 0.29/1.53 => ~ ! [Y2: a] :
% 0.29/1.53 ( ( member_a @ Y2 @ ( elemen49976720ball_a @ x2 @ D ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
% 0.29/1.53 & ( member_a @ Y2 @ x )
% 0.29/1.53 & ( ( f @ Y2 )
% 0.29/1.53 != zero_zero_a ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_14_transversal,axiom,
% 0.29/1.53 poinca522724647ment_a @ f @ x @ a2 @ b ).
% 0.29/1.53
% 0.29/1.53 thf(fact_15_inner__gt__zero__iff,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ ( inner_4346926r_real @ X @ X ) )
% 0.29/1.53 = ( X != zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_16_inner__gt__zero__iff,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) )
% 0.29/1.53 = ( X != zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_17_centre__in__cball,axiom,
% 0.29/1.53 ! [X: real,E: real] :
% 0.29/1.53 ( ( member_real @ X @ ( elemen1140313242l_real @ X @ E ) )
% 0.29/1.53 = ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_18_centre__in__cball,axiom,
% 0.29/1.53 ! [X: a,E: real] :
% 0.29/1.53 ( ( member_a @ X @ ( elemen49976720ball_a @ X @ E ) )
% 0.29/1.53 = ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_19_all__zero__iff,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( ! [U: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ X @ U )
% 0.29/1.53 = zero_zero_real ) )
% 0.29/1.53 = ( X = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_20_all__zero__iff,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( ! [U: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ X @ U )
% 0.29/1.53 = zero_zero_real ) )
% 0.29/1.53 = ( X = zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_21_inner__zero__left,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ zero_zero_real @ X )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_22_inner__zero__left,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ zero_zero_a @ X )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_23_inner__zero__right,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ X @ zero_zero_real )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_24_inner__zero__right,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ X @ zero_zero_a )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_25_inner__eq__zero__iff,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( ( inner_4346926r_real @ X @ X )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 = ( X = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_26_inner__eq__zero__iff,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( ( inner_1173012732nner_a @ X @ X )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 = ( X = zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_27_diff__gt__0__iff__gt,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
% 0.29/1.53 = ( ord_less_a @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_28_diff__gt__0__iff__gt,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.29/1.53 = ( ord_less_real @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_29_diff__ge__0__iff__ge,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
% 0.29/1.53 = ( ord_less_eq_a @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_30_diff__ge__0__iff__ge,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.29/1.53 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_31_rot__0,axiom,
% 0.29/1.53 ( ( poinca659159244_rot_a @ zero_zero_a )
% 0.29/1.53 = zero_zero_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_32_rot__eq__0__iff,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( ( poinca659159244_rot_a @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 = ( X = zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_33_seg_I3_J,axiom,
% 0.29/1.53 ! [Z: a] :
% 0.29/1.53 ( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_34_direction_I1_J,axiom,
% 0.29/1.53 member_a @ z @ ( line_c1152468841ment_a @ a2 @ b ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_35_transversal__segment__reverse,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_36_transversal__segment__commute,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 = ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_37_in__segment__inner__rot2,axiom,
% 0.29/1.53 ! [X: a,A: a,B: a,Y: a] :
% 0.29/1.53 ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.29/1.53 => ( ( member_a @ Y @ ( line_c1152468841ment_a @ A @ B ) )
% 0.29/1.53 => ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
% 0.29/1.53 = zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_38_in__segment__inner__rot,axiom,
% 0.29/1.53 ! [X: a,A: a,B: a] :
% 0.29/1.53 ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.29/1.53 => ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
% 0.29/1.53 = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_39_seg_I2_J,axiom,
% 0.29/1.53 ord_less_eq_set_a @ ( line_c1152468841ment_a @ a2 @ b ) @ x ).
% 0.29/1.53
% 0.29/1.53 thf(fact_40_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( minus_minus_a @ A @ A )
% 0.29/1.53 = zero_zero_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_41_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( minus_minus_real @ A @ A )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_42_diff__zero,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( minus_minus_a @ A @ zero_zero_a )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_43_diff__zero,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_44_diff__0__right,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( minus_minus_a @ A @ zero_zero_a )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_45_diff__0__right,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_46_diff__self,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( minus_minus_a @ A @ A )
% 0.29/1.53 = zero_zero_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_47_diff__self,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( minus_minus_real @ A @ A )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_48_subset__cball,axiom,
% 0.29/1.53 ! [D2: real,E: real,X: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ D2 @ E )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( elemen1140313242l_real @ X @ D2 ) @ ( elemen1140313242l_real @ X @ E ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_49_subset__cball,axiom,
% 0.29/1.53 ! [D2: real,E: real,X: a] :
% 0.29/1.53 ( ( ord_less_eq_real @ D2 @ E )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( elemen49976720ball_a @ X @ D2 ) @ ( elemen49976720ball_a @ X @ E ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_50_transversal__segment__def,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ A @ B )
% 0.29/1.53 = ( ( A != B )
% 0.29/1.53 & ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ A @ B ) @ x )
% 0.29/1.53 & ! [X3: a] :
% 0.29/1.53 ( ( member_a @ X3 @ ( line_c1152468841ment_a @ A @ B ) )
% 0.29/1.53 => ( ( inner_1173012732nner_a @ ( f @ X3 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
% 0.29/1.53 != zero_zero_real ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_51_transversal__segmentE,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( ( X != Y )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
% 0.29/1.53 => ~ ! [Z2: a] :
% 0.29/1.53 ( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) )
% 0.29/1.53 => ~ ( ( X != Y )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
% 0.29/1.53 => ~ ! [Z2: a] :
% 0.29/1.53 ( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_52_transversal__segment__posD_I2_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_53_transversal__segment__negD_I2_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_54_mem__Collect__eq,axiom,
% 0.29/1.53 ! [A: set_real,P: set_real > $o] :
% 0.29/1.53 ( ( member_set_real @ A @ ( collect_set_real @ P ) )
% 0.29/1.53 = ( P @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_55_mem__Collect__eq,axiom,
% 0.29/1.53 ! [A: set_a,P: set_a > $o] :
% 0.29/1.53 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 0.29/1.53 = ( P @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_56_mem__Collect__eq,axiom,
% 0.29/1.53 ! [A: a,P: a > $o] :
% 0.29/1.53 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.29/1.53 = ( P @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_57_mem__Collect__eq,axiom,
% 0.29/1.53 ! [A: real,P: real > $o] :
% 0.29/1.53 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.29/1.53 = ( P @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_58_Collect__mem__eq,axiom,
% 0.29/1.53 ! [A2: set_set_real] :
% 0.29/1.53 ( ( collect_set_real
% 0.29/1.53 @ ^ [X3: set_real] : ( member_set_real @ X3 @ A2 ) )
% 0.29/1.53 = A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_59_Collect__mem__eq,axiom,
% 0.29/1.53 ! [A2: set_set_a] :
% 0.29/1.53 ( ( collect_set_a
% 0.29/1.53 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
% 0.29/1.53 = A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_60_Collect__mem__eq,axiom,
% 0.29/1.53 ! [A2: set_a] :
% 0.29/1.53 ( ( collect_a
% 0.29/1.53 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.29/1.53 = A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_61_Collect__mem__eq,axiom,
% 0.29/1.53 ! [A2: set_real] :
% 0.29/1.53 ( ( collect_real
% 0.29/1.53 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.29/1.53 = A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_62_transversal__segment__sign__less,axiom,
% 0.29/1.53 ! [W: a,Y: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ W @ Y )
% 0.29/1.53 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real )
% 0.29/1.53 => ! [X2: a] :
% 0.29/1.53 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ W @ Y ) )
% 0.29/1.53 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_63_transversal__segment__pos,axiom,
% 0.29/1.53 ! [X: a,Y: a,W: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.29/1.53 => ! [X2: a] :
% 0.29/1.53 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_64_transversal__segment__neg,axiom,
% 0.29/1.53 ! [X: a,Y: a,W: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.29/1.53 => ! [X2: a] :
% 0.29/1.53 ( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_65_transversal__segment__posD_I1_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.29/1.53 => ( X != Y ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_66_transversal__segment__posD_I3_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Za: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_67_transversal__segment__negD_I1_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.29/1.53 => ( X != Y ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_68_transversal__segment__negD_I3_J,axiom,
% 0.29/1.53 ! [X: a,Y: a,Za: a,Z: a] :
% 0.29/1.53 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.29/1.53 => ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
% 0.29/1.53 => ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
% 0.29/1.53 => ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_69__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
% 0.29/1.53 ! [Z: a] :
% 0.29/1.53 ( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_70_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.29/1.53 poinca522724647ment_a = poinca522724647ment_a ).
% 0.29/1.53
% 0.29/1.53 thf(fact_71_zero__reorient,axiom,
% 0.29/1.53 ! [X: real] :
% 0.29/1.53 ( ( zero_zero_real = X )
% 0.29/1.53 = ( X = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_72_zero__reorient,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( zero_zero_a = X )
% 0.29/1.53 = ( X = zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_73_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.29/1.53 ! [A: a,C: a,B: a] :
% 0.29/1.53 ( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
% 0.29/1.53 = ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_74_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.29/1.53 ! [A: real,C: real,B: real] :
% 0.29/1.53 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.29/1.53 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_75_diff__eq__diff__eq,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a,D2: a] :
% 0.29/1.53 ( ( ( minus_minus_a @ A @ B )
% 0.29/1.53 = ( minus_minus_a @ C @ D2 ) )
% 0.29/1.53 => ( ( A = B )
% 0.29/1.53 = ( C = D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_76_diff__eq__diff__eq,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real,D2: real] :
% 0.29/1.53 ( ( ( minus_minus_real @ A @ B )
% 0.29/1.53 = ( minus_minus_real @ C @ D2 ) )
% 0.29/1.53 => ( ( A = B )
% 0.29/1.53 = ( C = D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_77_inner__commute,axiom,
% 0.29/1.53 ( inner_1173012732nner_a
% 0.29/1.53 = ( ^ [X3: a,Y3: a] : ( inner_1173012732nner_a @ Y3 @ X3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_78_eq__iff__diff__eq__0,axiom,
% 0.29/1.53 ( ( ^ [Y4: a,Z3: a] : ( Y4 = Z3 ) )
% 0.29/1.53 = ( ^ [A3: a,B2: a] :
% 0.29/1.53 ( ( minus_minus_a @ A3 @ B2 )
% 0.29/1.53 = zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_79_eq__iff__diff__eq__0,axiom,
% 0.29/1.53 ( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
% 0.29/1.53 = ( ^ [A3: real,B2: real] :
% 0.29/1.53 ( ( minus_minus_real @ A3 @ B2 )
% 0.29/1.53 = zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_80_diff__eq__diff__less__eq,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a,D2: a] :
% 0.29/1.53 ( ( ( minus_minus_a @ A @ B )
% 0.29/1.53 = ( minus_minus_a @ C @ D2 ) )
% 0.29/1.53 => ( ( ord_less_eq_a @ A @ B )
% 0.29/1.53 = ( ord_less_eq_a @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_81_diff__eq__diff__less__eq,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real,D2: real] :
% 0.29/1.53 ( ( ( minus_minus_real @ A @ B )
% 0.29/1.53 = ( minus_minus_real @ C @ D2 ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ A @ B )
% 0.29/1.53 = ( ord_less_eq_real @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_82_diff__right__mono,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ A @ B )
% 0.29/1.53 => ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_83_diff__right__mono,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ B )
% 0.29/1.53 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_84_diff__left__mono,axiom,
% 0.29/1.53 ! [B: a,A: a,C: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ B @ A )
% 0.29/1.53 => ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_85_diff__left__mono,axiom,
% 0.29/1.53 ! [B: real,A: real,C: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ B @ A )
% 0.29/1.53 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_86_diff__mono,axiom,
% 0.29/1.53 ! [A: a,B: a,D2: a,C: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ A @ B )
% 0.29/1.53 => ( ( ord_less_eq_a @ D2 @ C )
% 0.29/1.53 => ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_87_diff__mono,axiom,
% 0.29/1.53 ! [A: real,B: real,D2: real,C: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ B )
% 0.29/1.53 => ( ( ord_less_eq_real @ D2 @ C )
% 0.29/1.53 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_88_diff__strict__right__mono,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a] :
% 0.29/1.53 ( ( ord_less_a @ A @ B )
% 0.29/1.53 => ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_89_diff__strict__right__mono,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real] :
% 0.29/1.53 ( ( ord_less_real @ A @ B )
% 0.29/1.53 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_90_diff__strict__left__mono,axiom,
% 0.29/1.53 ! [B: a,A: a,C: a] :
% 0.29/1.53 ( ( ord_less_a @ B @ A )
% 0.29/1.53 => ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_91_diff__strict__left__mono,axiom,
% 0.29/1.53 ! [B: real,A: real,C: real] :
% 0.29/1.53 ( ( ord_less_real @ B @ A )
% 0.29/1.53 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_92_diff__eq__diff__less,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a,D2: a] :
% 0.29/1.53 ( ( ( minus_minus_a @ A @ B )
% 0.29/1.53 = ( minus_minus_a @ C @ D2 ) )
% 0.29/1.53 => ( ( ord_less_a @ A @ B )
% 0.29/1.53 = ( ord_less_a @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_93_diff__eq__diff__less,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real,D2: real] :
% 0.29/1.53 ( ( ( minus_minus_real @ A @ B )
% 0.29/1.53 = ( minus_minus_real @ C @ D2 ) )
% 0.29/1.53 => ( ( ord_less_real @ A @ B )
% 0.29/1.53 = ( ord_less_real @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_94_diff__strict__mono,axiom,
% 0.29/1.53 ! [A: a,B: a,D2: a,C: a] :
% 0.29/1.53 ( ( ord_less_a @ A @ B )
% 0.29/1.53 => ( ( ord_less_a @ D2 @ C )
% 0.29/1.53 => ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_95_diff__strict__mono,axiom,
% 0.29/1.53 ! [A: real,B: real,D2: real,C: real] :
% 0.29/1.53 ( ( ord_less_real @ A @ B )
% 0.29/1.53 => ( ( ord_less_real @ D2 @ C )
% 0.29/1.53 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_96_inner__diff__right,axiom,
% 0.29/1.53 ! [X: real,Y: real,Z: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ X @ ( minus_minus_real @ Y @ Z ) )
% 0.29/1.53 = ( minus_minus_real @ ( inner_4346926r_real @ X @ Y ) @ ( inner_4346926r_real @ X @ Z ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_97_inner__diff__right,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ X @ ( minus_minus_a @ Y @ Z ) )
% 0.29/1.53 = ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Y ) @ ( inner_1173012732nner_a @ X @ Z ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_98_inner__diff__left,axiom,
% 0.29/1.53 ! [X: real,Y: real,Z: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ ( minus_minus_real @ X @ Y ) @ Z )
% 0.29/1.53 = ( minus_minus_real @ ( inner_4346926r_real @ X @ Z ) @ ( inner_4346926r_real @ Y @ Z ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_99_inner__diff__left,axiom,
% 0.29/1.53 ! [X: a,Y: a,Z: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ Z )
% 0.29/1.53 = ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Z ) @ ( inner_1173012732nner_a @ Y @ Z ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_100_mem__cball__leI,axiom,
% 0.29/1.53 ! [X: real,Y: real,E: real,F: real] :
% 0.29/1.53 ( ( member_real @ X @ ( elemen1140313242l_real @ Y @ E ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ E @ F )
% 0.29/1.53 => ( member_real @ X @ ( elemen1140313242l_real @ Y @ F ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_101_mem__cball__leI,axiom,
% 0.29/1.53 ! [X: a,Y: a,E: real,F: real] :
% 0.29/1.53 ( ( member_a @ X @ ( elemen49976720ball_a @ Y @ E ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ E @ F )
% 0.29/1.53 => ( member_a @ X @ ( elemen49976720ball_a @ Y @ F ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_102_le__iff__diff__le__0,axiom,
% 0.29/1.53 ( ord_less_eq_a
% 0.29/1.53 = ( ^ [A3: a,B2: a] : ( ord_less_eq_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_103_le__iff__diff__le__0,axiom,
% 0.29/1.53 ( ord_less_eq_real
% 0.29/1.53 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_104_less__iff__diff__less__0,axiom,
% 0.29/1.53 ( ord_less_a
% 0.29/1.53 = ( ^ [A3: a,B2: a] : ( ord_less_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_105_less__iff__diff__less__0,axiom,
% 0.29/1.53 ( ord_less_real
% 0.29/1.53 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_106_inner__ge__zero,axiom,
% 0.29/1.53 ! [X: a] : ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_107_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( ( period1305449585riod_a @ f @ x @ X )
% 0.29/1.53 = zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_108_that,axiom,
% 0.29/1.53 ! [D2: real,B3: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.29/1.53 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.29/1.53 => ( ! [X4: a,Y5: a] :
% 0.29/1.53 ( ( member_a @ X4 @ ( line_c1152468841ment_a @ a2 @ b ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X4 @ Y5 ) @ D2 )
% 0.29/1.53 => ( ord_less_eq_real @ B3 @ ( inner_1173012732nner_a @ ( f @ Y5 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
% 0.29/1.53 => thesis ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_109_inner__nonneg__nonneg,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/1.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/1.53 => ( ord_less_eq_real @ zero_zero_real @ ( inner_4346926r_real @ A @ B ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_110_inner__nonneg__nonneg,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.29/1.53 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.29/1.53 => ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ A @ B ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_111_in__closed__segment__iff__rot,axiom,
% 0.29/1.53 ! [A: a,B: a,X: a] :
% 0.29/1.53 ( ( A != B )
% 0.29/1.53 => ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
% 0.29/1.53 = ( ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 & ( member_real @ ( inner_1173012732nner_a @ X @ ( minus_minus_a @ B @ A ) ) @ ( set_or656347191t_real @ ( inner_1173012732nner_a @ A @ ( minus_minus_a @ B @ A ) ) @ ( inner_1173012732nner_a @ B @ ( minus_minus_a @ B @ A ) ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_112_sectionD_I5_J,axiom,
% 0.29/1.53 ! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
% 0.29/1.53 ( ( reacha1191408304tion_a @ f @ x @ S2 @ Ds @ S3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ S3 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_113_transversal__segment__exists,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( f @ X )
% 0.29/1.53 != zero_zero_a )
% 0.29/1.53 => ~ ! [A4: a,B4: a] :
% 0.29/1.53 ( ( member_a @ X @ ( line_open_segment_a @ A4 @ B4 ) )
% 0.29/1.53 => ~ ( poinca522724647ment_a @ f @ x @ A4 @ B4 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_114_invariant__iff__compl__invariant,axiom,
% 0.29/1.53 ! [M: set_a] :
% 0.29/1.53 ( ( auto_l630715367iant_a @ f @ x @ M )
% 0.29/1.53 = ( auto_l630715367iant_a @ f @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_115_cball__eq__cball__iff,axiom,
% 0.29/1.53 ! [X: a,D2: real,Y: a,E: real] :
% 0.29/1.53 ( ( ( elemen49976720ball_a @ X @ D2 )
% 0.29/1.53 = ( elemen49976720ball_a @ Y @ E ) )
% 0.29/1.53 = ( ( ( ord_less_real @ D2 @ zero_zero_real )
% 0.29/1.53 & ( ord_less_real @ E @ zero_zero_real ) )
% 0.29/1.53 | ( ( X = Y )
% 0.29/1.53 & ( D2 = E ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_116_Bolzano,axiom,
% 0.29/1.53 ! [A: real,B: real,P: real > real > $o] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ B )
% 0.29/1.53 => ( ! [A4: real,B4: real,C2: real] :
% 0.29/1.53 ( ( P @ A4 @ B4 )
% 0.29/1.53 => ( ( P @ B4 @ C2 )
% 0.29/1.53 => ( ( ord_less_eq_real @ A4 @ B4 )
% 0.29/1.53 => ( ( ord_less_eq_real @ B4 @ C2 )
% 0.29/1.53 => ( P @ A4 @ C2 ) ) ) ) )
% 0.29/1.53 => ( ! [X4: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ X4 )
% 0.29/1.53 => ( ( ord_less_eq_real @ X4 @ B )
% 0.29/1.53 => ? [D3: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.29/1.53 & ! [A4: real,B4: real] :
% 0.29/1.53 ( ( ( ord_less_eq_real @ A4 @ X4 )
% 0.29/1.53 & ( ord_less_eq_real @ X4 @ B4 )
% 0.29/1.53 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
% 0.29/1.53 => ( P @ A4 @ B4 ) ) ) ) )
% 0.29/1.53 => ( P @ A @ B ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_117_mem__cball,axiom,
% 0.29/1.53 ! [Y: real,X: real,E: real] :
% 0.29/1.53 ( ( member_real @ Y @ ( elemen1140313242l_real @ X @ E ) )
% 0.29/1.53 = ( ord_less_eq_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_118_mem__cball,axiom,
% 0.29/1.53 ! [Y: a,X: a,E: real] :
% 0.29/1.53 ( ( member_a @ Y @ ( elemen49976720ball_a @ X @ E ) )
% 0.29/1.53 = ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_119_perfect__choose__dist,axiom,
% 0.29/1.53 ! [R: real,X: a] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ R )
% 0.29/1.53 => ? [A4: a] :
% 0.29/1.53 ( ( A4 != X )
% 0.29/1.53 & ( ord_less_real @ ( real_V1514887919dist_a @ A4 @ X ) @ R ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_120_interval__inner__leI_I2_J,axiom,
% 0.29/1.53 ! [X: a,A: a,B: a,I: a] :
% 0.29/1.53 ( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
% 0.29/1.53 => ( ( ord_less_eq_a @ zero_zero_a @ I )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ X @ I ) @ ( inner_1173012732nner_a @ B @ I ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_121_interval__inner__leI_I2_J,axiom,
% 0.29/1.53 ! [X: real,A: real,B: real,I: real] :
% 0.29/1.53 ( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ zero_zero_real @ I )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_4346926r_real @ X @ I ) @ ( inner_4346926r_real @ B @ I ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_122_interval__inner__leI_I1_J,axiom,
% 0.29/1.53 ! [X: a,A: a,B: a,I: a] :
% 0.29/1.53 ( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
% 0.29/1.53 => ( ( ord_less_eq_a @ zero_zero_a @ I )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_1173012732nner_a @ A @ I ) @ ( inner_1173012732nner_a @ X @ I ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_123_interval__inner__leI_I1_J,axiom,
% 0.29/1.53 ! [X: real,A: real,B: real,I: real] :
% 0.29/1.53 ( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
% 0.29/1.53 => ( ( ord_less_eq_real @ zero_zero_real @ I )
% 0.29/1.53 => ( ord_less_eq_real @ ( inner_4346926r_real @ A @ I ) @ ( inner_4346926r_real @ X @ I ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_124_eucl__less__le__not__le,axiom,
% 0.29/1.53 ( ord_less_real
% 0.29/1.53 = ( ^ [X3: real,Y3: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/1.53 & ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_125_invariant__def,axiom,
% 0.29/1.53 ! [M: set_a] :
% 0.29/1.53 ( ( auto_l630715367iant_a @ f @ x @ M )
% 0.29/1.53 = ( ! [X3: a] :
% 0.29/1.53 ( ( member_a @ X3 @ M )
% 0.29/1.53 => ( oDE_au1996717075pped_a @ f @ x @ X3 @ M ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_126_periodic__orbit__period_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period138238489rbit_a @ f @ x @ X )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_127_zero__less__dist__iff,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) )
% 0.29/1.53 = ( X != Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_128_dist__le__zero__iff,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real )
% 0.29/1.53 = ( X = Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_129_atLeastatMost__subset__iff,axiom,
% 0.29/1.53 ! [A: set_a,B: set_a,C: set_a,D2: set_a] :
% 0.29/1.53 ( ( ord_le318720350_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
% 0.29/1.53 = ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_set_a @ C @ A )
% 0.29/1.53 & ( ord_less_eq_set_a @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_130_atLeastatMost__subset__iff,axiom,
% 0.29/1.53 ! [A: set_real,B: set_real,C: set_real,D2: set_real] :
% 0.29/1.53 ( ( ord_le1733505144t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
% 0.29/1.53 = ( ~ ( ord_less_eq_set_real @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_set_real @ C @ A )
% 0.29/1.53 & ( ord_less_eq_set_real @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_131_atLeastatMost__subset__iff,axiom,
% 0.29/1.53 ! [A: a,B: a,C: a,D2: a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ ( set_or411607219Most_a @ A @ B ) @ ( set_or411607219Most_a @ C @ D2 ) )
% 0.29/1.53 = ( ~ ( ord_less_eq_a @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_a @ C @ A )
% 0.29/1.53 & ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_132_atLeastatMost__subset__iff,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real,D2: real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
% 0.29/1.53 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_real @ C @ A )
% 0.29/1.53 & ( ord_less_eq_real @ B @ D2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_133_dist__self,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( real_V1514887919dist_a @ X @ X )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_134_atLeastAtMost__iff,axiom,
% 0.29/1.53 ! [I: a,L: a,U2: a] :
% 0.29/1.53 ( ( member_a @ I @ ( set_or411607219Most_a @ L @ U2 ) )
% 0.29/1.53 = ( ( ord_less_eq_a @ L @ I )
% 0.29/1.53 & ( ord_less_eq_a @ I @ U2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_135_atLeastAtMost__iff,axiom,
% 0.29/1.53 ! [I: set_a,L: set_a,U2: set_a] :
% 0.29/1.53 ( ( member_set_a @ I @ ( set_or2094724627_set_a @ L @ U2 ) )
% 0.29/1.53 = ( ( ord_less_eq_set_a @ L @ I )
% 0.29/1.53 & ( ord_less_eq_set_a @ I @ U2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_136_atLeastAtMost__iff,axiom,
% 0.29/1.53 ! [I: set_real,L: set_real,U2: set_real] :
% 0.29/1.53 ( ( member_set_real @ I @ ( set_or1619725421t_real @ L @ U2 ) )
% 0.29/1.53 = ( ( ord_less_eq_set_real @ L @ I )
% 0.29/1.53 & ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_137_atLeastAtMost__iff,axiom,
% 0.29/1.53 ! [I: real,L: real,U2: real] :
% 0.29/1.53 ( ( member_real @ I @ ( set_or656347191t_real @ L @ U2 ) )
% 0.29/1.53 = ( ( ord_less_eq_real @ L @ I )
% 0.29/1.53 & ( ord_less_eq_real @ I @ U2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_138_Icc__eq__Icc,axiom,
% 0.29/1.53 ! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
% 0.29/1.53 ( ( ( set_or2094724627_set_a @ L @ H )
% 0.29/1.53 = ( set_or2094724627_set_a @ L2 @ H2 ) )
% 0.29/1.53 = ( ( ( L = L2 )
% 0.29/1.53 & ( H = H2 ) )
% 0.29/1.53 | ( ~ ( ord_less_eq_set_a @ L @ H )
% 0.29/1.53 & ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_139_Icc__eq__Icc,axiom,
% 0.29/1.53 ! [L: set_real,H: set_real,L2: set_real,H2: set_real] :
% 0.29/1.53 ( ( ( set_or1619725421t_real @ L @ H )
% 0.29/1.53 = ( set_or1619725421t_real @ L2 @ H2 ) )
% 0.29/1.53 = ( ( ( L = L2 )
% 0.29/1.53 & ( H = H2 ) )
% 0.29/1.53 | ( ~ ( ord_less_eq_set_real @ L @ H )
% 0.29/1.53 & ~ ( ord_less_eq_set_real @ L2 @ H2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_140_Icc__eq__Icc,axiom,
% 0.29/1.53 ! [L: real,H: real,L2: real,H2: real] :
% 0.29/1.53 ( ( ( set_or656347191t_real @ L @ H )
% 0.29/1.53 = ( set_or656347191t_real @ L2 @ H2 ) )
% 0.29/1.53 = ( ( ( L = L2 )
% 0.29/1.53 & ( H = H2 ) )
% 0.29/1.53 | ( ~ ( ord_less_eq_real @ L @ H )
% 0.29/1.53 & ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_141_dist__eq__0__iff,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( ( real_V1514887919dist_a @ X @ Y )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 = ( X = Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_142_dist__commute,axiom,
% 0.29/1.53 ( real_V1514887919dist_a
% 0.29/1.53 = ( ^ [X3: a,Y3: a] : ( real_V1514887919dist_a @ Y3 @ X3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_143_dist__commute__lessI,axiom,
% 0.29/1.53 ! [Y: a,X: a,E: real] :
% 0.29/1.53 ( ( ord_less_real @ ( real_V1514887919dist_a @ Y @ X ) @ E )
% 0.29/1.53 => ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_144_atLeastatMost__psubset__iff,axiom,
% 0.29/1.53 ! [A: set_a,B: set_a,C: set_a,D2: set_a] :
% 0.29/1.53 ( ( ord_less_set_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
% 0.29/1.53 = ( ( ~ ( ord_less_eq_set_a @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_set_a @ C @ A )
% 0.29/1.53 & ( ord_less_eq_set_a @ B @ D2 )
% 0.29/1.53 & ( ( ord_less_set_a @ C @ A )
% 0.29/1.53 | ( ord_less_set_a @ B @ D2 ) ) ) )
% 0.29/1.53 & ( ord_less_eq_set_a @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_145_atLeastatMost__psubset__iff,axiom,
% 0.29/1.53 ! [A: set_real,B: set_real,C: set_real,D2: set_real] :
% 0.29/1.53 ( ( ord_le861030508t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
% 0.29/1.53 = ( ( ~ ( ord_less_eq_set_real @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_set_real @ C @ A )
% 0.29/1.53 & ( ord_less_eq_set_real @ B @ D2 )
% 0.29/1.53 & ( ( ord_less_set_real @ C @ A )
% 0.29/1.53 | ( ord_less_set_real @ B @ D2 ) ) ) )
% 0.29/1.53 & ( ord_less_eq_set_real @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_146_atLeastatMost__psubset__iff,axiom,
% 0.29/1.53 ! [A: real,B: real,C: real,D2: real] :
% 0.29/1.53 ( ( ord_less_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
% 0.29/1.53 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.29/1.53 | ( ( ord_less_eq_real @ C @ A )
% 0.29/1.53 & ( ord_less_eq_real @ B @ D2 )
% 0.29/1.53 & ( ( ord_less_real @ C @ A )
% 0.29/1.53 | ( ord_less_real @ B @ D2 ) ) ) )
% 0.29/1.53 & ( ord_less_eq_real @ C @ D2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_147_zero__le__dist,axiom,
% 0.29/1.53 ! [X: a,Y: a] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_148_dist__not__less__zero,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ~ ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_149_dist__pos__lt,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( X != Y )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_150_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_151_periodic__orbit__def,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period138238489rbit_a @ f @ x @ X )
% 0.29/1.53 = ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_152_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = zero_zero_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_153_closed__orbit__period__zero__fixed__point,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 => ( ( ( period1305449585riod_a @ f @ x @ X )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 => ( ( f @ X )
% 0.29/1.53 = zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_154_closed__orbit__period__nonneg,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_155_closed__orbit__in__domain,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 => ( member_a @ X @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_156_add_Oinverse__inverse,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_157_add_Oinverse__inverse,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_158_neg__equal__iff__equal,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ( uminus_uminus_a @ A )
% 0.29/1.53 = ( uminus_uminus_a @ B ) )
% 0.29/1.53 = ( A = B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_159_neg__equal__iff__equal,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ( uminus_uminus_real @ A )
% 0.29/1.53 = ( uminus_uminus_real @ B ) )
% 0.29/1.53 = ( A = B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_160_rev_Oclosed__orbit__eq__rev,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = ( period720806154rbit_a @ f @ x @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_161_rev_Oclosed__orbit__in__domain,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( member_a @ X @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_162_rev_Oinvariant__eq__rev,axiom,
% 0.29/1.53 ! [M: set_a] :
% 0.29/1.53 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.29/1.53 = ( auto_l630715367iant_a @ f @ x @ M ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_163_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_164_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_165_rev_OsectionD_I5_J,axiom,
% 0.29/1.53 ! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
% 0.29/1.53 ( ( reacha1191408304tion_a @ ( uminus_uminus_a_a @ f ) @ x @ S2 @ Ds @ S3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ S3 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_166_rev_Oinvariant__iff__compl__invariant,axiom,
% 0.29/1.53 ! [M: set_a] :
% 0.29/1.53 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.29/1.53 = ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_167_closed__orbit__periodic,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 => ( ( ( f @ X )
% 0.29/1.53 != zero_zero_a )
% 0.29/1.53 => ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_168_rev_Oinvariant__def,axiom,
% 0.29/1.53 ! [M: set_a] :
% 0.29/1.53 ( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
% 0.29/1.53 = ( ! [X3: a] :
% 0.29/1.53 ( ( member_a @ X3 @ M )
% 0.29/1.53 => ( oDE_au1996717075pped_a @ ( uminus_uminus_a_a @ f ) @ x @ X3 @ M ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_169_rev_Oclosed__orbit__periodic,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.29/1.53 != zero_zero_a )
% 0.29/1.53 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_170_add_Oinverse__neutral,axiom,
% 0.29/1.53 ( ( uminus_uminus_a @ zero_zero_a )
% 0.29/1.53 = zero_zero_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_171_add_Oinverse__neutral,axiom,
% 0.29/1.53 ( ( uminus_uminus_real @ zero_zero_real )
% 0.29/1.53 = zero_zero_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_172_neg__0__equal__iff__equal,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( zero_zero_a
% 0.29/1.53 = ( uminus_uminus_a @ A ) )
% 0.29/1.53 = ( zero_zero_a = A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_173_neg__0__equal__iff__equal,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( zero_zero_real
% 0.29/1.53 = ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( zero_zero_real = A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_174_neg__equal__0__iff__equal,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( ( uminus_uminus_a @ A )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 = ( A = zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_175_neg__equal__0__iff__equal,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ( uminus_uminus_real @ A )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 = ( A = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_176_equal__neg__zero,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( A
% 0.29/1.53 = ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( A = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_177_neg__equal__zero,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ( uminus_uminus_real @ A )
% 0.29/1.53 = A )
% 0.29/1.53 = ( A = zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_178_neg__le__iff__le,axiom,
% 0.29/1.53 ! [B: a,A: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.29/1.53 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_179_neg__le__iff__le,axiom,
% 0.29/1.53 ! [B: real,A: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_180_neg__less__iff__less,axiom,
% 0.29/1.53 ! [B: a,A: a] :
% 0.29/1.53 ( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.29/1.53 = ( ord_less_a @ A @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_181_neg__less__iff__less,axiom,
% 0.29/1.53 ! [B: real,A: real] :
% 0.29/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_real @ A @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_182_minus__diff__eq,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
% 0.29/1.53 = ( minus_minus_a @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_183_minus__diff__eq,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.29/1.53 = ( minus_minus_real @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_184_inner__minus__left,axiom,
% 0.29/1.53 ! [X: real,Y: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ ( uminus_uminus_real @ X ) @ Y )
% 0.29/1.53 = ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_185_inner__minus__left,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ ( uminus_uminus_a @ X ) @ Y )
% 0.29/1.53 = ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_186_inner__minus__right,axiom,
% 0.29/1.53 ! [X: real,Y: real] :
% 0.29/1.53 ( ( inner_4346926r_real @ X @ ( uminus_uminus_real @ Y ) )
% 0.29/1.53 = ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_187_inner__minus__right,axiom,
% 0.29/1.53 ! [X: a,Y: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ X @ ( uminus_uminus_a @ Y ) )
% 0.29/1.53 = ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_188_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_189_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = zero_zero_real )
% 0.29/1.53 => ( ( uminus_uminus_a_a @ f @ X )
% 0.29/1.53 = zero_zero_a ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_190_rev_Operiodic__orbit__def,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_191_neg__0__le__iff__le,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.29/1.53 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_192_neg__0__le__iff__le,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_193_neg__le__0__iff__le,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.29/1.53 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_194_neg__le__0__iff__le,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/1.53 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_195_less__eq__neg__nonpos,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_196_neg__less__eq__nonneg,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/1.53 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_197_less__neg__neg,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_198_neg__less__pos,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/1.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_199_neg__0__less__iff__less,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.29/1.53 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_200_neg__0__less__iff__less,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/1.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_201_neg__less__0__iff__less,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.29/1.53 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_202_neg__less__0__iff__less,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/1.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_203_diff__0,axiom,
% 0.29/1.53 ! [A: a] :
% 0.29/1.53 ( ( minus_minus_a @ zero_zero_a @ A )
% 0.29/1.53 = ( uminus_uminus_a @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_204_diff__0,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.29/1.53 = ( uminus_uminus_real @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_205_le__imp__neg__le,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ A @ B )
% 0.29/1.53 => ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_206_le__imp__neg__le,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ B )
% 0.29/1.53 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_207_minus__le__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
% 0.29/1.53 = ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_208_minus__le__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/1.53 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_209_le__minus__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
% 0.29/1.53 = ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_210_le__minus__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/1.53 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_211_minus__less__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
% 0.29/1.53 = ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_212_minus__less__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.29/1.53 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_213_less__minus__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
% 0.29/1.53 = ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_214_less__minus__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.29/1.53 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_215_minus__diff__commute,axiom,
% 0.29/1.53 ! [B: a,A: a] :
% 0.29/1.53 ( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
% 0.29/1.53 = ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_216_minus__diff__commute,axiom,
% 0.29/1.53 ! [B: real,A: real] :
% 0.29/1.53 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.29/1.53 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_217_equation__minus__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( A
% 0.29/1.53 = ( uminus_uminus_a @ B ) )
% 0.29/1.53 = ( B
% 0.29/1.53 = ( uminus_uminus_a @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_218_equation__minus__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( A
% 0.29/1.53 = ( uminus_uminus_real @ B ) )
% 0.29/1.53 = ( B
% 0.29/1.53 = ( uminus_uminus_real @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_219_minus__equation__iff,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ( uminus_uminus_a @ A )
% 0.29/1.53 = B )
% 0.29/1.53 = ( ( uminus_uminus_a @ B )
% 0.29/1.53 = A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_220_minus__equation__iff,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ( uminus_uminus_real @ A )
% 0.29/1.53 = B )
% 0.29/1.53 = ( ( uminus_uminus_real @ B )
% 0.29/1.53 = A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_221_verit__minus__simplify_I3_J,axiom,
% 0.29/1.53 ! [B: a] :
% 0.29/1.53 ( ( minus_minus_a @ zero_zero_a @ B )
% 0.29/1.53 = ( uminus_uminus_a @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_222_verit__minus__simplify_I3_J,axiom,
% 0.29/1.53 ! [B: real] :
% 0.29/1.53 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.29/1.53 = ( uminus_uminus_real @ B ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_223_psubsetI,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( A2 != B3 )
% 0.29/1.53 => ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_224_psubsetI,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( A2 != B3 )
% 0.29/1.53 => ( ord_less_set_real @ A2 @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_225_uminus__apply,axiom,
% 0.29/1.53 ( uminus_uminus_a_a
% 0.29/1.53 = ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_226_compl__le__compl__iff,axiom,
% 0.29/1.53 ! [X: set_a,Y: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
% 0.29/1.53 = ( ord_less_eq_set_a @ Y @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_227_compl__le__compl__iff,axiom,
% 0.29/1.53 ! [X: set_real,Y: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ ( uminus773214379t_real @ Y ) )
% 0.29/1.53 = ( ord_less_eq_set_real @ Y @ X ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_228_rot__diff__commute,axiom,
% 0.29/1.53 ! [B: a,A: a] :
% 0.29/1.53 ( ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) )
% 0.29/1.53 = ( uminus_uminus_a @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_229_nrm__reverse,axiom,
% 0.29/1.53 ! [A: a,X: a,Y: a] :
% 0.29/1.53 ( ( inner_1173012732nner_a @ A @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
% 0.29/1.53 = ( inner_1173012732nner_a @ ( uminus_uminus_a @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_230_verit__minus__simplify_I4_J,axiom,
% 0.29/1.53 ! [B: a] :
% 0.29/1.53 ( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
% 0.29/1.53 = B ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_231_verit__minus__simplify_I4_J,axiom,
% 0.29/1.53 ! [B: real] :
% 0.29/1.53 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.29/1.53 = B ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_232_Compl__subset__Compl__iff,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
% 0.29/1.53 = ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_233_Compl__subset__Compl__iff,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ A2 ) @ ( uminus773214379t_real @ B3 ) )
% 0.29/1.53 = ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_234_Compl__anti__mono,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_235_Compl__anti__mono,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ B3 ) @ ( uminus773214379t_real @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_236_subset__antisym,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.29/1.53 => ( A2 = B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_237_subset__antisym,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_real @ B3 @ A2 )
% 0.29/1.53 => ( A2 = B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_238_subsetI,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ! [X4: a] :
% 0.29/1.53 ( ( member_a @ X4 @ A2 )
% 0.29/1.53 => ( member_a @ X4 @ B3 ) )
% 0.29/1.53 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_239_subsetI,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ! [X4: real] :
% 0.29/1.53 ( ( member_real @ X4 @ A2 )
% 0.29/1.53 => ( member_real @ X4 @ B3 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_240_Diff__idemp,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
% 0.29/1.53 = ( minus_minus_set_a @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_241_Diff__iff,axiom,
% 0.29/1.53 ! [C: real,A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.29/1.53 = ( ( member_real @ C @ A2 )
% 0.29/1.53 & ~ ( member_real @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_242_Diff__iff,axiom,
% 0.29/1.53 ! [C: a,A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.29/1.53 = ( ( member_a @ C @ A2 )
% 0.29/1.53 & ~ ( member_a @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_243_DiffI,axiom,
% 0.29/1.53 ! [C: real,A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( member_real @ C @ A2 )
% 0.29/1.53 => ( ~ ( member_real @ C @ B3 )
% 0.29/1.53 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_244_DiffI,axiom,
% 0.29/1.53 ! [C: a,A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( member_a @ C @ A2 )
% 0.29/1.53 => ( ~ ( member_a @ C @ B3 )
% 0.29/1.53 => ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_245_rot__rot,axiom,
% 0.29/1.53 ! [V: a] :
% 0.29/1.53 ( ( poinca659159244_rot_a @ ( poinca659159244_rot_a @ V ) )
% 0.29/1.53 = ( uminus_uminus_a @ V ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_246_verit__la__disequality,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( A = B )
% 0.29/1.53 | ~ ( ord_less_eq_real @ A @ B )
% 0.29/1.53 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_247_verit__comp__simplify1_I1_J,axiom,
% 0.29/1.53 ! [A: real] :
% 0.29/1.53 ~ ( ord_less_real @ A @ A ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_248_verit__negate__coefficient_I3_J,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( A = B )
% 0.29/1.53 => ( ( uminus_uminus_a @ A )
% 0.29/1.53 = ( uminus_uminus_a @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_249_verit__negate__coefficient_I3_J,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( A = B )
% 0.29/1.53 => ( ( uminus_uminus_real @ A )
% 0.29/1.53 = ( uminus_uminus_real @ B ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_250_Collect__mono__iff,axiom,
% 0.29/1.53 ! [P: a > $o,Q: a > $o] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.29/1.53 = ( ! [X3: a] :
% 0.29/1.53 ( ( P @ X3 )
% 0.29/1.53 => ( Q @ X3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_251_Collect__mono__iff,axiom,
% 0.29/1.53 ! [P: real > $o,Q: real > $o] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.29/1.53 = ( ! [X3: real] :
% 0.29/1.53 ( ( P @ X3 )
% 0.29/1.53 => ( Q @ X3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_252_set__eq__subset,axiom,
% 0.29/1.53 ( ( ^ [Y4: set_a,Z3: set_a] : ( Y4 = Z3 ) )
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.29/1.53 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_253_set__eq__subset,axiom,
% 0.29/1.53 ( ( ^ [Y4: set_real,Z3: set_real] : ( Y4 = Z3 ) )
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.29/1.53 & ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_254_subset__trans,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_255_subset__trans,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_256_Collect__mono,axiom,
% 0.29/1.53 ! [P: a > $o,Q: a > $o] :
% 0.29/1.53 ( ! [X4: a] :
% 0.29/1.53 ( ( P @ X4 )
% 0.29/1.53 => ( Q @ X4 ) )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_257_Collect__mono,axiom,
% 0.29/1.53 ! [P: real > $o,Q: real > $o] :
% 0.29/1.53 ( ! [X4: real] :
% 0.29/1.53 ( ( P @ X4 )
% 0.29/1.53 => ( Q @ X4 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_258_subset__refl,axiom,
% 0.29/1.53 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_259_subset__refl,axiom,
% 0.29/1.53 ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_260_subset__iff,axiom,
% 0.29/1.53 ( ord_less_eq_set_a
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ! [T: a] :
% 0.29/1.53 ( ( member_a @ T @ A5 )
% 0.29/1.53 => ( member_a @ T @ B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_261_subset__iff,axiom,
% 0.29/1.53 ( ord_less_eq_set_real
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ! [T: real] :
% 0.29/1.53 ( ( member_real @ T @ A5 )
% 0.29/1.53 => ( member_real @ T @ B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_262_equalityD2,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_263_equalityD2,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_264_equalityD1,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_265_equalityD1,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_266_subset__eq,axiom,
% 0.29/1.53 ( ord_less_eq_set_a
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ! [X3: a] :
% 0.29/1.53 ( ( member_a @ X3 @ A5 )
% 0.29/1.53 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_267_subset__eq,axiom,
% 0.29/1.53 ( ord_less_eq_set_real
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ! [X3: real] :
% 0.29/1.53 ( ( member_real @ X3 @ A5 )
% 0.29/1.53 => ( member_real @ X3 @ B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_268_equalityE,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_269_equalityE,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( A2 = B3 )
% 0.29/1.53 => ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ~ ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_270_subsetD,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C: a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( member_a @ C @ A2 )
% 0.29/1.53 => ( member_a @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_271_subsetD,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C: real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( member_real @ C @ A2 )
% 0.29/1.53 => ( member_real @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_272_in__mono,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,X: a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( member_a @ X @ A2 )
% 0.29/1.53 => ( member_a @ X @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_273_in__mono,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,X: real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( member_real @ X @ A2 )
% 0.29/1.53 => ( member_real @ X @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_274_DiffD2,axiom,
% 0.29/1.53 ! [C: real,A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.29/1.53 => ~ ( member_real @ C @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_275_DiffD2,axiom,
% 0.29/1.53 ! [C: a,A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.29/1.53 => ~ ( member_a @ C @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_276_DiffD1,axiom,
% 0.29/1.53 ! [C: real,A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.29/1.53 => ( member_real @ C @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_277_DiffD1,axiom,
% 0.29/1.53 ! [C: a,A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.29/1.53 => ( member_a @ C @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_278_DiffE,axiom,
% 0.29/1.53 ! [C: real,A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.29/1.53 => ~ ( ( member_real @ C @ A2 )
% 0.29/1.53 => ( member_real @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_279_DiffE,axiom,
% 0.29/1.53 ! [C: a,A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
% 0.29/1.53 => ~ ( ( member_a @ C @ A2 )
% 0.29/1.53 => ( member_a @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_280_psubsetD,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C: a] :
% 0.29/1.53 ( ( ord_less_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( member_a @ C @ A2 )
% 0.29/1.53 => ( member_a @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_281_psubsetD,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C: real] :
% 0.29/1.53 ( ( ord_less_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( member_real @ C @ A2 )
% 0.29/1.53 => ( member_real @ C @ B3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_282_verit__comp__simplify1_I3_J,axiom,
% 0.29/1.53 ! [B6: real,A6: real] :
% 0.29/1.53 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 0.29/1.53 = ( ord_less_real @ A6 @ B6 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_283_compl__le__swap2,axiom,
% 0.29/1.53 ! [Y: set_a,X: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_284_compl__le__swap2,axiom,
% 0.29/1.53 ! [Y: set_real,X: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ X )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ Y ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_285_compl__le__swap1,axiom,
% 0.29/1.53 ! [Y: set_a,X: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
% 0.29/1.53 => ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_286_compl__le__swap1,axiom,
% 0.29/1.53 ! [Y: set_real,X: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ Y @ ( uminus773214379t_real @ X ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ X @ ( uminus773214379t_real @ Y ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_287_compl__mono,axiom,
% 0.29/1.53 ! [X: set_a,Y: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_288_compl__mono,axiom,
% 0.29/1.53 ! [X: set_real,Y: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ X @ Y )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ ( uminus773214379t_real @ X ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_289_verit__negate__coefficient_I2_J,axiom,
% 0.29/1.53 ! [A: a,B: a] :
% 0.29/1.53 ( ( ord_less_a @ A @ B )
% 0.29/1.53 => ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_290_verit__negate__coefficient_I2_J,axiom,
% 0.29/1.53 ! [A: real,B: real] :
% 0.29/1.53 ( ( ord_less_real @ A @ B )
% 0.29/1.53 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_291_fun__Compl__def,axiom,
% 0.29/1.53 ( uminus_uminus_a_a
% 0.29/1.53 = ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_292_double__diff,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.29/1.53 => ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
% 0.29/1.53 = A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_293_double__diff,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.29/1.53 => ( ( minus_minus_set_real @ B3 @ ( minus_minus_set_real @ C3 @ A2 ) )
% 0.29/1.53 = A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_294_Diff__subset,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_295_Diff__subset,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ A2 ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_296_Diff__mono,axiom,
% 0.29/1.53 ! [A2: set_a,C3: set_a,D4: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ C3 )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ D4 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D4 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_297_Diff__mono,axiom,
% 0.29/1.53 ! [A2: set_real,C3: set_real,D4: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ C3 )
% 0.29/1.53 => ( ( ord_less_eq_set_real @ D4 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ ( minus_minus_set_real @ C3 @ D4 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_298_subset__iff__psubset__eq,axiom,
% 0.29/1.53 ( ord_less_eq_set_a
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ( ( ord_less_set_a @ A5 @ B5 )
% 0.29/1.53 | ( A5 = B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_299_subset__iff__psubset__eq,axiom,
% 0.29/1.53 ( ord_less_eq_set_real
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ( ( ord_less_set_real @ A5 @ B5 )
% 0.29/1.53 | ( A5 = B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_300_subset__psubset__trans,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_set_a @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_301_subset__psubset__trans,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_set_real @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_302_subset__not__subset__eq,axiom,
% 0.29/1.53 ( ord_less_set_a
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.29/1.53 & ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_303_subset__not__subset__eq,axiom,
% 0.29/1.53 ( ord_less_set_real
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.29/1.53 & ~ ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_304_psubset__subset__trans,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a,C3: set_a] :
% 0.29/1.53 ( ( ord_less_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_a @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_305_psubset__subset__trans,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real,C3: set_real] :
% 0.29/1.53 ( ( ord_less_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ( ord_less_eq_set_real @ B3 @ C3 )
% 0.29/1.53 => ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_306_psubset__imp__subset,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_307_psubset__imp__subset,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_308_psubset__eq,axiom,
% 0.29/1.53 ( ord_less_set_a
% 0.29/1.53 = ( ^ [A5: set_a,B5: set_a] :
% 0.29/1.53 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.29/1.53 & ( A5 != B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_309_psubset__eq,axiom,
% 0.29/1.53 ( ord_less_set_real
% 0.29/1.53 = ( ^ [A5: set_real,B5: set_real] :
% 0.29/1.53 ( ( ord_less_eq_set_real @ A5 @ B5 )
% 0.29/1.53 & ( A5 != B5 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_310_psubsetE,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_set_a @ A2 @ B3 )
% 0.29/1.53 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_311_psubsetE,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_set_real @ A2 @ B3 )
% 0.29/1.53 => ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.29/1.53 => ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_312_psubset__imp__ex__mem,axiom,
% 0.29/1.53 ! [A2: set_real,B3: set_real] :
% 0.29/1.53 ( ( ord_less_set_real @ A2 @ B3 )
% 0.29/1.53 => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_313_psubset__imp__ex__mem,axiom,
% 0.29/1.53 ! [A2: set_a,B3: set_a] :
% 0.29/1.53 ( ( ord_less_set_a @ A2 @ B3 )
% 0.29/1.53 => ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_314_rev_Oopen__existence__ivl_H,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ~ ! [A4: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ A4 )
% 0.29/1.53 => ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_315_open__existence__ivl_H,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ~ ! [A4: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ A4 )
% 0.29/1.53 => ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_316_rev_Oopen__existence__ivl0,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ? [A4: real,B4: real] :
% 0.29/1.53 ( ( ord_less_real @ A4 @ zero_zero_real )
% 0.29/1.53 & ( ord_less_real @ zero_zero_real @ B4 )
% 0.29/1.53 & ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_317_existence__ivl__zero,axiom,
% 0.29/1.53 ! [X0: a] :
% 0.29/1.53 ( ( member_a @ X0 @ x )
% 0.29/1.53 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_318_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/1.53 ! [T2: real,T0: real,X0: a] :
% 0.29/1.53 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( member_a @ X0 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_319_Compl__iff,axiom,
% 0.29/1.53 ! [C: a,A2: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
% 0.29/1.53 = ( ~ ( member_a @ C @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_320_Compl__iff,axiom,
% 0.29/1.53 ! [C: real,A2: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
% 0.29/1.53 = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_321_ComplI,axiom,
% 0.29/1.53 ! [C: a,A2: set_a] :
% 0.29/1.53 ( ~ ( member_a @ C @ A2 )
% 0.29/1.53 => ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_322_ComplI,axiom,
% 0.29/1.53 ! [C: real,A2: set_real] :
% 0.29/1.53 ( ~ ( member_real @ C @ A2 )
% 0.29/1.53 => ( member_real @ C @ ( uminus773214379t_real @ A2 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_323_rev_Oexistence__ivl__zero,axiom,
% 0.29/1.53 ! [X0: a] :
% 0.29/1.53 ( ( member_a @ X0 @ x )
% 0.29/1.53 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_324_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/1.53 ! [T2: real,T0: real,X0: a] :
% 0.29/1.53 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.29/1.53 => ( member_a @ X0 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_325_local_Oivl__subset__existence__ivl,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_326_local_Oivl__subset__existence__ivl_H,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_327_open__existence__ivl0,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ? [A4: real,B4: real] :
% 0.29/1.53 ( ( ord_less_real @ A4 @ zero_zero_real )
% 0.29/1.53 & ( ord_less_real @ zero_zero_real @ B4 )
% 0.29/1.53 & ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_328_local_Orev_Oivl__subset__existence__ivl,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_329_local_Orev_Oivl__subset__existence__ivl_H,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.29/1.53 => ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_330_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( member_a @ X0 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_331_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.29/1.53 => ( member_a @ X0 @ x ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_332_ComplD,axiom,
% 0.29/1.53 ! [C: a,A2: set_a] :
% 0.29/1.53 ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
% 0.29/1.53 => ~ ( member_a @ C @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_333_ComplD,axiom,
% 0.29/1.53 ! [C: real,A2: set_real] :
% 0.29/1.53 ( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
% 0.29/1.53 => ~ ( member_real @ C @ A2 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_334_rev_Omvar_Ointerval__axioms,axiom,
% 0.29/1.53 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_335_mvar_Ointerval__axioms,axiom,
% 0.29/1.53 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_336_rev_Oclosed__orbit__global__existence,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_337_rev_Ofixpoint__sol_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.29/1.53 = top_top_set_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_338_interval__axioms,axiom,
% 0.29/1.53 initia826609931terval @ top_top_set_real ).
% 0.29/1.53
% 0.29/1.53 thf(fact_339_UNIV__I,axiom,
% 0.29/1.53 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_340_UNIV__I,axiom,
% 0.29/1.53 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_341_local_Omem__existence__ivl__subset,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_342_local_Oexistence__ivl__initial__time,axiom,
% 0.29/1.53 ! [X0: a] :
% 0.29/1.53 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.29/1.53 => ( ( member_a @ X0 @ x )
% 0.29/1.53 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_343_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.29/1.53 ! [T2: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.29/1.53 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_344_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.29/1.53 ! [T2: real,T0: real,X0: a] :
% 0.29/1.53 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( member_real @ T0 @ top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_345_general_Oexistence__ivl__initial__time,axiom,
% 0.29/1.53 ! [T0: real,X0: a] :
% 0.29/1.53 ( ( member_real @ T0 @ top_top_set_real )
% 0.29/1.53 => ( ( member_a @ X0 @ x )
% 0.29/1.53 => ( member_real @ ( minus_minus_real @ T0 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_346_mem__existence__ivl__shift__autonomous2,axiom,
% 0.29/1.53 ! [T2: real,S2: real,X: a] :
% 0.29/1.53 ( ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.29/1.53 => ( ( member_a @ X @ x )
% 0.29/1.53 => ( ! [S: real,T3: real,X4: a] :
% 0.29/1.53 ( ( member_a @ X4 @ x )
% 0.29/1.53 => ( ( f @ X4 )
% 0.29/1.53 = ( f @ X4 ) ) )
% 0.29/1.53 => ( ( top_top_set_real = top_top_set_real )
% 0.29/1.53 => ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_347_general_Omem__existence__ivl__subset,axiom,
% 0.29/1.53 ! [T2: real,T0: real,X0: a] :
% 0.29/1.53 ( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.29/1.53 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_348_local_Oexistence__ivl__subset,axiom,
% 0.29/1.53 ! [X0: a] : ( ord_less_eq_set_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) @ top_top_set_real ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_349_fixpoint__sol_I1_J,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( member_a @ X @ x )
% 0.29/1.53 => ( ( ( f @ X )
% 0.29/1.53 = zero_zero_a )
% 0.29/1.53 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.29/1.53 = top_top_set_real ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_350_closed__orbit__global__existence,axiom,
% 0.29/1.53 ! [X: a] :
% 0.29/1.53 ( ( period720806154rbit_a @ f @ x @ X )
% 0.29/1.53 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.29/1.53 = top_top_set_real ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(fact_351_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.29/1.53 ! [X0: a] :
% 0.29/1.53 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.29/1.53 => ( ( member_a @ X0 @ x )
% 0.29/1.53 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.29/1.53
% 0.29/1.53 thf(conj_0,conjecture,
% 0.29/1.53 ? [D3: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.29/1.53 & ? [B7: real] :
% 0.29/1.53 ( ( ord_less_real @ zero_zero_real @ B7 )
% 0.29/1.53 & ! [X4: a] :
% 0.29/1.53 ( ( member_a @ X4 @ ( elemen49976720ball_a @ x2 @ D3 ) )
% 0.29/1.53 => ( ord_less_eq_real @ B7 @ ( inner_1173012732nner_a @ ( f @ X4 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ) ).
% 0.29/1.53 ------------------------
% 0.29/1.55 % (11784)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_11562 for (2999ds/27Mi)
% 0.29/1.55 % (11783)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_11562 for (2999ds/4Mi)
% 0.29/1.56 % (11783)Instruction limit reached!
% 0.29/1.56 % (11783)------------------------------
% 0.29/1.56 % (11783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.56 % (11783)Termination reason: Unknown
% 0.29/1.56 % (11783)Termination phase: shuffling
% 0.29/1.56
% 0.29/1.56 % (11783)Memory used [KB]: 1535
% 0.29/1.56 % (11783)Time elapsed: 0.003 s
% 0.29/1.56 % (11783)Instructions burned: 7 (million)
% 0.29/1.56 % (11783)------------------------------
% 0.29/1.56 % (11783)------------------------------
% 0.29/1.56 % (11782)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/183Mi)
% 0.29/1.56 % (11785)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/2Mi)
% 0.29/1.56 % (11786)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_11562 for (2999ds/2Mi)
% 0.29/1.56 % (11788)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/18Mi)
% 0.29/1.56 % (11787)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_11562 for (2999ds/275Mi)
% 0.29/1.56 % (11784)Instruction limit reached!
% 0.29/1.56 % (11784)------------------------------
% 0.29/1.56 % (11784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.56 % (11784)Termination reason: Unknown
% 0.29/1.56 % (11784)Termination phase: Property scanning
% 0.29/1.56
% 0.29/1.56 % (11784)Memory used [KB]: 1663
% 0.29/1.56 % (11784)Time elapsed: 0.011 s
% 0.29/1.56 % (11784)Instructions burned: 28 (million)
% 0.29/1.56 % (11784)------------------------------
% 0.29/1.56 % (11784)------------------------------
% 0.29/1.56 % (11785)Instruction limit reached!
% 0.29/1.56 % (11785)------------------------------
% 0.29/1.56 % (11785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.56 % (11785)Termination reason: Unknown
% 0.29/1.56 % (11785)Termination phase: shuffling
% 0.29/1.56
% 0.29/1.56 % (11785)Memory used [KB]: 1407
% 0.29/1.56 % (11785)Time elapsed: 0.004 s
% 0.29/1.56 % (11785)Instructions burned: 2 (million)
% 0.29/1.56 % (11785)------------------------------
% 0.29/1.56 % (11785)------------------------------
% 0.29/1.56 % (11786)Instruction limit reached!
% 0.29/1.56 % (11786)------------------------------
% 0.29/1.56 % (11786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.56 % (11786)Termination reason: Unknown
% 0.29/1.56 % (11786)Termination phase: shuffling
% 0.29/1.56
% 0.29/1.56 % (11786)Memory used [KB]: 1407
% 0.29/1.56 % (11786)Time elapsed: 0.004 s
% 0.29/1.56 % (11786)Instructions burned: 3 (million)
% 0.29/1.56 % (11786)------------------------------
% 0.29/1.56 % (11786)------------------------------
% 0.29/1.57 % (11789)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/3Mi)
% 0.29/1.57 % (11789)Instruction limit reached!
% 0.29/1.57 % (11789)------------------------------
% 0.29/1.57 % (11789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.57 % (11789)Termination reason: Unknown
% 0.29/1.57 % (11789)Termination phase: shuffling
% 0.29/1.57
% 0.29/1.57 % (11789)Memory used [KB]: 1407
% 0.29/1.57 % (11789)Time elapsed: 0.003 s
% 0.29/1.57 % (11789)Instructions burned: 6 (million)
% 0.29/1.57 % (11789)------------------------------
% 0.29/1.57 % (11789)------------------------------
% 0.29/1.57 % (11790)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_11562 for (2999ds/37Mi)
% 0.29/1.57 % (11788)Instruction limit reached!
% 0.29/1.57 % (11788)------------------------------
% 0.29/1.57 % (11788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.57 % (11788)Termination reason: Unknown
% 0.29/1.57 % (11788)Termination phase: shuffling
% 0.29/1.57
% 0.29/1.57 % (11788)Memory used [KB]: 1535
% 0.29/1.57 % (11788)Time elapsed: 0.014 s
% 0.29/1.57 % (11788)Instructions burned: 19 (million)
% 0.29/1.57 % (11788)------------------------------
% 0.29/1.57 % (11788)------------------------------
% 0.29/1.58 % (11793)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_11562 for (2999ds/1041Mi)
% 0.29/1.58 % (11791)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_11562 for (2999ds/15Mi)
% 0.29/1.58 % (11792)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_11562 for (2999ds/3Mi)
% 0.29/1.58 % (11790)Instruction limit reached!
% 0.29/1.58 % (11790)------------------------------
% 0.29/1.58 % (11790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.58 % (11790)Termination reason: Unknown
% 0.29/1.58 % (11790)Termination phase: shuffling
% 0.29/1.58
% 0.29/1.58 % (11790)Memory used [KB]: 2046
% 0.29/1.58 % (11790)Time elapsed: 0.011 s
% 0.29/1.58 % (11790)Instructions burned: 37 (million)
% 0.29/1.58 % (11790)------------------------------
% 0.29/1.58 % (11790)------------------------------
% 0.29/1.58 % (11792)Instruction limit reached!
% 0.29/1.58 % (11792)------------------------------
% 0.29/1.58 % (11792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.58 % (11792)Termination reason: Unknown
% 0.29/1.58 % (11792)Termination phase: shuffling
% 0.29/1.58
% 0.29/1.58 % (11792)Memory used [KB]: 1407
% 0.29/1.58 % (11792)Time elapsed: 0.004 s
% 0.29/1.58 % (11792)Instructions burned: 3 (million)
% 0.29/1.58 % (11792)------------------------------
% 0.29/1.58 % (11792)------------------------------
% 0.29/1.59 % (11791)Instruction limit reached!
% 0.29/1.59 % (11791)------------------------------
% 0.29/1.59 % (11791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.59 % (11791)Termination reason: Unknown
% 0.29/1.59 % (11791)Termination phase: shuffling
% 0.29/1.59
% 0.29/1.59 % (11791)Memory used [KB]: 1535
% 0.29/1.59 % (11791)Time elapsed: 0.011 s
% 0.29/1.59 % (11791)Instructions burned: 15 (million)
% 0.29/1.59 % (11791)------------------------------
% 0.29/1.59 % (11791)------------------------------
% 0.29/1.59 % (11794)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/7Mi)
% 0.29/1.59 % (11795)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/16Mi)
% 0.29/1.60 % (11794)Instruction limit reached!
% 0.29/1.60 % (11794)------------------------------
% 0.29/1.60 % (11794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.60 % (11794)Termination reason: Unknown
% 0.29/1.60 % (11794)Termination phase: shuffling
% 0.29/1.60
% 0.29/1.60 % (11794)Memory used [KB]: 1407
% 0.29/1.60 % (11794)Time elapsed: 0.006 s
% 0.29/1.60 % (11794)Instructions burned: 7 (million)
% 0.29/1.60 % (11794)------------------------------
% 0.29/1.60 % (11794)------------------------------
% 2.72/1.60 % (11795)Instruction limit reached!
% 2.72/1.60 % (11795)------------------------------
% 2.72/1.60 % (11795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (11795)Termination reason: Unknown
% 2.72/1.60 % (11795)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (11795)Memory used [KB]: 1663
% 2.72/1.60 % (11795)Time elapsed: 0.007 s
% 2.72/1.60 % (11795)Instructions burned: 18 (million)
% 2.72/1.60 % (11795)------------------------------
% 2.72/1.60 % (11795)------------------------------
% 2.72/1.60 % (11796)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_11562 for (2999ds/3Mi)
% 2.72/1.60 % (11796)Instruction limit reached!
% 2.72/1.60 % (11796)------------------------------
% 2.72/1.60 % (11796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (11796)Termination reason: Unknown
% 2.72/1.60 % (11796)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (11796)Memory used [KB]: 1407
% 2.72/1.60 % (11796)Time elapsed: 0.003 s
% 2.72/1.60 % (11796)Instructions burned: 3 (million)
% 2.72/1.60 % (11796)------------------------------
% 2.72/1.60 % (11796)------------------------------
% 2.72/1.61 % (11799)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/3Mi)
% 2.72/1.61 % (11797)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_11562 for (2999ds/3Mi)
% 2.72/1.61 % (11799)Instruction limit reached!
% 2.72/1.61 % (11799)------------------------------
% 2.72/1.61 % (11799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (11799)Termination reason: Unknown
% 2.72/1.61 % (11799)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (11799)Memory used [KB]: 1407
% 2.72/1.61 % (11799)Time elapsed: 0.002 s
% 2.72/1.61 % (11799)Instructions burned: 4 (million)
% 2.72/1.61 % (11799)------------------------------
% 2.72/1.61 % (11799)------------------------------
% 2.72/1.61 % (11797)Instruction limit reached!
% 2.72/1.61 % (11797)------------------------------
% 2.72/1.61 % (11797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (11797)Termination reason: Unknown
% 2.72/1.61 % (11797)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (11797)Memory used [KB]: 1407
% 2.72/1.61 % (11797)Time elapsed: 0.004 s
% 2.72/1.61 % (11797)Instructions burned: 3 (million)
% 2.72/1.61 % (11797)------------------------------
% 2.72/1.61 % (11797)------------------------------
% 2.72/1.61 % (11798)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/7Mi)
% 2.72/1.62 % (11798)Instruction limit reached!
% 2.72/1.62 % (11798)------------------------------
% 2.72/1.62 % (11798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (11798)Termination reason: Unknown
% 2.72/1.62 % (11798)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (11798)Memory used [KB]: 1407
% 2.72/1.62 % (11798)Time elapsed: 0.006 s
% 2.72/1.62 % (11798)Instructions burned: 9 (million)
% 2.72/1.62 % (11798)------------------------------
% 2.72/1.62 % (11798)------------------------------
% 2.72/1.62 % (11801)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/18Mi)
% 2.72/1.62 % (11800)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/4Mi)
% 2.72/1.62 % (11787)Refutation not found, incomplete strategy
% 2.72/1.62 % (11787)------------------------------
% 2.72/1.62 % (11787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (11787)Termination reason: Refutation not found, incomplete strategy
% 2.72/1.62
% 2.72/1.62
% 2.72/1.62 % (11787)Memory used [KB]: 6524
% 2.72/1.62 % (11787)Time elapsed: 0.061 s
% 2.72/1.62 % (11787)Instructions burned: 98 (million)
% 2.72/1.62 % (11787)------------------------------
% 2.72/1.62 % (11787)------------------------------
% 2.72/1.62 % (11800)Instruction limit reached!
% 2.72/1.62 % (11800)------------------------------
% 2.72/1.62 % (11800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (11800)Termination reason: Unknown
% 2.72/1.62 % (11800)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (11800)Memory used [KB]: 1407
% 2.72/1.62 % (11800)Time elapsed: 0.005 s
% 2.72/1.62 % (11800)Instructions burned: 5 (million)
% 2.72/1.62 % (11800)------------------------------
% 2.72/1.62 % (11800)------------------------------
% 2.72/1.62 % (11801)Instruction limit reached!
% 2.72/1.62 % (11801)------------------------------
% 2.72/1.62 % (11801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (11801)Termination reason: Unknown
% 2.72/1.62 % (11801)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (11801)Memory used [KB]: 1663
% 2.72/1.62 % (11801)Time elapsed: 0.007 s
% 2.72/1.62 % (11801)Instructions burned: 19 (million)
% 2.72/1.62 % (11801)------------------------------
% 2.72/1.62 % (11801)------------------------------
% 2.72/1.63 % (11803)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/6Mi)
% 2.72/1.63 % (11802)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/710Mi)
% 2.72/1.63 % (11803)Instruction limit reached!
% 2.72/1.63 % (11803)------------------------------
% 2.72/1.63 % (11803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (11803)Termination reason: Unknown
% 2.72/1.63 % (11803)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (11803)Memory used [KB]: 1407
% 2.72/1.63 % (11803)Time elapsed: 0.004 s
% 2.72/1.63 % (11803)Instructions burned: 8 (million)
% 2.72/1.63 % (11803)------------------------------
% 2.72/1.63 % (11803)------------------------------
% 2.72/1.63 % (11806)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/5Mi)
% 2.72/1.64 % (11806)Instruction limit reached!
% 2.72/1.64 % (11806)------------------------------
% 2.72/1.64 % (11806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (11806)Termination reason: Unknown
% 2.72/1.64 % (11806)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (11806)Memory used [KB]: 1535
% 2.72/1.64 % (11806)Time elapsed: 0.003 s
% 2.72/1.64 % (11806)Instructions burned: 7 (million)
% 2.72/1.64 % (11806)------------------------------
% 2.72/1.64 % (11806)------------------------------
% 2.72/1.64 % (11807)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_11562 for (2999ds/6Mi)
% 2.72/1.64 % (11804)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_11562 for (2999ds/902Mi)
% 2.72/1.64 % (11807)Instruction limit reached!
% 2.72/1.64 % (11807)------------------------------
% 2.72/1.64 % (11807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (11807)Termination reason: Unknown
% 2.72/1.64 % (11807)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (11807)Memory used [KB]: 1535
% 2.72/1.64 % (11807)Time elapsed: 0.003 s
% 2.72/1.64 % (11807)Instructions burned: 6 (million)
% 2.72/1.64 % (11807)------------------------------
% 2.72/1.64 % (11807)------------------------------
% 2.72/1.64 % (11805)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_11562 for (2999ds/21Mi)
% 2.72/1.64 % (11808)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11562 for (2999ds/377Mi)
% 2.72/1.65 % (11809)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_11562 for (2999ds/779Mi)
% 2.72/1.65 % (11805)Instruction limit reached!
% 2.72/1.65 % (11805)------------------------------
% 2.72/1.65 % (11805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (11805)Termination reason: Unknown
% 2.72/1.65 % (11805)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (11805)Memory used [KB]: 1663
% 2.72/1.65 % (11805)Time elapsed: 0.015 s
% 2.72/1.65 % (11805)Instructions burned: 22 (million)
% 2.72/1.65 % (11805)------------------------------
% 2.72/1.65 % (11805)------------------------------
% 2.72/1.66 % (11782)Instruction limit reached!
% 2.72/1.66 % (11782)------------------------------
% 2.72/1.66 % (11782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (11782)Termination reason: Unknown
% 2.72/1.66 % (11782)Termination phase: Saturation
% 2.72/1.66
% 2.72/1.66 % (11782)Memory used [KB]: 7164
% 2.72/1.66 % (11782)Time elapsed: 0.099 s
% 2.72/1.66 % (11782)Instructions burned: 184 (million)
% 2.72/1.66 % (11782)------------------------------
% 2.72/1.66 % (11782)------------------------------
% 2.72/1.67 % (11810)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_11562 for (2998ds/19Mi)
% 2.72/1.67 % (11811)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11562 for (2998ds/879Mi)
% 2.72/1.67 % (11810)Instruction limit reached!
% 2.72/1.67 % (11810)------------------------------
% 2.72/1.67 % (11810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.67 % (11810)Termination reason: Unknown
% 2.72/1.67 % (11810)Termination phase: shuffling
% 2.72/1.67
% 2.72/1.67 % (11810)Memory used [KB]: 1791
% 2.72/1.67 % (11810)Time elapsed: 0.008 s
% 2.72/1.67 % (11810)Instructions burned: 21 (million)
% 2.72/1.67 % (11810)------------------------------
% 2.72/1.67 % (11810)------------------------------
% 2.72/1.68 % (11812)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11562 for (2998ds/17Mi)
% 2.72/1.69 % (11812)Instruction limit reached!
% 2.72/1.69 % (11812)------------------------------
% 2.72/1.69 % (11812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.69 % (11812)Termination reason: Unknown
% 2.72/1.69 % (11812)Termination phase: shuffling
% 2.72/1.69
% 2.72/1.69 % (11812)Memory used [KB]: 1663
% 2.72/1.69 % (11812)Time elapsed: 0.008 s
% 2.72/1.69 % (11812)Instructions burned: 18 (million)
% 2.72/1.69 % (11812)------------------------------
% 2.72/1.69 % (11812)------------------------------
% 2.72/1.70 % (11813)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11562 for (2998ds/3Mi)
% 2.72/1.70 % (11813)Instruction limit reached!
% 2.72/1.70 % (11813)------------------------------
% 2.72/1.70 % (11813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.70 % (11813)Termination reason: Unknown
% 2.72/1.70 % (11813)Termination phase: shuffling
% 2.72/1.70
% 2.72/1.70 % (11813)Memory used [KB]: 1407
% 2.72/1.70 % (11813)Time elapsed: 0.002 s
% 2.72/1.70 % (11813)Instructions burned: 3 (million)
% 2.72/1.70 % (11813)------------------------------
% 2.72/1.70 % (11813)------------------------------
% 2.72/1.71 % (11804)First to succeed.
% 2.72/1.71 % (11814)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11562 for (2998ds/30Mi)
% 2.72/1.71 % (11804)Refutation found. Thanks to Tanya!
% 2.72/1.71 % SZS status Theorem for DTF2THF_11562
% 2.72/1.71 % SZS output start Proof for DTF2THF_11562
% 2.72/1.71 thf(type_def_5, type, real: $tType).
% 2.72/1.71 thf(type_def_6, type, set_real: $tType).
% 2.72/1.71 thf(type_def_8, type, a: $tType).
% 2.72/1.71 thf(type_def_9, type, set_a: $tType).
% 2.72/1.71 thf(type_def_10, type, set_set_real: $tType).
% 2.72/1.71 thf(type_def_11, type, set_set_a: $tType).
% 2.72/1.71 thf(type_def_12, type, bounde7994401a_real: $tType).
% 2.72/1.71 thf(func_def_0, type, bounde7994401a_real: $tType).
% 2.72/1.71 thf(func_def_1, type, set_set_real: $tType).
% 2.72/1.71 thf(func_def_2, type, set_set_a: $tType).
% 2.72/1.71 thf(func_def_3, type, set_real: $tType).
% 2.72/1.71 thf(func_def_4, type, set_a: $tType).
% 2.72/1.71 thf(func_def_5, type, real: $tType).
% 2.72/1.71 thf(func_def_6, type, a: $tType).
% 2.72/1.71 thf(func_def_7, type, elemen1140313242l_real: real > real > set_real).
% 2.72/1.71 thf(func_def_8, type, elemen49976720ball_a: a > real > set_a).
% 2.72/1.71 thf(func_def_9, type, auto_l612940ivl0_a: (a > a) > set_a > a > set_real).
% 2.72/1.71 thf(func_def_10, type, minus_minus_real: real > real > real).
% 2.72/1.71 thf(func_def_11, type, minus_minus_set_real: set_real > set_real > set_real).
% 2.72/1.71 thf(func_def_12, type, minus_minus_set_a: set_a > set_a > set_a).
% 2.72/1.71 thf(func_def_13, type, minus_minus_a: a > a > a).
% 2.72/1.71 thf(func_def_14, type, uminus_uminus_a_a: (a > a) > a > a).
% 2.72/1.71 thf(func_def_15, type, uminus_uminus_real: real > real).
% 2.72/1.71 thf(func_def_16, type, uminus773214379t_real: set_real > set_real).
% 2.72/1.71 thf(func_def_17, type, uminus_uminus_set_a: set_a > set_a).
% 2.72/1.71 thf(func_def_18, type, uminus_uminus_a: a > a).
% 2.72/1.71 thf(func_def_19, type, zero_zero_real: real).
% 2.72/1.71 thf(func_def_20, type, zero_zero_a: a).
% 2.72/1.71 thf(func_def_21, type, initia826609931terval: set_real > $o).
% 2.72/1.71 thf(func_def_22, type, inner_4346926r_real: real > real > real).
% 2.72/1.71 thf(func_def_23, type, inner_1173012732nner_a: a > a > real).
% 2.72/1.71 thf(func_def_24, type, auto_l630715367iant_a: (a > a) > set_a > set_a > $o).
% 2.72/1.71 thf(func_def_25, type, line_c1152468841ment_a: a > a > set_a).
% 2.72/1.71 thf(func_def_26, type, line_open_segment_a: a > a > set_a).
% 2.72/1.71 thf(func_def_27, type, oDE_au1996717075pped_a: (a > a) > set_a > a > set_a > $o).
% 2.72/1.71 thf(func_def_28, type, ord_less_real: real > real > $o).
% 2.72/1.71 thf(func_def_29, type, ord_less_set_real: set_real > set_real > $o).
% 2.72/1.71 thf(func_def_30, type, ord_le861030508t_real: set_set_real > set_set_real > $o).
% 2.72/1.71 thf(func_def_31, type, ord_less_set_set_a: set_set_a > set_set_a > $o).
% 2.72/1.71 thf(func_def_32, type, ord_less_set_a: set_a > set_a > $o).
% 2.72/1.71 thf(func_def_33, type, ord_less_a: a > a > $o).
% 2.72/1.71 thf(func_def_34, type, ord_less_eq_real: real > real > $o).
% 2.72/1.71 thf(func_def_35, type, ord_less_eq_set_real: set_real > set_real > $o).
% 2.72/1.71 thf(func_def_36, type, ord_le1733505144t_real: set_set_real > set_set_real > $o).
% 2.72/1.71 thf(func_def_37, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 2.72/1.71 thf(func_def_38, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.72/1.71 thf(func_def_39, type, ord_less_eq_a: a > a > $o).
% 2.72/1.71 thf(func_def_40, type, top_top_set_real: set_real).
% 2.72/1.71 thf(func_def_41, type, top_top_set_a: set_a).
% 2.72/1.71 thf(func_def_42, type, period720806154rbit_a: (a > a) > set_a > a > $o).
% 2.72/1.71 thf(func_def_43, type, period1305449585riod_a: (a > a) > set_a > a > real).
% 2.72/1.71 thf(func_def_44, type, period138238489rbit_a: (a > a) > set_a > a > $o).
% 2.72/1.71 thf(func_def_45, type, poinca659159244_rot_a: a > a).
% 2.72/1.71 thf(func_def_46, type, poinca522724647ment_a: (a > a) > set_a > a > a > $o).
% 2.72/1.71 thf(func_def_47, type, reacha1191408304tion_a: (a > a) > set_a > (a > real) > (a > bounde7994401a_real) > set_a > $o).
% 2.72/1.71 thf(func_def_48, type, real_V1934908667t_real: real > real > real).
% 2.72/1.71 thf(func_def_49, type, real_V1514887919dist_a: a > a > real).
% 2.72/1.71 thf(func_def_50, type, collect_real: (real > $o) > set_real).
% 2.72/1.71 thf(func_def_51, type, collect_set_real: (set_real > $o) > set_set_real).
% 2.72/1.71 thf(func_def_52, type, collect_set_a: (set_a > $o) > set_set_a).
% 2.72/1.71 thf(func_def_53, type, collect_a: (a > $o) > set_a).
% 2.72/1.71 thf(func_def_54, type, set_or656347191t_real: real > real > set_real).
% 2.72/1.71 thf(func_def_55, type, set_or1619725421t_real: set_real > set_real > set_set_real).
% 2.72/1.71 thf(func_def_56, type, set_or2094724627_set_a: set_a > set_a > set_set_a).
% 2.72/1.71 thf(func_def_57, type, set_or411607219Most_a: a > a > set_a).
% 2.72/1.71 thf(func_def_58, type, member_real: real > set_real > $o).
% 2.72/1.71 thf(func_def_59, type, member_set_real: set_real > set_set_real > $o).
% 2.72/1.71 thf(func_def_60, type, member_set_a: set_a > set_set_a > $o).
% 2.72/1.71 thf(func_def_61, type, member_a: a > set_a > $o).
% 2.72/1.71 thf(func_def_62, type, x: set_a).
% 2.72/1.71 thf(func_def_63, type, a2: a).
% 2.72/1.71 thf(func_def_64, type, b: a).
% 2.72/1.71 thf(func_def_65, type, d: real).
% 2.72/1.71 thf(func_def_66, type, f: a > a).
% 2.72/1.71 thf(func_def_67, type, s: a).
% 2.72/1.71 thf(func_def_69, type, x2: a).
% 2.72/1.71 thf(func_def_70, type, z: a).
% 2.72/1.71 thf(func_def_84, type, sP0: real > (real > real > $o) > real > $o).
% 2.72/1.71 thf(func_def_85, type, sP1: a > a > $o).
% 2.72/1.71 thf(func_def_86, type, sK2: a).
% 2.72/1.71 thf(func_def_87, type, sK3: a > a).
% 2.72/1.71 thf(func_def_88, type, sK4: a > a).
% 2.72/1.71 thf(func_def_89, type, sK5: a > real).
% 2.72/1.71 thf(func_def_90, type, sK6: a > real).
% 2.72/1.71 thf(func_def_91, type, sK7: real > (real > real > $o) > real > real).
% 2.72/1.71 thf(func_def_92, type, sK8: real > real > (real > real > $o) > real > real).
% 2.72/1.71 thf(func_def_93, type, sK9: real > real > (real > real > $o) > real > real).
% 2.72/1.71 thf(func_def_94, type, sK10: (real > real > $o) > real).
% 2.72/1.71 thf(func_def_95, type, sK11: (real > real > $o) > real).
% 2.72/1.71 thf(func_def_96, type, sK12: (real > real > $o) > real).
% 2.72/1.71 thf(func_def_97, type, sK13: set_real > set_real > real).
% 2.72/1.71 thf(func_def_98, type, sK14: real > real).
% 2.72/1.71 thf(func_def_99, type, sK15: set_a > set_a > a).
% 2.72/1.71 thf(func_def_100, type, sK16: a > real).
% 2.72/1.71 thf(func_def_101, type, sK17: real > real > a).
% 2.72/1.71 thf(func_def_102, type, sK18: (real > $o) > (real > $o) > real).
% 2.72/1.71 thf(func_def_103, type, sK19: (a > $o) > (a > $o) > a).
% 2.72/1.71 thf(func_def_104, type, sK20: a > a).
% 2.72/1.71 thf(func_def_105, type, sK21: a).
% 2.72/1.71 thf(func_def_106, type, sK22: real).
% 2.72/1.71 thf(func_def_107, type, sK23: a > real).
% 2.72/1.71 thf(func_def_108, type, sK24: set_a > a).
% 2.72/1.71 thf(func_def_109, type, sK25: (a > $o) > (a > $o) > a).
% 2.72/1.71 thf(func_def_110, type, sK26: set_real > set_real > real).
% 2.72/1.71 thf(func_def_111, type, sK27: a > a > a).
% 2.72/1.71 thf(func_def_112, type, sK28: real > real > a).
% 2.72/1.71 thf(func_def_113, type, sK29: real > real > a).
% 2.72/1.71 thf(func_def_114, type, sK30: (real > $o) > (real > $o) > real).
% 2.72/1.71 thf(func_def_115, type, sK31: a > real).
% 2.72/1.71 thf(func_def_116, type, sK32: a > real).
% 2.72/1.71 thf(func_def_117, type, sK33: set_a > a).
% 2.72/1.71 thf(func_def_118, type, sK34: real > a > a).
% 2.72/1.71 thf(func_def_119, type, sK35: set_a > set_a > a).
% 2.72/1.71 thf(func_def_121, type, ph37: !>[X0: $tType]:(X0)).
% 2.72/1.71 thf(f2177,plain,(
% 2.72/1.71 $false),
% 2.72/1.71 inference(avatar_sat_refutation,[status(thm)],[f1949,f2136,f2175])).
% 2.72/1.71 thf(f2175,plain,(
% 2.72/1.71 ~spl36_1 | ~spl36_2),
% 2.72/1.71 inference(avatar_split_clause,[status(thm)],[f2174,f1947,f1943])).
% 2.72/1.71 thf(f1943,definition,(
% 2.72/1.71 spl36_1 <=> ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true)),
% 2.72/1.71 introduced(definition,[new_symbols(naming,[spl36_1])],[avatar_definition])).
% 2.72/1.71 thf(f1947,definition,(
% 2.72/1.71 spl36_2 <=> ! [X0 : real] : (($true != (member_a @ (sK17 @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ X0) @ (elemen49976720ball_a @ x2 @ d))) | ((ord_less_real @ zero_zero_real @ X0) != $true))),
% 2.72/1.71 introduced(definition,[new_symbols(naming,[spl36_2])],[avatar_definition])).
% 2.72/1.71 thf(f2174,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ~spl36_2),
% 2.72/1.71 inference(trivial_inequality_removal,[status(thm)],[f2173])).
% 2.72/1.71 thf(f2173,plain,(
% 2.72/1.71 ($true != $true) | ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ~spl36_2),
% 2.72/1.71 inference(forward_demodulation,[status(thm)],[f2165,f1638])).
% 2.72/1.71 thf(f1638,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ d) = $true)),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f630])).
% 2.72/1.71 thf(f630,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ d) = $true)),
% 2.72/1.71 inference(fool_elimination,[status(thm)],[f629])).
% 2.72/1.71 thf(f629,plain,(
% 2.72/1.71 (ord_less_real @ zero_zero_real @ d)),
% 2.72/1.71 inference(rectify,[status(thm)],[f1])).
% 2.72/1.71 thf(f1,axiom,(
% 2.72/1.71 (ord_less_real @ zero_zero_real @ d)),
% 2.72/1.71 file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_0_d_I1_J)).
% 2.72/1.71 thf(f2165,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ((ord_less_real @ zero_zero_real @ d) != $true) | ~spl36_2),
% 2.72/1.71 inference(trivial_inequality_removal,[status(thm)],[f2164])).
% 2.72/1.71 thf(f2164,plain,(
% 2.72/1.71 ($true != $true) | ((ord_less_real @ zero_zero_real @ d) != $true) | ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ~spl36_2),
% 2.72/1.71 inference(duplicate_literal_removal,[status(thm)],[f2159])).
% 2.72/1.71 thf(f2159,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ((ord_less_real @ zero_zero_real @ d) != $true) | ((ord_less_real @ zero_zero_real @ d) != $true) | ($true != $true) | ~spl36_2),
% 2.72/1.71 inference(superposition,[status(thm)],[f1948,f1508])).
% 2.72/1.71 thf(f1508,plain,(
% 2.72/1.71 ( ! [X0 : real,X1 : real] : (((member_a @ (sK17 @ X1 @ X0) @ (elemen49976720ball_a @ x2 @ X0)) = $true) | ((ord_less_real @ zero_zero_real @ X1) != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true)) )),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f1248])).
% 2.72/1.71 thf(f1248,plain,(
% 2.72/1.71 ! [X0 : real] : (! [X1 : real] : ((($true != (ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ (sK17 @ X1 @ X0)) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))) & ((member_a @ (sK17 @ X1 @ X0) @ (elemen49976720ball_a @ x2 @ X0)) = $true)) | ((ord_less_real @ zero_zero_real @ X1) != $true)) | ((ord_less_real @ zero_zero_real @ X0) != $true))),
% 2.72/1.71 inference(skolemisation,[status(esa),new_symbols(skolem,[sK17])],[f1005,f1247])).
% 2.72/1.71 thf(f1247,plain,(
% 2.72/1.71 ! [X0 : real,X1 : real] : (? [X2 : a] : (((ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ X2) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) & ((member_a @ X2 @ (elemen49976720ball_a @ x2 @ X0)) = $true)) => (($true != (ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ (sK17 @ X1 @ X0)) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))) & ((member_a @ (sK17 @ X1 @ X0) @ (elemen49976720ball_a @ x2 @ X0)) = $true)))),
% 2.72/1.71 introduced(definition,[],[choice_axiom])).
% 2.72/1.71 thf(f1005,plain,(
% 2.72/1.71 ! [X0 : real] : (! [X1 : real] : (? [X2 : a] : (((ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ X2) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) & ((member_a @ X2 @ (elemen49976720ball_a @ x2 @ X0)) = $true)) | ((ord_less_real @ zero_zero_real @ X1) != $true)) | ((ord_less_real @ zero_zero_real @ X0) != $true))),
% 2.72/1.71 inference(ennf_transformation,[status(thm)],[f763])).
% 2.72/1.71 thf(f763,plain,(
% 2.72/1.71 ~? [X0 : real] : (? [X1 : real] : (! [X2 : a] : (((member_a @ X2 @ (elemen49976720ball_a @ x2 @ X0)) = $true) => ((ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ X2) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true)) & ((ord_less_real @ zero_zero_real @ X1) = $true)) & ((ord_less_real @ zero_zero_real @ X0) = $true))),
% 2.72/1.71 inference(fool_elimination,[status(thm)],[f762])).
% 2.72/1.71 thf(f762,plain,(
% 2.72/1.71 ~? [X0 : real] : (? [X1 : real] : ((ord_less_real @ zero_zero_real @ X1) & ! [X2 : a] : ((member_a @ X2 @ (elemen49976720ball_a @ x2 @ X0)) => (ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ X2) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))))) & (ord_less_real @ zero_zero_real @ X0))),
% 2.72/1.71 inference(rectify,[status(thm)],[f354])).
% 2.72/1.71 thf(f354,negated_conjecture,(
% 2.72/1.71 ~? [X35 : real] : (? [X56 : real] : ((ord_less_real @ zero_zero_real @ X56) & ! [X26 : a] : ((member_a @ X26 @ (elemen49976720ball_a @ x2 @ X35)) => (ord_less_eq_real @ X56 @ (inner_1173012732nner_a @ (f @ X26) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))))) & (ord_less_real @ zero_zero_real @ X35))),
% 2.72/1.71 inference(negated_conjecture,[status(cth)],[f353])).
% 2.72/1.71 thf(f353,conjecture,(
% 2.72/1.71 ? [X35 : real] : (? [X56 : real] : ((ord_less_real @ zero_zero_real @ X56) & ! [X26 : a] : ((member_a @ X26 @ (elemen49976720ball_a @ x2 @ X35)) => (ord_less_eq_real @ X56 @ (inner_1173012732nner_a @ (f @ X26) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))))) & (ord_less_real @ zero_zero_real @ X35))),
% 2.72/1.71 file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',conj_0)).
% 2.72/1.71 thf(f1948,plain,(
% 2.72/1.71 ( ! [X0 : real] : (($true != (member_a @ (sK17 @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ X0) @ (elemen49976720ball_a @ x2 @ d))) | ((ord_less_real @ zero_zero_real @ X0) != $true)) ) | ~spl36_2),
% 2.72/1.71 inference(avatar_component_clause,[status(thm)],[f1947])).
% 2.72/1.71 thf(f2136,plain,(
% 2.72/1.71 spl36_1),
% 2.72/1.71 inference(avatar_contradiction_clause,[status(thm)],[f2135])).
% 2.72/1.71 thf(f2135,plain,(
% 2.72/1.71 $false | spl36_1),
% 2.72/1.71 inference(trivial_inequality_removal,[status(thm)],[f2128])).
% 2.72/1.71 thf(f2128,plain,(
% 2.72/1.71 ($true != $true) | spl36_1),
% 2.72/1.71 inference(superposition,[status(thm)],[f1992,f1569])).
% 2.72/1.71 thf(f1569,plain,(
% 2.72/1.71 ((member_a @ sK21 @ (elemen49976720ball_a @ x2 @ d)) = $true)),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f1271])).
% 2.72/1.71 thf(f1271,plain,(
% 2.72/1.71 ((member_a @ sK21 @ (elemen49976720ball_a @ x2 @ d)) = $true) & ! [X1 : a] : (((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) != $true) | ((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true))),
% 2.72/1.71 inference(skolemisation,[status(esa),new_symbols(skolem,[sK21])],[f1193,f1270])).
% 2.72/1.71 thf(f1270,plain,(
% 2.72/1.71 ? [X0 : a] : (((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) = $true) & ! [X1 : a] : (((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) != $true) | ((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true))) => (((member_a @ sK21 @ (elemen49976720ball_a @ x2 @ d)) = $true) & ! [X1 : a] : (((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) != $true) | ((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true)))),
% 2.72/1.71 introduced(definition,[],[choice_axiom])).
% 2.72/1.71 thf(f1193,plain,(
% 2.72/1.71 ? [X0 : a] : (((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) = $true) & ! [X1 : a] : (((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) != $true) | ((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true)))),
% 2.72/1.71 inference(ennf_transformation,[status(thm)],[f735])).
% 2.72/1.71 thf(f735,plain,(
% 2.72/1.71 ~! [X0 : a] : (((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) = $true) => ~! [X1 : a] : (((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) = $true) => ((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true)))),
% 2.72/1.71 inference(fool_elimination,[status(thm)],[f734])).
% 2.72/1.71 thf(f734,plain,(
% 2.72/1.71 ~! [X0 : a] : ((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) => ~! [X1 : a] : ((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) => (ord_less_eq_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))))),
% 2.72/1.71 inference(rectify,[status(thm)],[f12])).
% 2.72/1.71 thf(f12,axiom,(
% 2.72/1.71 ~! [X3 : a] : ((member_a @ X3 @ (elemen49976720ball_a @ x2 @ d)) => ~! [X2 : a] : ((member_a @ X2 @ (elemen49976720ball_a @ x2 @ d)) => (ord_less_eq_real @ (inner_1173012732nner_a @ (f @ X3) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X2) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))))),
% 2.72/1.71 file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062)).
% 2.72/1.71 thf(f1992,plain,(
% 2.72/1.71 ((member_a @ sK21 @ (elemen49976720ball_a @ x2 @ d)) != $true) | spl36_1),
% 2.72/1.71 inference(trivial_inequality_removal,[status(thm)],[f1980])).
% 2.72/1.71 thf(f1980,plain,(
% 2.72/1.71 ($true != $true) | ((member_a @ sK21 @ (elemen49976720ball_a @ x2 @ d)) != $true) | spl36_1),
% 2.72/1.71 inference(superposition,[status(thm)],[f1945,f1670])).
% 2.72/1.71 thf(f1670,plain,(
% 2.72/1.71 ( ! [X0 : a] : (($true = (ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))) | ((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) != $true)) )),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f1177])).
% 2.72/1.71 thf(f1177,plain,(
% 2.72/1.71 ! [X0 : a] : (((zero_zero_a != (f @ X0)) & ((member_a @ X0 @ x) = $true) & ($true = (ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))))) | ((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) != $true))),
% 2.72/1.71 inference(ennf_transformation,[status(thm)],[f530])).
% 2.72/1.71 thf(f530,plain,(
% 2.72/1.71 ! [X0 : a] : (((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) = $true) => ((zero_zero_a != (f @ X0)) & ((member_a @ X0 @ x) = $true) & ($true = (ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))))))),
% 2.72/1.71 inference(fool_elimination,[status(thm)],[f529])).
% 2.72/1.71 thf(f529,plain,(
% 2.72/1.71 ! [X0 : a] : ((member_a @ X0 @ (elemen49976720ball_a @ x2 @ d)) => ((member_a @ X0 @ x) & (ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ X0) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) & (zero_zero_a != (f @ X0))))),
% 2.72/1.71 inference(rectify,[status(thm)],[f9])).
% 2.72/1.71 thf(f9,axiom,(
% 2.72/1.71 ! [X1 : a] : ((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) => ((member_a @ X1 @ x) & (ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) & (zero_zero_a != (f @ X1))))),
% 2.72/1.71 file('/export/starexec/sandbox2/tmp/tmp.mEou7nFUz1/DTF2THF_11562.p',fact_8_d_I2_J)).
% 2.72/1.71 thf(f1945,plain,(
% 2.72/1.71 ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | spl36_1),
% 2.72/1.71 inference(avatar_component_clause,[status(thm)],[f1943])).
% 2.72/1.71 thf(f1949,plain,(
% 2.72/1.71 ~spl36_1 | spl36_2),
% 2.72/1.71 inference(avatar_split_clause,[status(thm)],[f1941,f1947,f1943])).
% 2.72/1.71 thf(f1941,plain,(
% 2.72/1.71 ( ! [X0 : real] : (($true != (member_a @ (sK17 @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ X0) @ (elemen49976720ball_a @ x2 @ d))) | ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true)) )),
% 2.72/1.71 inference(trivial_inequality_removal,[status(thm)],[f1929])).
% 2.72/1.71 thf(f1929,plain,(
% 2.72/1.71 ( ! [X0 : real] : (($true != (member_a @ (sK17 @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ X0) @ (elemen49976720ball_a @ x2 @ d))) | ((ord_less_real @ zero_zero_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) != $true) | ($true != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true)) )),
% 2.72/1.71 inference(superposition,[status(thm)],[f1509,f1568])).
% 2.72/1.71 thf(f1568,plain,(
% 2.72/1.71 ( ! [X1 : a] : (((ord_less_eq_real @ (inner_1173012732nner_a @ (f @ sK21) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))) @ (inner_1173012732nner_a @ (f @ X1) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b)))) = $true) | ((member_a @ X1 @ (elemen49976720ball_a @ x2 @ d)) != $true)) )),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f1271])).
% 2.72/1.71 thf(f1509,plain,(
% 2.72/1.71 ( ! [X0 : real,X1 : real] : (($true != (ord_less_eq_real @ X1 @ (inner_1173012732nner_a @ (f @ (sK17 @ X1 @ X0)) @ (poinca659159244_rot_a @ (minus_minus_a @ a2 @ b))))) | ((ord_less_real @ zero_zero_real @ X1) != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true)) )),
% 2.72/1.71 inference(cnf_transformation,[status(thm)],[f1248])).
% 2.72/1.71 % SZS output end Proof for DTF2THF_11562
% 2.72/1.71 % (11804)------------------------------
% 2.72/1.71 % (11804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.71 % (11804)Termination reason: Refutation
% 2.72/1.71
% 2.72/1.71 % (11804)Memory used [KB]: 7291
% 2.72/1.71 % (11804)Time elapsed: 0.075 s
% 2.72/1.71 % (11804)Instructions burned: 189 (million)
% 2.72/1.71 % (11804)------------------------------
% 2.72/1.71 % (11804)------------------------------
% 2.72/1.71 % (11781)Success in time 0.166 s
%------------------------------------------------------------------------------