%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP147^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n009.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 08:55:55 AM UTC 2026
% Result : Theorem 3.40s 1.92s
% Output : Refutation 3.40s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 12
% Syntax : Number of formulae : 74 ( 12 unt; 0 typ; 5 def)
% Number of atoms : 885 ( 290 equ; 0 cnn)
% Maximal formula atoms : 16 ( 11 avg)
% Number of connectives : 1841 ( 231 ~; 219 |; 45 &;1295 @)
% ( 5 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 36 ( 36 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 60 usr; 20 con; 0-5 aty)
% Number of variables : 107 ( 0 ^; 91 !; 16 ?; 107 :)
% 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(func_def_0,type,
set_real: $tType ).
thf(func_def_1,type,
set_a: $tType ).
thf(func_def_2,type,
real: $tType ).
thf(func_def_3,type,
a: $tType ).
thf(func_def_4,type,
elemen341675809l_real: real > real > set_real ).
thf(func_def_5,type,
elemen154694473ball_a: a > real > set_a ).
thf(func_def_6,type,
auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
thf(func_def_7,type,
auto_ll_on_flow0_a: ( a > a ) > set_a > a > real > a ).
thf(func_def_8,type,
abs_abs_real: real > real ).
thf(func_def_9,type,
abs_abs_a: a > a ).
thf(func_def_10,type,
one_one_real: real ).
thf(func_def_11,type,
plus_plus_real: real > real > real ).
thf(func_def_12,type,
plus_plus_a: a > a > a ).
thf(func_def_13,type,
uminus_uminus_a_a: ( a > a ) > a > a ).
thf(func_def_14,type,
uminus_uminus_real: real > real ).
thf(func_def_15,type,
uminus_uminus_a: a > a ).
thf(func_def_16,type,
zero_zero_real: real ).
thf(func_def_17,type,
zero_zero_a: a ).
thf(func_def_18,type,
if_real: $o > real > real > real ).
thf(func_def_19,type,
initia826609931terval: set_real > $o ).
thf(func_def_20,type,
limit_1865182413oint_a: ( a > a ) > set_a > a > a > $o ).
thf(func_def_21,type,
limit_94460170oint_a: ( a > a ) > set_a > a > a > $o ).
thf(func_def_22,type,
line_open_segment_a: a > a > set_a ).
thf(func_def_23,type,
bot_bot_set_real: set_real ).
thf(func_def_24,type,
bot_bot_set_a: set_a ).
thf(func_def_25,type,
ord_less_real: real > real > $o ).
thf(func_def_26,type,
ord_less_a: a > a > $o ).
thf(func_def_27,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_28,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_29,type,
top_top_set_real: set_real ).
thf(func_def_30,type,
period720806154rbit_a: ( a > a ) > set_a > a > $o ).
thf(func_def_31,type,
period1305449585riod_a: ( a > a ) > set_a > a > real ).
thf(func_def_32,type,
period138238489rbit_a: ( a > a ) > set_a > a > $o ).
thf(func_def_33,type,
poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
thf(func_def_34,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_35,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_36,type,
topolo1710226732a_real: set_a > ( a > real ) > $o ).
thf(func_def_37,type,
member_real: real > set_real > $o ).
thf(func_def_38,type,
member_a: a > set_a > $o ).
thf(func_def_39,type,
x: set_a ).
thf(func_def_40,type,
a2: a ).
thf(func_def_41,type,
b: a ).
thf(func_def_42,type,
f: a > a ).
thf(func_def_43,type,
p: a ).
thf(func_def_45,type,
x2: a ).
thf(func_def_52,type,
sK0: real ).
thf(func_def_53,type,
sK1: a > real ).
thf(func_def_54,type,
sK2: ( a > real ) > real > a ).
thf(func_def_55,type,
sK3: ( a > real ) > real > a ).
thf(func_def_56,type,
sK4: ( a > real ) > real > a ).
thf(func_def_57,type,
sK5: ( a > real ) > real > a ).
thf(func_def_58,type,
sK6: real > a > a > a > real ).
thf(func_def_59,type,
sK7: real > a > a > a > a > real ).
thf(func_def_60,type,
sK8: a > a ).
thf(func_def_61,type,
sK9: a > a ).
thf(func_def_62,type,
sK10: a > a ).
thf(func_def_63,type,
sK11: a > a ).
thf(func_def_65,type,
ph13:
!>[X0: $tType] : X0 ).
thf(f1623,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1409,f1410,f1414,f1418,f1422,f1622]) ).
thf(f1622,plain,
( ~ spl12_1
| ~ spl12_3
| ~ spl12_4
| ~ spl12_5 ),
inference(avatar_contradiction_clause,[status(thm)],[f1621]) ).
thf(f1621,plain,
( $false
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4
| ~ spl12_5 ),
inference(subsumption_resolution,[status(thm)],[f1620,f1473]) ).
thf(f1473,plain,
( ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ~ spl12_1
| ~ spl12_4 ),
inference(subsumption_resolution,[status(thm)],[f1472,f1261]) ).
thf(f1261,plain,
( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
= $true ),
inference(definition_unfolding,[status(thm)],[f1054,f1059]) ).
thf(f1059,plain,
( zero_zero_real
= ( sK1 @ p ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f980,plain,
( ( zero_zero_real
= ( sK1 @ p ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( sK1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ sK0 ) )
!= $true ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ X3 ) ) @ one_one_real )
= $true ) )
& ( $true
= ( ord_less_real @ zero_zero_real @ sK0 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f887,f979,f978]) ).
thf(f978,plain,
( ? [X0: real] :
( ? [X1: a > real] :
( ( zero_zero_real
= ( X1 @ p ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( $true
!= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) ) )
& ( ( ord_less_real @ zero_zero_real @ X0 )
= $true ) )
=> ( ? [X1: a > real] :
( ( zero_zero_real
= ( X1 @ p ) )
& ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X1 )
= $true )
& ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X1 )
= $true )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ sK0 ) )
!= $true ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) ) )
& ( $true
= ( ord_less_real @ zero_zero_real @ sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f979,plain,
( ? [X1: a > real] :
( ( zero_zero_real
= ( X1 @ p ) )
& ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X1 )
= $true )
& ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X1 )
= $true )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ sK0 ) )
!= $true ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) ) )
=> ( ( zero_zero_real
= ( sK1 @ p ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( sK1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ sK0 ) )
!= $true ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ X3 ) ) @ one_one_real )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f887,plain,
? [X0: real] :
( ? [X1: a > real] :
( ( zero_zero_real
= ( X1 @ p ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( $true
!= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) ) )
& ( ( ord_less_real @ zero_zero_real @ X0 )
= $true ) ),
inference(flattening,[status(thm)],[f886]) ).
thf(f886,plain,
? [X0: real] :
( ? [X1: a > real] :
( ( zero_zero_real
= ( X1 @ p ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
& ! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) )
& ! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( $true
!= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
& ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) ) )
& ( ( ord_less_real @ zero_zero_real @ X0 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f429]) ).
thf(f429,plain,
~ ! [X0: real] :
( ( ( ord_less_real @ zero_zero_real @ X0 )
= $true )
=> ! [X1: a > real] :
( ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
=> ( ! [X2: a] :
( ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) )
=> ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true ) )
=> ( ! [X3: a] :
( ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
=> ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) )
=> ( ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
=> ( zero_zero_real
!= ( X1 @ p ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f428]) ).
thf(f428,plain,
~ ! [X0: real] :
( ( ord_less_real @ zero_zero_real @ X0 )
=> ! [X1: a > real] :
( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) )
=> ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 )
=> ( zero_zero_real
!= ( X1 @ p ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
~ ! [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
=> ! [X5: a > real] :
( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X4 ) @ X5 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ ( elemen154694473ball_a @ p @ X4 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X6 @ ( X5 @ X6 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
=> ( ! [X6: a] :
( ( member_a @ X6 @ ( elemen154694473ball_a @ p @ X4 ) )
=> ( ord_less_real @ ( abs_abs_real @ ( X5 @ X6 ) ) @ one_one_real ) )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X4 ) @ X5 )
=> ( zero_zero_real
!= ( X5 @ p ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062) ).
thf(f1054,plain,
( $true
= ( ord_less_real @ zero_zero_real @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f1472,plain,
( ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ~ spl12_1
| ~ spl12_4 ),
inference(subsumption_resolution,[status(thm)],[f1471,f1058]) ).
thf(f1058,plain,
( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f1471,plain,
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ~ spl12_1
| ~ spl12_4 ),
inference(trivial_inequality_removal,[status(thm)],[f1470]) ).
thf(f1470,plain,
( ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( $true != $true )
| ( ( sK1 @ p )
!= ( sK1 @ p ) )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ~ spl12_1
| ~ spl12_4 ),
inference(duplicate_literal_removal,[status(thm)],[f1468]) ).
thf(f1468,plain,
( ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( $true != $true )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
!= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( ( sK1 @ p )
!= ( sK1 @ p ) )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ~ spl12_1
| ~ spl12_4 ),
inference(superposition,[status(thm)],[f1445,f1404]) ).
thf(f1404,plain,
( ! [X0: real,X1: a > real] :
( ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) ) )
| ~ spl12_1 ),
inference(avatar_component_clause,[status(thm)],[f1403]) ).
thf(f1403,definition,
( spl12_1
<=> ! [X0: real,X1: a > real] :
( ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) ) ) ),
introduced(definition,[new_symbols(naming,[spl12_1])],[avatar_definition]) ).
thf(f1445,plain,
( ! [X0: real] :
( ( $true
!= ( member_a @ ( sK2 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ sK1 ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ X0 ) ) ) @ one_one_real ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) ) )
| ~ spl12_4 ),
inference(trivial_inequality_removal,[status(thm)],[f1441]) ).
thf(f1441,plain,
( ! [X0: real] :
( ( ( sK1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ X0 ) ) ) @ one_one_real ) )
| ( $true
!= ( member_a @ ( sK2 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ sK1 ) )
| ( $true != $true ) )
| ~ spl12_4 ),
inference(superposition,[status(thm)],[f1417,f1056]) ).
thf(f1056,plain,
! [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( sK1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
= $true )
| ( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ sK0 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f1417,plain,
( ! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) ) )
| ~ spl12_4 ),
inference(avatar_component_clause,[status(thm)],[f1416]) ).
thf(f1416,definition,
( spl12_4
<=> ! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl12_4])],[avatar_definition]) ).
thf(f1620,plain,
( ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
= $true )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4
| ~ spl12_5 ),
inference(subsumption_resolution,[status(thm)],[f1618,f1058]) ).
thf(f1618,plain,
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
= $true )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4
| ~ spl12_5 ),
inference(trivial_inequality_removal,[status(thm)],[f1616]) ).
thf(f1616,plain,
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( $true != $true )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
= $true )
| ( ( sK1 @ p )
!= ( sK1 @ p ) )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4
| ~ spl12_5 ),
inference(superposition,[status(thm)],[f1478,f1432]) ).
thf(f1432,plain,
( ! [X0: a > real] :
( ( ( member_a @ ( sK2 @ X0 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) )
= $true )
| ( ( X0 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X0 ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ X0 @ sK0 ) ) ) @ one_one_real ) ) )
| ~ spl12_5 ),
inference(subsumption_resolution,[status(thm)],[f1431,f1261]) ).
thf(f1431,plain,
( ! [X0: a > real] :
( ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( ( member_a @ ( sK2 @ X0 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) )
= $true )
| ( ( X0 @ p )
!= ( sK1 @ p ) )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ X0 @ sK0 ) ) ) @ one_one_real ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X0 ) ) )
| ~ spl12_5 ),
inference(trivial_inequality_removal,[status(thm)],[f1430]) ).
thf(f1430,plain,
( ! [X0: a > real] :
( ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( ( member_a @ ( sK2 @ X0 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) )
= $true )
| ( $true
= ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ X0 @ sK0 ) ) ) @ one_one_real ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ X0 ) )
| ( ( X0 @ p )
!= ( sK1 @ p ) )
| ( $true != $true ) )
| ~ spl12_5 ),
inference(superposition,[status(thm)],[f1055,f1421]) ).
thf(f1421,plain,
( ! [X0: real,X1: a > real] :
( ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( ( X1 @ p )
!= ( sK1 @ p ) ) )
| ~ spl12_5 ),
inference(avatar_component_clause,[status(thm)],[f1420]) ).
thf(f1420,definition,
( spl12_5
<=> ! [X0: real,X1: a > real] :
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl12_5])],[avatar_definition]) ).
thf(f1055,plain,
! [X3: a] :
( ( $true
!= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ X3 ) ) @ one_one_real )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f980]) ).
thf(f1478,plain,
( ( $true
!= ( member_a @ ( sK2 @ sK1 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4 ),
inference(subsumption_resolution,[status(thm)],[f1477,f1058]) ).
thf(f1477,plain,
( ( $true
!= ( member_a @ ( sK2 @ sK1 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4 ),
inference(subsumption_resolution,[status(thm)],[f1476,f1261]) ).
thf(f1476,plain,
( ( $true
!= ( member_a @ ( sK2 @ sK1 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ~ spl12_1
| ~ spl12_3
| ~ spl12_4 ),
inference(subsumption_resolution,[status(thm)],[f1475,f1473]) ).
thf(f1475,plain,
( ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
= $true )
| ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ( $true
!= ( member_a @ ( sK2 @ sK1 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ~ spl12_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1474]) ).
thf(f1474,plain,
( ( ( ord_less_real @ ( sK1 @ p ) @ sK0 )
!= $true )
| ( ( ord_less_real @ ( abs_abs_real @ ( sK1 @ ( sK3 @ sK1 @ sK0 ) ) ) @ one_one_real )
= $true )
| ( $true
!= ( member_a @ ( sK2 @ sK1 @ sK0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true != $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ sK0 ) @ sK1 ) )
| ~ spl12_3 ),
inference(superposition,[status(thm)],[f1055,f1455]) ).
thf(f1455,plain,
( ! [X0: real] :
( ( ( member_a @ ( sK3 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ sK1 ) )
| ( $true
!= ( member_a @ ( sK2 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) ) )
| ~ spl12_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1451]) ).
thf(f1451,plain,
( ! [X0: real] :
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ sK1 ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( member_a @ ( sK3 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( $true != $true )
| ( ( sK1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( member_a @ ( sK2 @ sK1 @ X0 ) @ ( elemen154694473ball_a @ p @ sK0 ) ) ) )
| ~ spl12_3 ),
inference(superposition,[status(thm)],[f1413,f1056]) ).
thf(f1413,plain,
( ! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) ) )
| ~ spl12_3 ),
inference(avatar_component_clause,[status(thm)],[f1412]) ).
thf(f1412,definition,
( spl12_3
<=> ! [X0: real,X1: a > real] :
( ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl12_3])],[avatar_definition]) ).
thf(f1422,plain,
( spl12_2
| spl12_5 ),
inference(avatar_split_clause,[status(thm)],[f1278,f1420,f1406]) ).
thf(f1406,definition,
( spl12_2
<=> ( thesisa = $true ) ),
introduced(definition,[new_symbols(naming,[spl12_2])],[avatar_definition]) ).
thf(f1278,plain,
! [X0: real,X1: a > real] :
( ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( thesisa = $true )
| ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1085,f1059,f1059]) ).
thf(f1085,plain,
! [X0: real,X1: a > real] :
( ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f990]) ).
thf(f990,plain,
! [X0: real,X1: a > real] :
( ( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
& ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f987,f989,f988]) ).
thf(f988,plain,
! [X0: real,X1: a > real] :
( ? [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
& ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
=> ( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
& ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f989,plain,
! [X0: real,X1: a > real] :
( ? [X3: a] :
( ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) )
=> ( ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f987,plain,
! [X0: real,X1: a > real] :
( ? [X2: a] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X2 @ ( X1 @ X2 ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
& ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ? [X3: a] :
( ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X3 ) ) @ one_one_real ) ) ) ),
inference(rectify,[status(thm)],[f870]) ).
thf(f870,plain,
! [X0: real,X1: a > real] :
( ? [X3: a] :
( ( $true
!= ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X3 @ ( X1 @ X3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
& ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ? [X2: a] :
( ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X2 ) ) @ one_one_real ) ) ) ),
inference(flattening,[status(thm)],[f869]) ).
thf(f869,plain,
! [X0: real,X1: a > real] :
( ( thesisa = $true )
| ? [X3: a] :
( ( $true
!= ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X3 @ ( X1 @ X3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
& ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) ) )
| ? [X2: a] :
( ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) )
& ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X2 ) ) @ one_one_real ) ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f397]) ).
thf(f397,plain,
! [X0: real,X1: a > real] :
( ( ( ord_less_real @ zero_zero_real @ X0 )
= $true )
=> ( ( $true
= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
=> ( ( zero_zero_real
= ( X1 @ p ) )
=> ( ! [X2: a] :
( ( $true
= ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) ) )
=> ( $true
= ( ord_less_real @ ( abs_abs_real @ ( X1 @ X2 ) ) @ one_one_real ) ) )
=> ( ! [X3: a] :
( ( $true
= ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) ) )
=> ( $true
= ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X3 @ ( X1 @ X3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) ) )
=> ( thesisa = $true ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f396]) ).
thf(f396,plain,
! [X0: real,X1: a > real] :
( ( ord_less_real @ zero_zero_real @ X0 )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 )
=> ( ( zero_zero_real
= ( X1 @ p ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( elemen154694473ball_a @ p @ X0 ) )
=> ( ord_less_real @ ( abs_abs_real @ ( X1 @ X2 ) ) @ one_one_real ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( elemen154694473ball_a @ p @ X0 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X3 @ ( X1 @ X3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
=> thesisa ) ) ) ) ),
inference(rectify,[status(thm)],[f348]) ).
thf(f348,axiom,
! [X44: real,X45: a > real] :
( ( ord_less_real @ zero_zero_real @ X44 )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X44 ) @ X45 )
=> ( ( zero_zero_real
= ( X45 @ p ) )
=> ( ! [X9: a] :
( ( member_a @ X9 @ ( elemen154694473ball_a @ p @ X44 ) )
=> ( ord_less_real @ ( abs_abs_real @ ( X45 @ X9 ) ) @ one_one_real ) )
=> ( ! [X9: a] :
( ( member_a @ X9 @ ( elemen154694473ball_a @ p @ X44 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X9 @ ( X45 @ X9 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
=> thesisa ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',conj_0) ).
thf(f1418,plain,
( spl12_4
| spl12_2 ),
inference(avatar_split_clause,[status(thm)],[f1277,f1406,f1416]) ).
thf(f1277,plain,
! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( thesisa = $true ) ),
inference(definition_unfolding,[status(thm)],[f1086,f1059,f1059]) ).
thf(f1086,plain,
! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) ) ),
inference(cnf_transformation,[status(thm)],[f990]) ).
thf(f1414,plain,
( spl12_3
| spl12_2 ),
inference(avatar_split_clause,[status(thm)],[f1276,f1406,f1412]) ).
thf(f1276,plain,
! [X0: real,X1: a > real] :
( ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( thesisa = $true ) ),
inference(definition_unfolding,[status(thm)],[f1087,f1059,f1059]) ).
thf(f1087,plain,
! [X0: real,X1: a > real] :
( ( ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ ( sK2 @ X1 @ X0 ) @ ( X1 @ ( sK2 @ X1 @ X0 ) ) ) @ ( line_open_segment_a @ a2 @ b ) )
!= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( $true
= ( member_a @ ( sK3 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f990]) ).
thf(f1410,plain,
~ spl12_2,
inference(avatar_split_clause,[status(thm)],[f1113,f1406]) ).
thf(f1113,plain,
thesisa != $true,
inference(cnf_transformation,[status(thm)],[f791]) ).
thf(f791,plain,
thesisa != $true,
inference(flattening,[status(thm)],[f631]) ).
thf(f631,plain,
thesisa != $true,
inference(fool_elimination,[status(thm)],[f630]) ).
thf(f630,plain,
~ thesisa,
inference(rectify,[status(thm)],[f350]) ).
thf(f350,negated_conjecture,
~ thesisa,
inference(negated_conjecture,[status(cth)],[f349]) ).
thf(f349,conjecture,
thesisa,
file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',conj_1) ).
thf(f1409,plain,
( spl12_1
| spl12_2 ),
inference(avatar_split_clause,[status(thm)],[f1279,f1406,f1403]) ).
thf(f1279,plain,
! [X0: real,X1: a > real] :
( ( thesisa = $true )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) )
| ( ( X1 @ p )
!= ( sK1 @ p ) )
| ( $true
!= ( ord_less_real @ ( sK1 @ p ) @ X0 ) )
| ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f1084,f1059,f1059]) ).
thf(f1084,plain,
! [X0: real,X1: a > real] :
( ( ( member_a @ ( sK2 @ X1 @ X0 ) @ ( elemen154694473ball_a @ p @ X0 ) )
= $true )
| ( ( ord_less_real @ zero_zero_real @ X0 )
!= $true )
| ( thesisa = $true )
| ( $true
!= ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ X0 ) @ X1 ) )
| ( zero_zero_real
!= ( X1 @ p ) )
| ( $true
!= ( ord_less_real @ ( abs_abs_real @ ( X1 @ ( sK3 @ X1 @ X0 ) ) ) @ one_one_real ) ) ),
inference(cnf_transformation,[status(thm)],[f990]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP147^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n009.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:27:40 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP147^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer Poincare_Bendixson problem prob_2118__19618702_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : Poincare_Bendixson/prob_2118__19618702_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.44 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 395 ( 195 unt; 46 typ; 0 def)
% 0.19/0.46 % Number of atoms : 895 ( 396 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 11 ( 2 avg)
% 0.19/0.46 % Number of connectives : 2765 ( 48 ~; 6 |; 31 &;2414 @)
% 0.19/0.46 % ( 0 <=>; 266 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 22 ( 5 avg)
% 0.19/0.46 % Number of types : 5 ( 4 usr)
% 0.19/0.46 % Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 45 ( 42 usr; 14 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 663 ( 11 ^; 650 !; 2 ?; 663 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:49:33.538
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (4)
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.19/0.46 set_real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 set_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.19/0.46 real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_tf__a,type,
% 0.19/0.46 a: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (42)
% 0.19/0.46 thf(sy_c_Elementary__Metric__Spaces_Oball_001t__Real__Oreal,type,
% 0.19/0.46 elemen341675809l_real: real > real > set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Elementary__Metric__Spaces_Oball_001tf__a,type,
% 0.19/0.46 elemen154694473ball_a: a > real > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
% 0.19/0.46 auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Flow_Oauto__ll__on__open_Oflow0_001tf__a,type,
% 0.19/0.46 auto_ll_on_flow0_a: ( a > a ) > set_a > a > real > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.19/0.46 abs_abs_real: real > real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001tf__a,type,
% 0.19/0.46 abs_abs_a: a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.19/0.46 one_one_real: real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.19/0.46 plus_plus_real: real > real > real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
% 0.19/0.46 plus_plus_a: a > a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
% 0.19/0.46 uminus_uminus_a_a: ( a > a ) > a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.19/0.46 uminus_uminus_real: real > real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
% 0.19/0.46 uminus_uminus_a: a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.19/0.46 zero_zero_real: real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 0.19/0.46 zero_zero_a: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_If_001t__Real__Oreal,type,
% 0.19/0.46 if_real: $o > real > real > real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.19/0.46 initia826609931terval: set_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point_001tf__a,type,
% 0.19/0.46 limit_1865182413oint_a: ( a > a ) > set_a > a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point_001tf__a,type,
% 0.19/0.46 limit_94460170oint_a: ( a > a ) > set_a > a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
% 0.19/0.46 line_open_segment_a: a > a > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.19/0.46 bot_bot_set_real: set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 bot_bot_set_a: set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.19/0.46 ord_less_real: real > real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 0.19/0.46 ord_less_a: a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.19/0.46 ord_less_eq_real: real > real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.19/0.46 ord_less_eq_a: a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.19/0.46 top_top_set_real: set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
% 0.19/0.46 period720806154rbit_a: ( a > a ) > set_a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
% 0.19/0.46 period1305449585riod_a: ( a > a ) > set_a > a > real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
% 0.19/0.46 period138238489rbit_a: ( a > a ) > set_a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
% 0.19/0.46 poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.19/0.46 collect_real: ( real > $o ) > set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.19/0.46 collect_a: ( a > $o ) > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001t__Real__Oreal,type,
% 0.19/0.46 topolo1710226732a_real: set_a > ( a > real ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Real__Oreal,type,
% 0.19/0.46 member_real: real > set_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001tf__a,type,
% 0.19/0.46 member_a: a > set_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_X,type,
% 0.19/0.46 x: set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_a,type,
% 0.19/0.46 a2: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_b,type,
% 0.19/0.46 b: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_f,type,
% 0.19/0.46 f: a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_p,type,
% 0.19/0.46 p: a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_thesisa____,type,
% 0.19/0.46 thesisa: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_x,type,
% 0.19/0.46 x2: a ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (344)
% 0.19/0.46 thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
% 0.19/0.46 ! [X: a,A: a,B: a,Y: a] :
% 0.19/0.46 ( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
% 0.19/0.46 => ( ( member_a @ Y @ ( line_open_segment_a @ A @ B ) )
% 0.19/0.46 => ( ( X = Y )
% 0.19/0.46 | ( member_a @ Y @ ( line_open_segment_a @ X @ B ) )
% 0.19/0.46 | ( member_a @ Y @ ( line_open_segment_a @ A @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.open_segment_trichotomy
% 0.19/0.46 thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062,axiom,
% 0.19/0.46 member_a @ p @ x ).
% 0.19/0.46
% 0.19/0.46 % \<open>p \<in> X\<close>
% 0.19/0.46 thf(fact_2_assms_I4_J,axiom,
% 0.19/0.46 member_a @ p @ ( line_open_segment_a @ a2 @ b ) ).
% 0.19/0.46
% 0.19/0.46 % assms(4)
% 0.19/0.46 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.19/0.46 ~ ! [D: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ D )
% 0.19/0.46 => ! [T: a > real] :
% 0.19/0.46 ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ D ) @ T )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ p @ D ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ p @ D ) )
% 0.19/0.46 => ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ one_one_real ) )
% 0.19/0.46 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ D ) @ T )
% 0.19/0.46 => ( ( T @ p )
% 0.19/0.46 != zero_zero_real ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<And>thesis. (\<And>d t. \<lbrakk>0 < d; continuous_on (ball p d) t; \<And>y. y \<in> ball p d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}; \<And>y. y \<in> ball p d \<Longrightarrow> \<bar>t y\<bar> < 1; continuous_on (ball p d) t; t p = 0\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.19/0.46 thf(fact_4_that,axiom,
% 0.19/0.46 ! [Delta: real,Tau: a > real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ Delta )
% 0.19/0.46 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta ) @ Tau )
% 0.19/0.46 => ( ( ( Tau @ p )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 => ( ! [Y3: a] :
% 0.19/0.46 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta ) )
% 0.19/0.46 => ( ord_less_real @ ( abs_abs_real @ ( Tau @ Y3 ) ) @ one_one_real ) )
% 0.19/0.46 => ( ! [Y3: a] :
% 0.19/0.46 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y3 @ ( Tau @ Y3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.19/0.46 => thesisa ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % that
% 0.19/0.46 thf(fact_5_assms_I1_J,axiom,
% 0.19/0.46 poinca522724647ment_a @ f @ x @ a2 @ b ).
% 0.19/0.46
% 0.19/0.46 % assms(1)
% 0.19/0.46 thf(fact_6_assms_I3_J,axiom,
% 0.19/0.46 limit_94460170oint_a @ f @ x @ x2 @ p ).
% 0.19/0.46
% 0.19/0.46 % assms(3)
% 0.19/0.46 thf(fact_7_centre__in__ball,axiom,
% 0.19/0.46 ! [X: real,E: real] :
% 0.19/0.46 ( ( member_real @ X @ ( elemen341675809l_real @ X @ E ) )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ E ) ) ).
% 0.19/0.46
% 0.19/0.46 % centre_in_ball
% 0.19/0.46 thf(fact_8_centre__in__ball,axiom,
% 0.19/0.46 ! [X: a,E: real] :
% 0.19/0.46 ( ( member_a @ X @ ( elemen154694473ball_a @ X @ E ) )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ E ) ) ).
% 0.19/0.46
% 0.19/0.46 % centre_in_ball
% 0.19/0.46 thf(fact_9_zero__less__abs__iff,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_a @ zero_zero_a @ ( abs_abs_a @ A ) )
% 0.19/0.46 = ( A != zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_less_abs_iff
% 0.19/0.46 thf(fact_10_zero__less__abs__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.19/0.46 = ( A != zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_less_abs_iff
% 0.19/0.46 thf(fact_11_cross__time__continuous,axiom,
% 0.19/0.46 ! [A: a,B: a,X: a,E: real] :
% 0.19/0.46 ( ( poinca522724647ment_a @ f @ x @ A @ B )
% 0.19/0.46 => ( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
% 0.19/0.46 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.19/0.46 => ~ ! [D: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ D )
% 0.19/0.46 => ! [T: a > real] :
% 0.19/0.46 ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment_a @ A @ B ) ) )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
% 0.19/0.46 => ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ E ) )
% 0.19/0.46 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
% 0.19/0.46 => ( ( T @ X )
% 0.19/0.46 != zero_zero_real ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % cross_time_continuous
% 0.19/0.46 thf(fact_12_abs__1,axiom,
% 0.19/0.46 ( ( abs_abs_real @ one_one_real )
% 0.19/0.46 = one_one_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_1
% 0.19/0.46 thf(fact_13_abs__0,axiom,
% 0.19/0.46 ( ( abs_abs_real @ zero_zero_real )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_0
% 0.19/0.46 thf(fact_14_abs__0__eq,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( zero_zero_real
% 0.19/0.46 = ( abs_abs_real @ A ) )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_0_eq
% 0.19/0.46 thf(fact_15_abs__0__eq,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( zero_zero_a
% 0.19/0.46 = ( abs_abs_a @ A ) )
% 0.19/0.46 = ( A = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_0_eq
% 0.19/0.46 thf(fact_16_abs__eq__0,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( abs_abs_real @ A )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_eq_0
% 0.19/0.46 thf(fact_17_abs__eq__0,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ( abs_abs_a @ A )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 = ( A = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_eq_0
% 0.19/0.46 thf(fact_18_abs__zero,axiom,
% 0.19/0.46 ( ( abs_abs_real @ zero_zero_real )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_zero
% 0.19/0.46 thf(fact_19_abs__zero,axiom,
% 0.19/0.46 ( ( abs_abs_a @ zero_zero_a )
% 0.19/0.46 = zero_zero_a ) ).
% 0.19/0.46
% 0.19/0.46 % abs_zero
% 0.19/0.46 thf(fact_20_fixpoint__sol_I2_J,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % fixpoint_sol(2)
% 0.19/0.46 thf(fact_21_gt__one__absI,axiom,
% 0.19/0.46 ! [K: real] :
% 0.19/0.46 ( ( ord_less_real @ ( abs_abs_real @ K ) @ one_one_real )
% 0.19/0.46 => ( ord_less_real @ K @ one_one_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % gt_one_absI
% 0.19/0.46 thf(fact_22_transversal__segment__reverse,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.19/0.46 => ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % transversal_segment_reverse
% 0.19/0.46 thf(fact_23_transversal__segment__commute,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.19/0.46 = ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % transversal_segment_commute
% 0.19/0.46 thf(fact_24_abs__idempotent,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.19/0.46 = ( abs_abs_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_idempotent
% 0.19/0.46 thf(fact_25_abs__idempotent,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( abs_abs_a @ ( abs_abs_a @ A ) )
% 0.19/0.46 = ( abs_abs_a @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_idempotent
% 0.19/0.46 thf(fact_26_abs__abs,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.19/0.46 = ( abs_abs_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_abs
% 0.19/0.46 thf(fact_27_transversal__segment__exists,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ~ ! [A2: a,B2: a] :
% 0.19/0.46 ( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
% 0.19/0.46 => ~ ( poinca522724647ment_a @ f @ x @ A2 @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % transversal_segment_exists
% 0.19/0.46 thf(fact_28_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.19/0.46 poinca522724647ment_a = poinca522724647ment_a ).
% 0.19/0.46
% 0.19/0.46 % c1_on_open_R2.transversal_segment.cong
% 0.19/0.46 thf(fact_29_zero__reorient,axiom,
% 0.19/0.46 ! [X: real] :
% 0.19/0.46 ( ( zero_zero_real = X )
% 0.19/0.46 = ( X = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_reorient
% 0.19/0.46 thf(fact_30_zero__reorient,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( zero_zero_a = X )
% 0.19/0.46 = ( X = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_reorient
% 0.19/0.46 thf(fact_31_linorder__neqE__linordered__idom,axiom,
% 0.19/0.46 ! [X: real,Y: real] :
% 0.19/0.46 ( ( X != Y )
% 0.19/0.46 => ( ~ ( ord_less_real @ X @ Y )
% 0.19/0.46 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_neqE_linordered_idom
% 0.19/0.46 thf(fact_32_one__reorient,axiom,
% 0.19/0.46 ! [X: real] :
% 0.19/0.46 ( ( one_one_real = X )
% 0.19/0.46 = ( X = one_one_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % one_reorient
% 0.19/0.46 thf(fact_33_zero__neq__one,axiom,
% 0.19/0.46 zero_zero_real != one_one_real ).
% 0.19/0.46
% 0.19/0.46 % zero_neq_one
% 0.19/0.46 thf(fact_34_abs__eq__0__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( abs_abs_real @ A )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_eq_0_iff
% 0.19/0.46 thf(fact_35_abs__one,axiom,
% 0.19/0.46 ( ( abs_abs_real @ one_one_real )
% 0.19/0.46 = one_one_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_one
% 0.19/0.46 thf(fact_36_not__one__less__zero,axiom,
% 0.19/0.46 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % not_one_less_zero
% 0.19/0.46 thf(fact_37_zero__less__one,axiom,
% 0.19/0.46 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.19/0.46
% 0.19/0.46 % zero_less_one
% 0.19/0.46 thf(fact_38_abs__not__less__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_not_less_zero
% 0.19/0.46 thf(fact_39_abs__not__less__zero,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ~ ( ord_less_a @ ( abs_abs_a @ A ) @ zero_zero_a ) ).
% 0.19/0.46
% 0.19/0.46 % abs_not_less_zero
% 0.19/0.46 thf(fact_40_abs__of__pos,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.19/0.46 => ( ( abs_abs_real @ A )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_pos
% 0.19/0.46 thf(fact_41_abs__of__pos,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_a @ zero_zero_a @ A )
% 0.19/0.46 => ( ( abs_abs_a @ A )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_pos
% 0.19/0.46 thf(fact_42_periodic__orbit__imp__flow0__regular,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ( f @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) )
% 0.19/0.46 != zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % periodic_orbit_imp_flow0_regular
% 0.19/0.46 thf(fact_43_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( period1305449585riod_a @ f @ x @ X )
% 0.19/0.46 = zero_zero_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % fixed_point_imp_closed_orbit_period_zero(2)
% 0.19/0.46 thf(fact_44_local_Oflow__initial__time__if,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = X0 ) )
% 0.19/0.46 & ( ~ ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = zero_zero_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.flow_initial_time_if
% 0.19/0.46 thf(fact_45_rev_Otransversal__segment__exists,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ~ ! [A2: a,B2: a] :
% 0.19/0.46 ( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
% 0.19/0.46 => ~ ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A2 @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.transversal_segment_exists
% 0.19/0.46 thf(fact_46_rev_Ofixpoint__sol_I2_J,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.fixpoint_sol(2)
% 0.19/0.46 thf(fact_47_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: a,P: a > $o] :
% 0.19/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_48_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: real,P: real > $o] :
% 0.19/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_49_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_a] :
% 0.19/0.46 ( ( collect_a
% 0.19/0.46 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_50_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_real] :
% 0.19/0.46 ( ( collect_real
% 0.19/0.46 @ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_51_Collect__cong,axiom,
% 0.19/0.46 ! [P: real > $o,Q: real > $o] :
% 0.19/0.46 ( ! [X3: real] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 = ( Q @ X3 ) )
% 0.19/0.46 => ( ( collect_real @ P )
% 0.19/0.46 = ( collect_real @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_52_Collect__cong,axiom,
% 0.19/0.46 ! [P: a > $o,Q: a > $o] :
% 0.19/0.46 ( ! [X3: a] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 = ( Q @ X3 ) )
% 0.19/0.46 => ( ( collect_a @ P )
% 0.19/0.46 = ( collect_a @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_53_local_Oflow__undefined0,axiom,
% 0.19/0.46 ! [T2: real,X: a] :
% 0.19/0.46 ( ~ ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
% 0.19/0.46 = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.flow_undefined0
% 0.19/0.46 thf(fact_54_rev__transversal__segment,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A @ B )
% 0.19/0.46 = ( poinca522724647ment_a @ f @ x @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_transversal_segment
% 0.19/0.46 thf(fact_55_rev_Otransversal__segment__reverse,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
% 0.19/0.46 => ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.transversal_segment_reverse
% 0.19/0.46 thf(fact_56_rev_Otransversal__segment__commute,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
% 0.19/0.46 = ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.transversal_segment_commute
% 0.19/0.46 thf(fact_57_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % fixed_point_imp_closed_orbit_period_zero(1)
% 0.19/0.46 thf(fact_58_closed__orbit__in__domain,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( member_a @ X @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_in_domain
% 0.19/0.46 thf(fact_59_add_Oinverse__inverse,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_inverse
% 0.19/0.46 thf(fact_60_add_Oinverse__inverse,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_inverse
% 0.19/0.46 thf(fact_61_neg__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ( uminus_uminus_a @ A )
% 0.19/0.46 = ( uminus_uminus_a @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_iff_equal
% 0.19/0.46 thf(fact_62_neg__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_iff_equal
% 0.19/0.46 thf(fact_63_existence__ivl__zero,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % existence_ivl_zero
% 0.19/0.46 thf(fact_64_local_Omem__existence__ivl__subset,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.mem_existence_ivl_subset
% 0.19/0.46 thf(fact_65_rev_Oclosed__orbit__eq__rev,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = ( period720806154rbit_a @ f @ x @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_eq_rev
% 0.19/0.46 thf(fact_66_rev_Oclosed__orbit__in__domain,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( member_a @ X @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_in_domain
% 0.19/0.46 thf(fact_67_closed__orbit__flow0,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( period720806154rbit_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_flow0
% 0.19/0.46 thf(fact_68_rev_Oexistence__ivl__zero,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.existence_ivl_zero
% 0.19/0.46 thf(fact_69_local_Oexistence__ivl__initial__time,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_initial_time
% 0.19/0.46 thf(fact_70_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.mem_existence_ivl_subset
% 0.19/0.46 thf(fact_71_fixpoint__sol_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.19/0.46 = top_top_set_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % fixpoint_sol(1)
% 0.19/0.46 thf(fact_72_rev_Oclosed__orbit__flow0,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_flow0
% 0.19/0.46 thf(fact_73_closed__orbit__global__existence,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.19/0.46 = top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_global_existence
% 0.19/0.46 thf(fact_74_closed__orbitI,axiom,
% 0.19/0.46 ! [T2: real,T3: real,Y: a] :
% 0.19/0.46 ( ( T2 != T3 )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.19/0.46 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
% 0.19/0.46 => ( period720806154rbit_a @ f @ x @ Y ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbitI
% 0.19/0.46 thf(fact_75_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(1)
% 0.19/0.46 thf(fact_76_periodic__orbit__period_I2_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( period1305449585riod_a @ f @ x @ X ) )
% 0.19/0.46 = X ) ) ).
% 0.19/0.46
% 0.19/0.46 % periodic_orbit_period(2)
% 0.19/0.46 thf(fact_77_closed__orbit__periodic,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ( ( f @ X )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_periodic
% 0.19/0.46 thf(fact_78_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_initial_time
% 0.19/0.46 thf(fact_79_rev_Ofixpoint__sol_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = top_top_set_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.fixpoint_sol(1)
% 0.19/0.46 thf(fact_80_local_Orev_Oflow__undefined0,axiom,
% 0.19/0.46 ! [T2: real,X: a] :
% 0.19/0.46 ( ~ ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 )
% 0.19/0.46 = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.flow_undefined0
% 0.19/0.46 thf(fact_81_closed__orbit__def,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 = ( ? [X2: real] :
% 0.19/0.46 ( ( member_real @ X2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.19/0.46 & ( X2 != zero_zero_real )
% 0.19/0.46 & ( ( auto_ll_on_flow0_a @ f @ x @ X @ X2 )
% 0.19/0.46 = X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_def
% 0.19/0.46 thf(fact_82_rev_Oclosed__orbit__global__existence,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_global_existence
% 0.19/0.46 thf(fact_83_rev_Oclosed__orbitI,axiom,
% 0.19/0.46 ! [T2: real,T3: real,Y: a] :
% 0.19/0.46 ( ( T2 != T3 )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.19/0.46 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T3 ) )
% 0.19/0.46 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbitI
% 0.19/0.46 thf(fact_84_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( member_a @ X @ x )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 => ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = zero_zero_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.fixed_point_imp_closed_orbit_period_zero(2)
% 0.19/0.46 thf(fact_85_rev_Operiodic__orbit__imp__flow0__regular,axiom,
% 0.19/0.46 ! [X: a,T2: real] :
% 0.19/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ( uminus_uminus_a_a @ f @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 ) )
% 0.19/0.46 != zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.periodic_orbit_imp_flow0_regular
% 0.19/0.46 thf(fact_86_periodic__orbitI,axiom,
% 0.19/0.46 ! [T2: real,T3: real,Y: a] :
% 0.19/0.46 ( ( T2 != T3 )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.19/0.46 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
% 0.19/0.46 => ( ( ( f @ Y )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ( period138238489rbit_a @ f @ x @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % periodic_orbitI
% 0.19/0.46 thf(fact_87_periodic__orbit__period_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % periodic_orbit_period(1)
% 0.19/0.46 thf(fact_88_closed__orbit__period__zero__fixed__point,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ( ( period1305449585riod_a @ f @ x @ X )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 => ( ( f @ X )
% 0.19/0.46 = zero_zero_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_period_zero_fixed_point
% 0.19/0.46 thf(fact_89_rev_Operiodic__orbit__period_I2_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.19/0.46 = X ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.periodic_orbit_period(2)
% 0.19/0.46 thf(fact_90_rev_Oclosed__orbit__periodic,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_periodic
% 0.19/0.46 thf(fact_91_add_Oinverse__neutral,axiom,
% 0.19/0.46 ( ( uminus_uminus_a @ zero_zero_a )
% 0.19/0.46 = zero_zero_a ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_neutral
% 0.19/0.46 thf(fact_92_add_Oinverse__neutral,axiom,
% 0.19/0.46 ( ( uminus_uminus_real @ zero_zero_real )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_neutral
% 0.19/0.46 thf(fact_93_neg__0__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( zero_zero_a
% 0.19/0.46 = ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( zero_zero_a = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_equal_iff_equal
% 0.19/0.46 thf(fact_94_neg__0__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( zero_zero_real
% 0.19/0.46 = ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( zero_zero_real = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_equal_iff_equal
% 0.19/0.46 thf(fact_95_neg__equal__0__iff__equal,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ( uminus_uminus_a @ A )
% 0.19/0.46 = zero_zero_a )
% 0.19/0.46 = ( A = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_0_iff_equal
% 0.19/0.46 thf(fact_96_neg__equal__0__iff__equal,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_0_iff_equal
% 0.19/0.46 thf(fact_97_equal__neg__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % equal_neg_zero
% 0.19/0.46 thf(fact_98_neg__equal__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = A )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_zero
% 0.19/0.46 thf(fact_99_local_Orev_Oflow__initial__time__if,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = X0 ) )
% 0.19/0.46 & ( ~ ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = zero_zero_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.flow_initial_time_if
% 0.19/0.46 thf(fact_100_neg__less__iff__less,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( ord_less_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_iff_less
% 0.19/0.46 thf(fact_101_neg__less__iff__less,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_iff_less
% 0.19/0.46 thf(fact_102_rev_Oclosed__orbit__def,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = ( ? [X2: real] :
% 0.19/0.46 ( ( member_real @ X2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.19/0.46 & ( X2 != zero_zero_real )
% 0.19/0.46 & ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ X2 )
% 0.19/0.46 = X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_def
% 0.19/0.46 thf(fact_103_abs__minus,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( abs_abs_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_minus
% 0.19/0.46 thf(fact_104_abs__minus__cancel,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( abs_abs_a @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( abs_abs_a @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_minus_cancel
% 0.19/0.46 thf(fact_105_abs__minus__cancel,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( abs_abs_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_minus_cancel
% 0.19/0.46 thf(fact_106_rev_Operiodic__orbitI,axiom,
% 0.19/0.46 ! [T2: real,T3: real,Y: a] :
% 0.19/0.46 ( ( T2 != T3 )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.19/0.46 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T3 ) )
% 0.19/0.46 => ( ( ( uminus_uminus_a_a @ f @ Y )
% 0.19/0.46 != zero_zero_a )
% 0.19/0.46 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.periodic_orbitI
% 0.19/0.46 thf(fact_107_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.periodic_orbit_period(1)
% 0.19/0.46 thf(fact_108_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 => ( ( uminus_uminus_a_a @ f @ X )
% 0.19/0.46 = zero_zero_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_period_zero_fixed_point
% 0.19/0.46 thf(fact_109_periodic__orbit__def,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ f @ x @ X )
% 0.19/0.46 = ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % periodic_orbit_def
% 0.19/0.46 thf(fact_110_rev_Operiodic__orbit__def,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 = ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.periodic_orbit_def
% 0.19/0.46 thf(fact_111_less__neg__neg,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_neg_neg
% 0.19/0.46 thf(fact_112_neg__less__pos,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_pos
% 0.19/0.46 thf(fact_113_neg__0__less__iff__less,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_less_iff_less
% 0.19/0.46 thf(fact_114_neg__0__less__iff__less,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_less_iff_less
% 0.19/0.46 thf(fact_115_neg__less__0__iff__less,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.19/0.46 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_0_iff_less
% 0.19/0.46 thf(fact_116_neg__less__0__iff__less,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_0_iff_less
% 0.19/0.46 thf(fact_117_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ X0 @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.mem_existence_ivl_iv_defined(2)
% 0.19/0.46 thf(fact_118_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ X0 @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.mem_existence_ivl_iv_defined(2)
% 0.19/0.46 thf(fact_119_flow0__defined,axiom,
% 0.19/0.46 ! [Xa: real,X0: a] :
% 0.19/0.46 ( ( member_real @ Xa @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % flow0_defined
% 0.19/0.46 thf(fact_120_local_Oflow__in__domain,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.flow_in_domain
% 0.19/0.46 thf(fact_121_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.mem_existence_ivl_iv_defined(1)
% 0.19/0.46 thf(fact_122_local_Oexistence__ivl__initial__time__iff,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 = ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_initial_time_iff
% 0.19/0.46 thf(fact_123_local_Oflow__initial__time,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = X0 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.flow_initial_time
% 0.19/0.46 thf(fact_124_rev_Oflow0__defined,axiom,
% 0.19/0.46 ! [Xa: real,X0: a] :
% 0.19/0.46 ( ( member_real @ Xa @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ Xa ) @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.flow0_defined
% 0.19/0.46 thf(fact_125_local_Orev_Oflow__in__domain,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_a @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ T2 ) @ x ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.flow_in_domain
% 0.19/0.46 thf(fact_126_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.19/0.46 ! [T2: real,X0: a] :
% 0.19/0.46 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.mem_existence_ivl_iv_defined(1)
% 0.19/0.46 thf(fact_127_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 = ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 & ( member_a @ X0 @ x ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_initial_time_iff
% 0.19/0.46 thf(fact_128_local_Orev_Oflow__initial__time,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.19/0.46 = X0 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.flow_initial_time
% 0.19/0.46 thf(fact_129_equation__minus__iff,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_a @ B ) )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( uminus_uminus_a @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equation_minus_iff
% 0.19/0.46 thf(fact_130_equation__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equation_minus_iff
% 0.19/0.46 thf(fact_131_minus__equation__iff,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ( uminus_uminus_a @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( ( uminus_uminus_a @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_equation_iff
% 0.19/0.46 thf(fact_132_minus__equation__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( ( uminus_uminus_real @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_equation_iff
% 0.19/0.46 thf(fact_133_less__minus__iff,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
% 0.19/0.46 = ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_minus_iff
% 0.19/0.46 thf(fact_134_less__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_minus_iff
% 0.19/0.46 thf(fact_135_minus__less__iff,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
% 0.19/0.46 = ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_less_iff
% 0.19/0.46 thf(fact_136_minus__less__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.19/0.46 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_less_iff
% 0.19/0.46 thf(fact_137_abs__eq__iff,axiom,
% 0.19/0.46 ! [X: real,Y: real] :
% 0.19/0.46 ( ( ( abs_abs_real @ X )
% 0.19/0.46 = ( abs_abs_real @ Y ) )
% 0.19/0.46 = ( ( X = Y )
% 0.19/0.46 | ( X
% 0.19/0.46 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_eq_iff
% 0.19/0.46 thf(fact_138_abs__less__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.19/0.46 = ( ( ord_less_real @ A @ B )
% 0.19/0.46 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_less_iff
% 0.19/0.46 thf(fact_139_abs__if,axiom,
% 0.19/0.46 ( abs_abs_real
% 0.19/0.46 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_if
% 0.19/0.46 thf(fact_140_abs__of__neg,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_a @ A @ zero_zero_a )
% 0.19/0.46 => ( ( abs_abs_a @ A )
% 0.19/0.46 = ( uminus_uminus_a @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_neg
% 0.19/0.46 thf(fact_141_abs__of__neg,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.19/0.46 => ( ( abs_abs_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_neg
% 0.19/0.46 thf(fact_142__092_060alpha_062__limit__point__eq__rev,axiom,
% 0.19/0.46 ! [X: a,P2: a] :
% 0.19/0.46 ( ( limit_1865182413oint_a @ f @ x @ X @ P2 )
% 0.19/0.46 = ( limit_94460170oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<alpha>_limit_point_eq_rev
% 0.19/0.46 thf(fact_143_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
% 0.19/0.46 ! [X: a,P2: a] :
% 0.19/0.46 ( ( limit_1865182413oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 )
% 0.19/0.46 = ( limit_94460170oint_a @ f @ x @ X @ P2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.\<alpha>_limit_point_eq_rev
% 0.19/0.46 thf(fact_144_rev_Omvar_Ointerval__axioms,axiom,
% 0.19/0.46 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.mvar.interval_axioms
% 0.19/0.46 thf(fact_145_mvar_Ointerval__axioms,axiom,
% 0.19/0.46 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% 0.19/0.46
% 0.19/0.46 % mvar.interval_axioms
% 0.19/0.46 thf(fact_146_abs__neg__one,axiom,
% 0.19/0.46 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.19/0.46 = one_one_real ) ).
% 0.19/0.46
% 0.19/0.46 % abs_neg_one
% 0.19/0.46 thf(fact_147_rev_Oclosed__orbitE,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ~ ! [T4: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.19/0.46 => ~ ! [T5: real] :
% 0.19/0.46 ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( plus_plus_real @ T5 @ T4 ) )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbitE
% 0.19/0.46 thf(fact_148_local_Orev_Oexistence__ivl__notempty,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.19/0.46 != bot_bot_set_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_notempty
% 0.19/0.46 thf(fact_149_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.19/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.closed_orbit_period_nonneg
% 0.19/0.46 thf(fact_150_rev_Ovec__simps_I11_J,axiom,
% 0.19/0.46 ! [Ae: a,Be: a,Cd: a] :
% 0.19/0.46 ( ( plus_plus_a @ ( plus_plus_a @ Ae @ Be ) @ Cd )
% 0.19/0.46 = ( plus_plus_a @ Ae @ ( plus_plus_a @ Be @ Cd ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(11)
% 0.19/0.46 thf(fact_151_rev_Ovec__simps_I11_J,axiom,
% 0.19/0.46 ! [Ae: real,Be: real,Cd: real] :
% 0.19/0.46 ( ( plus_plus_real @ ( plus_plus_real @ Ae @ Be ) @ Cd )
% 0.19/0.46 = ( plus_plus_real @ Ae @ ( plus_plus_real @ Be @ Cd ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(11)
% 0.19/0.46 thf(fact_152_rev_Ovec__simps_I12_J,axiom,
% 0.19/0.46 ( plus_plus_a
% 0.19/0.46 = ( ^ [Ag: a,Bg: a] : ( plus_plus_a @ Bg @ Ag ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(12)
% 0.19/0.46 thf(fact_153_rev_Ovec__simps_I12_J,axiom,
% 0.19/0.46 ( plus_plus_real
% 0.19/0.46 = ( ^ [Ag: real,Bg: real] : ( plus_plus_real @ Bg @ Ag ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(12)
% 0.19/0.46 thf(fact_154_rev_Ovec__simps_I13_J,axiom,
% 0.19/0.46 ! [Bg2: a,Ag2: a,Cf: a] :
% 0.19/0.46 ( ( plus_plus_a @ Bg2 @ ( plus_plus_a @ Ag2 @ Cf ) )
% 0.19/0.46 = ( plus_plus_a @ Ag2 @ ( plus_plus_a @ Bg2 @ Cf ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(13)
% 0.19/0.46 thf(fact_155_rev_Ovec__simps_I13_J,axiom,
% 0.19/0.46 ! [Bg2: real,Ag2: real,Cf: real] :
% 0.19/0.46 ( ( plus_plus_real @ Bg2 @ ( plus_plus_real @ Ag2 @ Cf ) )
% 0.19/0.46 = ( plus_plus_real @ Ag2 @ ( plus_plus_real @ Bg2 @ Cf ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.vec_simps(13)
% 0.19/0.46 thf(fact_156_interval__axioms,axiom,
% 0.19/0.46 initia826609931terval @ top_top_set_real ).
% 0.19/0.46
% 0.19/0.46 % interval_axioms
% 0.19/0.46 thf(fact_157_add__right__cancel,axiom,
% 0.19/0.46 ! [B: a,A: a,C: a] :
% 0.19/0.46 ( ( ( plus_plus_a @ B @ A )
% 0.19/0.46 = ( plus_plus_a @ C @ A ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_cancel
% 0.19/0.46 thf(fact_158_add__right__cancel,axiom,
% 0.19/0.46 ! [B: real,A: real,C: real] :
% 0.19/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.19/0.46 = ( plus_plus_real @ C @ A ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_cancel
% 0.19/0.46 thf(fact_159_add__left__cancel,axiom,
% 0.19/0.46 ! [A: a,B: a,C: a] :
% 0.19/0.46 ( ( ( plus_plus_a @ A @ B )
% 0.19/0.46 = ( plus_plus_a @ A @ C ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_cancel
% 0.19/0.46 thf(fact_160_add__left__cancel,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.19/0.46 = ( plus_plus_real @ A @ C ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_cancel
% 0.19/0.46 thf(fact_161_rev__eq__flow,axiom,
% 0.19/0.46 ! [Y: a,T2: real] :
% 0.19/0.46 ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ f @ x @ Y @ ( uminus_uminus_real @ T2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_eq_flow
% 0.19/0.46 thf(fact_162_rev_Orev__eq__flow,axiom,
% 0.19/0.46 ! [Y: a,T2: real] :
% 0.19/0.46 ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ ( uminus_uminus_real @ T2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.rev_eq_flow
% 0.19/0.46 thf(fact_163_recurrence__time__flip__sign_I1_J,axiom,
% 0.19/0.46 ! [T6: real,X: a] :
% 0.19/0.46 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T6 )
% 0.19/0.46 = X )
% 0.19/0.46 => ( member_real @ ( uminus_uminus_real @ T6 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % recurrence_time_flip_sign(1)
% 0.19/0.46 thf(fact_164_recurrence__time__flip__sign_I2_J,axiom,
% 0.19/0.46 ! [T6: real,X: a] :
% 0.19/0.46 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T6 )
% 0.19/0.46 = X )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( uminus_uminus_real @ T6 ) )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % recurrence_time_flip_sign(2)
% 0.19/0.46 thf(fact_165_rev_Orecurrence__time__flip__sign_I1_J,axiom,
% 0.19/0.46 ! [T6: real,X: a] :
% 0.19/0.46 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T6 )
% 0.19/0.46 = X )
% 0.19/0.46 => ( member_real @ ( uminus_uminus_real @ T6 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.recurrence_time_flip_sign(1)
% 0.19/0.46 thf(fact_166_rev_Orecurrence__time__flip__sign_I2_J,axiom,
% 0.19/0.46 ! [T6: real,X: a] :
% 0.19/0.46 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.19/0.46 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T6 )
% 0.19/0.46 = X )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( uminus_uminus_real @ T6 ) )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev.recurrence_time_flip_sign(2)
% 0.19/0.46 thf(fact_167_local_Oflow__trans,axiom,
% 0.19/0.46 ! [S: real,X0: a,T2: real] :
% 0.19/0.46 ( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ ( plus_plus_real @ S @ T2 ) )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.flow_trans
% 0.19/0.46 thf(fact_168_local_Oexistence__ivl__trans_H,axiom,
% 0.19/0.46 ! [T2: real,S: real,X0: a] :
% 0.19/0.46 ( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_trans'
% 0.19/0.46 thf(fact_169_local_Oexistence__ivl__trans,axiom,
% 0.19/0.46 ! [S: real,X0: a,T2: real] :
% 0.19/0.46 ( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
% 0.19/0.46 => ( member_real @ ( plus_plus_real @ S @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_trans
% 0.19/0.46 thf(fact_170_local_Orev_Oflow__trans,axiom,
% 0.19/0.46 ! [S: real,X0: a,T2: real] :
% 0.19/0.46 ( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) ) )
% 0.19/0.46 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ ( plus_plus_real @ S @ T2 ) )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.flow_trans
% 0.19/0.46 thf(fact_171_local_Orev_Oexistence__ivl__trans_H,axiom,
% 0.19/0.46 ! [T2: real,S: real,X0: a] :
% 0.19/0.46 ( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_trans'
% 0.19/0.46 thf(fact_172_local_Orev_Oexistence__ivl__trans,axiom,
% 0.19/0.46 ! [S: real,X0: a,T2: real] :
% 0.19/0.46 ( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.19/0.46 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) ) )
% 0.19/0.46 => ( member_real @ ( plus_plus_real @ S @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_trans
% 0.19/0.46 thf(fact_173_local_Oexistence__ivl__notempty,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.19/0.46 != bot_bot_set_real ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_notempty
% 0.19/0.46 thf(fact_174_closed__orbit__period__nonneg,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbit_period_nonneg
% 0.19/0.46 thf(fact_175_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_176_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_177_add__cancel__right__right,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = ( B = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_right_right
% 0.19/0.46 thf(fact_178_add__cancel__right__right,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( plus_plus_a @ A @ B ) )
% 0.19/0.46 = ( B = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_right_right
% 0.19/0.46 thf(fact_179_add__cancel__right__left,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( plus_plus_real @ B @ A ) )
% 0.19/0.46 = ( B = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_right_left
% 0.19/0.46 thf(fact_180_add__cancel__right__left,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( plus_plus_a @ B @ A ) )
% 0.19/0.46 = ( B = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_right_left
% 0.19/0.46 thf(fact_181_add__cancel__left__right,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.19/0.46 = A )
% 0.19/0.46 = ( B = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_left_right
% 0.19/0.46 thf(fact_182_add__cancel__left__right,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ( plus_plus_a @ A @ B )
% 0.19/0.46 = A )
% 0.19/0.46 = ( B = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_left_right
% 0.19/0.46 thf(fact_183_add__cancel__left__left,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.19/0.46 = A )
% 0.19/0.46 = ( B = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_left_left
% 0.19/0.46 thf(fact_184_add__cancel__left__left,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ( plus_plus_a @ B @ A )
% 0.19/0.46 = A )
% 0.19/0.46 = ( B = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_cancel_left_left
% 0.19/0.46 thf(fact_185_double__zero__sym,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( zero_zero_real
% 0.19/0.46 = ( plus_plus_real @ A @ A ) )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % double_zero_sym
% 0.19/0.46 thf(fact_186_linordered__ab__group__add__class_Odouble__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( plus_plus_real @ A @ A )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % linordered_ab_group_add_class.double_zero
% 0.19/0.46 thf(fact_187_add_Oright__neutral,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.right_neutral
% 0.19/0.46 thf(fact_188_add_Oright__neutral,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.right_neutral
% 0.19/0.46 thf(fact_189_add_Oleft__neutral,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_neutral
% 0.19/0.46 thf(fact_190_add_Oleft__neutral,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_neutral
% 0.19/0.46 thf(fact_191_add__le__cancel__right,axiom,
% 0.19/0.46 ! [A: a,C: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.19/0.46 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_right
% 0.19/0.46 thf(fact_192_add__le__cancel__right,axiom,
% 0.19/0.46 ! [A: real,C: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_right
% 0.19/0.46 thf(fact_193_add__le__cancel__left,axiom,
% 0.19/0.46 ! [C: a,A: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_left
% 0.19/0.46 thf(fact_194_add__le__cancel__left,axiom,
% 0.19/0.46 ! [C: real,A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_left
% 0.19/0.46 thf(fact_195_add__less__cancel__right,axiom,
% 0.19/0.46 ! [A: real,C: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.19/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_right
% 0.19/0.46 thf(fact_196_add__less__cancel__right,axiom,
% 0.19/0.46 ! [A: a,C: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.19/0.46 = ( ord_less_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_right
% 0.19/0.46 thf(fact_197_add__less__cancel__left,axiom,
% 0.19/0.46 ! [C: real,A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.19/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_left
% 0.19/0.46 thf(fact_198_add__less__cancel__left,axiom,
% 0.19/0.46 ! [C: a,A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.19/0.46 = ( ord_less_a @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_left
% 0.19/0.46 thf(fact_199_closed__orbitE,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( period720806154rbit_a @ f @ x @ X )
% 0.19/0.46 => ~ ! [T4: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.19/0.46 => ~ ! [T5: real] :
% 0.19/0.46 ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( plus_plus_real @ T5 @ T4 ) )
% 0.19/0.46 = ( auto_ll_on_flow0_a @ f @ x @ X @ T5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % closed_orbitE
% 0.19/0.46 thf(fact_200_minus__add__distrib,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
% 0.19/0.46 = ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_distrib
% 0.19/0.46 thf(fact_201_minus__add__distrib,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_distrib
% 0.19/0.46 thf(fact_202_minus__add__cancel,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_cancel
% 0.19/0.46 thf(fact_203_minus__add__cancel,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % minus_add_cancel
% 0.19/0.46 thf(fact_204_add__minus__cancel,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % add_minus_cancel
% 0.19/0.46 thf(fact_205_add__minus__cancel,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % add_minus_cancel
% 0.19/0.46 thf(fact_206_abs__add__abs,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( abs_abs_a @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) )
% 0.19/0.46 = ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_add_abs
% 0.19/0.46 thf(fact_207_abs__add__abs,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.19/0.46 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_add_abs
% 0.19/0.46 thf(fact_208_neg__0__le__iff__le,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_le_iff_le
% 0.19/0.46 thf(fact_209_neg__0__le__iff__le,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_le_iff_le
% 0.19/0.46 thf(fact_210_neg__le__0__iff__le,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.19/0.46 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_0_iff_le
% 0.19/0.46 thf(fact_211_neg__le__0__iff__le,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_0_iff_le
% 0.19/0.46 thf(fact_212_less__eq__neg__nonpos,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_neg_nonpos
% 0.19/0.46 thf(fact_213_neg__less__eq__nonneg,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_eq_nonneg
% 0.19/0.46 thf(fact_214_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
% 0.19/0.46 thf(fact_215_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
% 0.19/0.46 thf(fact_216_le__add__same__cancel2,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.19/0.46 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_same_cancel2
% 0.19/0.46 thf(fact_217_le__add__same__cancel2,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_same_cancel2
% 0.19/0.46 thf(fact_218_le__add__same__cancel1,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.19/0.46 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_same_cancel1
% 0.19/0.46 thf(fact_219_le__add__same__cancel1,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_same_cancel1
% 0.19/0.46 thf(fact_220_add__le__same__cancel2,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.19/0.46 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_same_cancel2
% 0.19/0.46 thf(fact_221_add__le__same__cancel2,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_same_cancel2
% 0.19/0.46 thf(fact_222_add__le__same__cancel1,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_same_cancel1
% 0.19/0.46 thf(fact_223_add__le__same__cancel1,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_same_cancel1
% 0.19/0.46 thf(fact_224_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
% 0.19/0.46 thf(fact_225_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.19/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.19/0.46 thf(fact_226_less__add__same__cancel2,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_same_cancel2
% 0.19/0.46 thf(fact_227_less__add__same__cancel2,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.19/0.46 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_same_cancel2
% 0.19/0.46 thf(fact_228_less__add__same__cancel1,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_same_cancel1
% 0.19/0.46 thf(fact_229_less__add__same__cancel1,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.19/0.46 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_same_cancel1
% 0.19/0.46 thf(fact_230_add__less__same__cancel2,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.19/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_same_cancel2
% 0.19/0.46 thf(fact_231_add__less__same__cancel2,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.19/0.46 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_same_cancel2
% 0.19/0.46 thf(fact_232_add__less__same__cancel1,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.19/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_same_cancel1
% 0.19/0.46 thf(fact_233_add__less__same__cancel1,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.19/0.46 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_same_cancel1
% 0.19/0.46 thf(fact_234_add_Oleft__inverse,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
% 0.19/0.46 = zero_zero_a ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_inverse
% 0.19/0.46 thf(fact_235_add_Oleft__inverse,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_inverse
% 0.19/0.46 thf(fact_236_add_Oright__inverse,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( plus_plus_a @ A @ ( uminus_uminus_a @ A ) )
% 0.19/0.46 = zero_zero_a ) ).
% 0.19/0.46
% 0.19/0.46 % add.right_inverse
% 0.19/0.46 thf(fact_237_add_Oright__inverse,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add.right_inverse
% 0.19/0.46 thf(fact_238_abs__le__zero__iff,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ zero_zero_a )
% 0.19/0.46 = ( A = zero_zero_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_le_zero_iff
% 0.19/0.46 thf(fact_239_abs__le__zero__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_le_zero_iff
% 0.19/0.46 thf(fact_240_abs__le__self__iff,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ A )
% 0.19/0.46 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_le_self_iff
% 0.19/0.46 thf(fact_241_abs__le__self__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_le_self_iff
% 0.19/0.46 thf(fact_242_abs__of__nonneg,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.19/0.46 => ( ( abs_abs_a @ A )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_nonneg
% 0.19/0.46 thf(fact_243_abs__of__nonneg,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.19/0.46 => ( ( abs_abs_real @ A )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_nonneg
% 0.19/0.46 thf(fact_244_ball__trivial,axiom,
% 0.19/0.46 ! [X: a] :
% 0.19/0.46 ( ( elemen154694473ball_a @ X @ zero_zero_real )
% 0.19/0.46 = bot_bot_set_a ) ).
% 0.19/0.46
% 0.19/0.46 % ball_trivial
% 0.19/0.46 thf(fact_245_ball__trivial,axiom,
% 0.19/0.46 ! [X: real] :
% 0.19/0.46 ( ( elemen341675809l_real @ X @ zero_zero_real )
% 0.19/0.46 = bot_bot_set_real ) ).
% 0.19/0.46
% 0.19/0.46 % ball_trivial
% 0.19/0.46 thf(fact_246_add__neg__numeral__special_I7_J,axiom,
% 0.19/0.46 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add_neg_numeral_special(7)
% 0.19/0.46 thf(fact_247_add__neg__numeral__special_I8_J,axiom,
% 0.19/0.46 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add_neg_numeral_special(8)
% 0.19/0.46 thf(fact_248_abs__of__nonpos,axiom,
% 0.19/0.46 ! [A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.19/0.46 => ( ( abs_abs_a @ A )
% 0.19/0.46 = ( uminus_uminus_a @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_nonpos
% 0.19/0.46 thf(fact_249_abs__of__nonpos,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.19/0.46 => ( ( abs_abs_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_of_nonpos
% 0.19/0.46 thf(fact_250_ball__eq__empty,axiom,
% 0.19/0.46 ! [X: a,E: real] :
% 0.19/0.46 ( ( ( elemen154694473ball_a @ X @ E )
% 0.19/0.46 = bot_bot_set_a )
% 0.19/0.46 = ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % ball_eq_empty
% 0.19/0.46 thf(fact_251_ball__eq__empty,axiom,
% 0.19/0.46 ! [X: real,E: real] :
% 0.19/0.46 ( ( ( elemen341675809l_real @ X @ E )
% 0.19/0.46 = bot_bot_set_real )
% 0.19/0.46 = ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % ball_eq_empty
% 0.19/0.46 thf(fact_252_local_Oexistence__ivl__undefined,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ~ ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_undefined
% 0.19/0.46 thf(fact_253_local_Orev_Oexistence__ivl__undefined,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ~ ( member_a @ X0 @ x )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_undefined
% 0.19/0.46 thf(fact_254_local_Oexistence__ivl__empty__iff,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real )
% 0.19/0.46 = ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 | ~ ( member_a @ X0 @ x ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_empty_iff
% 0.19/0.46 thf(fact_255_local_Oexistence__ivl__empty1,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.existence_ivl_empty1
% 0.19/0.46 thf(fact_256_local_Orev_Oexistence__ivl__empty__iff,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real )
% 0.19/0.46 = ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 | ~ ( member_a @ X0 @ x ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_empty_iff
% 0.19/0.46 thf(fact_257_local_Orev_Oexistence__ivl__empty1,axiom,
% 0.19/0.46 ! [X0: a] :
% 0.19/0.46 ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.19/0.46 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.19/0.46 = bot_bot_set_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % local.rev.existence_ivl_empty1
% 0.19/0.46 thf(fact_258_is__num__normalize_I8_J,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.19/0.46 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_num_normalize(8)
% 0.19/0.46 thf(fact_259_abs__triangle__ineq,axiom,
% 0.19/0.46 ! [A: a,B: a] : ( ord_less_eq_a @ ( abs_abs_a @ ( plus_plus_a @ A @ B ) ) @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_triangle_ineq
% 0.19/0.46 thf(fact_260_abs__triangle__ineq,axiom,
% 0.19/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_triangle_ineq
% 0.19/0.46 thf(fact_261_le__numeral__extra_I3_J,axiom,
% 0.19/0.46 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.19/0.46
% 0.19/0.46 % le_numeral_extra(3)
% 0.19/0.46 thf(fact_262_le__numeral__extra_I4_J,axiom,
% 0.19/0.46 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.19/0.46
% 0.19/0.46 % le_numeral_extra(4)
% 0.19/0.46 thf(fact_263_is__num__normalize_I1_J,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.19/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_num_normalize(1)
% 0.19/0.47 thf(fact_264_add__le__imp__le__right,axiom,
% 0.19/0.47 ! [A: a,C: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.19/0.47 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_imp_le_right
% 0.19/0.47 thf(fact_265_add__le__imp__le__right,axiom,
% 0.19/0.47 ! [A: real,C: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.19/0.47 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_imp_le_right
% 0.19/0.47 thf(fact_266_add__le__imp__le__left,axiom,
% 0.19/0.47 ! [C: a,A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.19/0.47 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_imp_le_left
% 0.19/0.47 thf(fact_267_add__le__imp__le__left,axiom,
% 0.19/0.47 ! [C: real,A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.19/0.47 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_imp_le_left
% 0.19/0.47 thf(fact_268_add__less__le__mono,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a,D2: a] :
% 0.19/0.47 ( ( ord_less_a @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_a @ C @ D2 )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_less_le_mono
% 0.19/0.47 thf(fact_269_add__less__le__mono,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real,D2: real] :
% 0.19/0.47 ( ( ord_less_real @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_real @ C @ D2 )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_less_le_mono
% 0.19/0.47 thf(fact_270_add__le__less__mono,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a,D2: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.47 => ( ( ord_less_a @ C @ D2 )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_less_mono
% 0.19/0.47 thf(fact_271_add__le__less__mono,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real,D2: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.47 => ( ( ord_less_real @ C @ D2 )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_less_mono
% 0.19/0.47 thf(fact_272_add__nonpos__eq__0__iff,axiom,
% 0.19/0.47 ! [X: a,Y: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ X @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_eq_a @ Y @ zero_zero_a )
% 0.19/0.47 => ( ( ( plus_plus_a @ X @ Y )
% 0.19/0.47 = zero_zero_a )
% 0.19/0.47 = ( ( X = zero_zero_a )
% 0.19/0.47 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_eq_0_iff
% 0.19/0.47 thf(fact_273_add__nonpos__eq__0__iff,axiom,
% 0.19/0.47 ! [X: real,Y: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.19/0.47 => ( ( ( plus_plus_real @ X @ Y )
% 0.19/0.47 = zero_zero_real )
% 0.19/0.47 = ( ( X = zero_zero_real )
% 0.19/0.47 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_eq_0_iff
% 0.19/0.47 thf(fact_274_add__nonneg__eq__0__iff,axiom,
% 0.19/0.47 ! [X: a,Y: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ X )
% 0.19/0.47 => ( ( ord_less_eq_a @ zero_zero_a @ Y )
% 0.19/0.47 => ( ( ( plus_plus_a @ X @ Y )
% 0.19/0.47 = zero_zero_a )
% 0.19/0.47 = ( ( X = zero_zero_a )
% 0.19/0.47 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_eq_0_iff
% 0.19/0.47 thf(fact_275_add__nonneg__eq__0__iff,axiom,
% 0.19/0.47 ! [X: real,Y: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.19/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.19/0.47 => ( ( ( plus_plus_real @ X @ Y )
% 0.19/0.47 = zero_zero_real )
% 0.19/0.47 = ( ( X = zero_zero_real )
% 0.19/0.47 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_eq_0_iff
% 0.19/0.47 thf(fact_276_add__nonpos__nonpos,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_eq_a @ B @ zero_zero_a )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_nonpos
% 0.19/0.47 thf(fact_277_add__nonpos__nonpos,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_nonpos
% 0.19/0.47 thf(fact_278_add__nonneg__nonneg,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.19/0.47 => ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_nonneg
% 0.19/0.47 thf(fact_279_add__nonneg__nonneg,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.19/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_nonneg
% 0.19/0.47 thf(fact_280_add__increasing2,axiom,
% 0.19/0.47 ! [C: a,B: a,A: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ C )
% 0.19/0.47 => ( ( ord_less_eq_a @ B @ A )
% 0.19/0.47 => ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_increasing2
% 0.19/0.47 thf(fact_281_add__increasing2,axiom,
% 0.19/0.47 ! [C: real,B: real,A: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.19/0.47 => ( ( ord_less_eq_real @ B @ A )
% 0.19/0.47 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_increasing2
% 0.19/0.47 thf(fact_282_add__decreasing2,axiom,
% 0.19/0.47 ! [C: a,A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ C @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_eq_a @ A @ B )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_decreasing2
% 0.19/0.47 thf(fact_283_add__decreasing2,axiom,
% 0.19/0.47 ! [C: real,A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_eq_real @ A @ B )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_decreasing2
% 0.19/0.47 thf(fact_284_add__right__mono,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_right_mono
% 0.19/0.47 thf(fact_285_add__right__mono,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_right_mono
% 0.19/0.47 thf(fact_286_add__increasing,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.47 => ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_increasing
% 0.19/0.47 thf(fact_287_add__increasing,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.47 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_increasing
% 0.19/0.47 thf(fact_288_add__decreasing,axiom,
% 0.19/0.47 ! [A: a,C: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_eq_a @ C @ B )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_decreasing
% 0.19/0.47 thf(fact_289_add__decreasing,axiom,
% 0.19/0.47 ! [A: real,C: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_eq_real @ C @ B )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_decreasing
% 0.19/0.47 thf(fact_290_add__left__mono,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_left_mono
% 0.19/0.47 thf(fact_291_add__left__mono,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_left_mono
% 0.19/0.47 thf(fact_292_add__right__imp__eq,axiom,
% 0.19/0.47 ! [B: a,A: a,C: a] :
% 0.19/0.47 ( ( ( plus_plus_a @ B @ A )
% 0.19/0.47 = ( plus_plus_a @ C @ A ) )
% 0.19/0.47 => ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_right_imp_eq
% 0.19/0.47 thf(fact_293_add__right__imp__eq,axiom,
% 0.19/0.47 ! [B: real,A: real,C: real] :
% 0.19/0.47 ( ( ( plus_plus_real @ B @ A )
% 0.19/0.47 = ( plus_plus_real @ C @ A ) )
% 0.19/0.47 => ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_right_imp_eq
% 0.19/0.47 thf(fact_294_add__left__imp__eq,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ( plus_plus_a @ A @ B )
% 0.19/0.47 = ( plus_plus_a @ A @ C ) )
% 0.19/0.47 => ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_left_imp_eq
% 0.19/0.47 thf(fact_295_add__left__imp__eq,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ( plus_plus_real @ A @ B )
% 0.19/0.47 = ( plus_plus_real @ A @ C ) )
% 0.19/0.47 => ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_left_imp_eq
% 0.19/0.47 thf(fact_296_add__mono,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a,D2: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_a @ C @ D2 )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono
% 0.19/0.47 thf(fact_297_add__mono,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real,D2: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_real @ C @ D2 )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono
% 0.19/0.47 thf(fact_298_add_Oleft__commute,axiom,
% 0.19/0.47 ! [B: a,A: a,C: a] :
% 0.19/0.47 ( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
% 0.19/0.47 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.left_commute
% 0.19/0.47 thf(fact_299_add_Oleft__commute,axiom,
% 0.19/0.47 ! [B: real,A: real,C: real] :
% 0.19/0.47 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.19/0.47 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.left_commute
% 0.19/0.47 thf(fact_300_add_Ocommute,axiom,
% 0.19/0.47 ( plus_plus_a
% 0.19/0.47 = ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.commute
% 0.19/0.47 thf(fact_301_add_Ocommute,axiom,
% 0.19/0.47 ( plus_plus_real
% 0.19/0.47 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.commute
% 0.19/0.47 thf(fact_302_add_Oright__cancel,axiom,
% 0.19/0.47 ! [B: a,A: a,C: a] :
% 0.19/0.47 ( ( ( plus_plus_a @ B @ A )
% 0.19/0.47 = ( plus_plus_a @ C @ A ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.right_cancel
% 0.19/0.47 thf(fact_303_add_Oright__cancel,axiom,
% 0.19/0.47 ! [B: real,A: real,C: real] :
% 0.19/0.47 ( ( ( plus_plus_real @ B @ A )
% 0.19/0.47 = ( plus_plus_real @ C @ A ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.right_cancel
% 0.19/0.47 thf(fact_304_add_Oleft__cancel,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ( plus_plus_a @ A @ B )
% 0.19/0.47 = ( plus_plus_a @ A @ C ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.left_cancel
% 0.19/0.47 thf(fact_305_add_Oleft__cancel,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ( plus_plus_real @ A @ B )
% 0.19/0.47 = ( plus_plus_real @ A @ C ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.left_cancel
% 0.19/0.47 thf(fact_306_add_Oassoc,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.19/0.47 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.assoc
% 0.19/0.47 thf(fact_307_add_Oassoc,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.19/0.47 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.assoc
% 0.19/0.47 thf(fact_308_group__cancel_Oadd2,axiom,
% 0.19/0.47 ! [B4: a,K: a,B: a,A: a] :
% 0.19/0.47 ( ( B4
% 0.19/0.47 = ( plus_plus_a @ K @ B ) )
% 0.19/0.47 => ( ( plus_plus_a @ A @ B4 )
% 0.19/0.47 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % group_cancel.add2
% 0.19/0.47 thf(fact_309_group__cancel_Oadd2,axiom,
% 0.19/0.47 ! [B4: real,K: real,B: real,A: real] :
% 0.19/0.47 ( ( B4
% 0.19/0.47 = ( plus_plus_real @ K @ B ) )
% 0.19/0.47 => ( ( plus_plus_real @ A @ B4 )
% 0.19/0.47 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % group_cancel.add2
% 0.19/0.47 thf(fact_310_group__cancel_Oadd1,axiom,
% 0.19/0.47 ! [A3: a,K: a,A: a,B: a] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( plus_plus_a @ K @ A ) )
% 0.19/0.47 => ( ( plus_plus_a @ A3 @ B )
% 0.19/0.47 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % group_cancel.add1
% 0.19/0.47 thf(fact_311_group__cancel_Oadd1,axiom,
% 0.19/0.47 ! [A3: real,K: real,A: real,B: real] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( plus_plus_real @ K @ A ) )
% 0.19/0.47 => ( ( plus_plus_real @ A3 @ B )
% 0.19/0.47 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % group_cancel.add1
% 0.19/0.47 thf(fact_312_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( ord_less_eq_a @ I @ J )
% 0.19/0.47 & ( ord_less_eq_a @ K @ L ) )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(1)
% 0.19/0.47 thf(fact_313_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( ord_less_eq_real @ I @ J )
% 0.19/0.47 & ( ord_less_eq_real @ K @ L ) )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(1)
% 0.19/0.47 thf(fact_314_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( I = J )
% 0.19/0.47 & ( ord_less_eq_a @ K @ L ) )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(2)
% 0.19/0.47 thf(fact_315_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( I = J )
% 0.19/0.47 & ( ord_less_eq_real @ K @ L ) )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(2)
% 0.19/0.47 thf(fact_316_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( ord_less_eq_a @ I @ J )
% 0.19/0.47 & ( K = L ) )
% 0.19/0.47 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(3)
% 0.19/0.47 thf(fact_317_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( ord_less_eq_real @ I @ J )
% 0.19/0.47 & ( K = L ) )
% 0.19/0.47 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(3)
% 0.19/0.47 thf(fact_318_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( I = J )
% 0.19/0.47 & ( K = L ) )
% 0.19/0.47 => ( ( plus_plus_a @ I @ K )
% 0.19/0.47 = ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(4)
% 0.19/0.47 thf(fact_319_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( I = J )
% 0.19/0.47 & ( K = L ) )
% 0.19/0.47 => ( ( plus_plus_real @ I @ K )
% 0.19/0.47 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_semiring(4)
% 0.19/0.47 thf(fact_320_add__mono__thms__linordered__field_I3_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( ord_less_a @ I @ J )
% 0.19/0.47 & ( ord_less_eq_a @ K @ L ) )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_field(3)
% 0.19/0.47 thf(fact_321_add__mono__thms__linordered__field_I3_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( ord_less_real @ I @ J )
% 0.19/0.47 & ( ord_less_eq_real @ K @ L ) )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_field(3)
% 0.19/0.47 thf(fact_322_add__mono__thms__linordered__field_I4_J,axiom,
% 0.19/0.47 ! [I: a,J: a,K: a,L: a] :
% 0.19/0.47 ( ( ( ord_less_eq_a @ I @ J )
% 0.19/0.47 & ( ord_less_a @ K @ L ) )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_field(4)
% 0.19/0.47 thf(fact_323_add__mono__thms__linordered__field_I4_J,axiom,
% 0.19/0.47 ! [I: real,J: real,K: real,L: real] :
% 0.19/0.47 ( ( ( ord_less_eq_real @ I @ J )
% 0.19/0.47 & ( ord_less_real @ K @ L ) )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_mono_thms_linordered_field(4)
% 0.19/0.47 thf(fact_324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.19/0.47 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.47 thf(fact_325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.19/0.47 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.47 thf(fact_326_add__strict__increasing2,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_a @ B @ C )
% 0.19/0.47 => ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_strict_increasing2
% 0.19/0.47 thf(fact_327_add__strict__increasing2,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_real @ B @ C )
% 0.19/0.47 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_strict_increasing2
% 0.19/0.47 thf(fact_328_add__strict__increasing,axiom,
% 0.19/0.47 ! [A: a,B: a,C: a] :
% 0.19/0.47 ( ( ord_less_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.47 => ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_strict_increasing
% 0.19/0.47 thf(fact_329_add__strict__increasing,axiom,
% 0.19/0.47 ! [A: real,B: real,C: real] :
% 0.19/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.47 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_strict_increasing
% 0.19/0.47 thf(fact_330_add__pos__nonneg,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.19/0.47 => ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_pos_nonneg
% 0.19/0.47 thf(fact_331_add__pos__nonneg,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.19/0.47 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_pos_nonneg
% 0.19/0.47 thf(fact_332_add__nonpos__neg,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_a @ B @ zero_zero_a )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_neg
% 0.19/0.47 thf(fact_333_add__nonpos__neg,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonpos_neg
% 0.19/0.47 thf(fact_334_add__nonneg__pos,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.19/0.47 => ( ( ord_less_a @ zero_zero_a @ B )
% 0.19/0.47 => ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_pos
% 0.19/0.47 thf(fact_335_add__nonneg__pos,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.19/0.47 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.19/0.47 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_nonneg_pos
% 0.19/0.47 thf(fact_336_add__neg__nonpos,axiom,
% 0.19/0.47 ! [A: a,B: a] :
% 0.19/0.47 ( ( ord_less_a @ A @ zero_zero_a )
% 0.19/0.47 => ( ( ord_less_eq_a @ B @ zero_zero_a )
% 0.19/0.47 => ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_neg_nonpos
% 0.19/0.47 thf(fact_337_add__neg__nonpos,axiom,
% 0.19/0.47 ! [A: real,B: real] :
% 0.19/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.19/0.47 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.19/0.47 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_neg_nonpos
% 0.19/0.47 thf(fact_338_Elementary__Metric__Spaces_Oball__empty,axiom,
% 0.19/0.47 ! [E: real,X: a] :
% 0.19/0.47 ( ( ord_less_eq_real @ E @ zero_zero_real )
% 0.19/0.47 => ( ( elemen154694473ball_a @ X @ E )
% 0.19/0.47 = bot_bot_set_a ) ) ).
% 0.19/0.47
% 0.19/0.47 % Elementary_Metric_Spaces.ball_empty
% 0.19/0.47 thf(fact_339_Elementary__Metric__Spaces_Oball__empty,axiom,
% 0.19/0.47 ! [E: real,X: real] :
% 0.19/0.47 ( ( ord_less_eq_real @ E @ zero_zero_real )
% 0.19/0.47 => ( ( elemen341675809l_real @ X @ E )
% 0.19/0.47 = bot_bot_set_real ) ) ).
% 0.19/0.47
% 0.19/0.47 % Elementary_Metric_Spaces.ball_empty
% 0.19/0.47 thf(fact_340_add_Ogroup__left__neutral,axiom,
% 0.19/0.47 ! [A: real] :
% 0.19/0.47 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.19/0.47 = A ) ).
% 0.19/0.47
% 0.19/0.47 % add.group_left_neutral
% 0.19/0.47 thf(fact_341_add_Ogroup__left__neutral,axiom,
% 0.19/0.47 ! [A: a] :
% 0.19/0.47 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.19/0.47 = A ) ).
% 0.19/0.47
% 0.19/0.47 % add.group_left_neutral
% 0.19/0.47 thf(fact_342_add_Ocomm__neutral,axiom,
% 0.19/0.47 ! [A: real] :
% 0.19/0.47 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.19/0.47 = A ) ).
% 0.19/0.47
% 0.19/0.47 % add.comm_neutral
% 0.19/0.47 thf(fact_343_add_Ocomm__neutral,axiom,
% 0.19/0.47 ! [A: a] :
% 0.19/0.47 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.19/0.47 = A ) ).
% 0.19/0.47
% 0.19/0.47 % add.comm_neutral
% 0.19/0.47
% 0.19/0.47 % Helper facts (3)
% 0.19/0.47 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.19/0.47 ! [P: $o] :
% 0.19/0.47 ( ( P = $true )
% 0.19/0.47 | ( P = $false ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.19/0.47 ! [X: real,Y: real] :
% 0.19/0.47 ( ( if_real @ $false @ X @ Y )
% 0.19/0.47 = Y ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.19/0.47 ! [X: real,Y: real] :
% 0.19/0.47 ( ( if_real @ $true @ X @ Y )
% 0.19/0.47 = X ) ).
% 0.19/0.47
% 0.19/0.47 % Conjectures (2)
% 0.19/0.47 thf(conj_0,hypothesis,
% 0.19/0.47 ! [Delta2: real,Tau2: a > real] :
% 0.19/0.47 ( ( ord_less_real @ zero_zero_real @ Delta2 )
% 0.19/0.47 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta2 ) @ Tau2 )
% 0.19/0.47 => ( ( ( Tau2 @ p )
% 0.19/0.47 = zero_zero_real )
% 0.19/0.47 => ( ! [Y3: a] :
% 0.19/0.47 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta2 ) )
% 0.19/0.47 => ( ord_less_real @ ( abs_abs_real @ ( Tau2 @ Y3 ) ) @ one_one_real ) )
% 0.19/0.47 => ( ! [Y3: a] :
% 0.19/0.47 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta2 ) )
% 0.19/0.47 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y3 @ ( Tau2 @ Y3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.19/0.47 => thesisa ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(conj_1,conjecture,
% 0.19/0.47 thesisa ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 0.30/1.71 ---- Embedded in THF ---
% 0.30/1.71 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.71 %%% Generated on Tue Feb 24 08:27:41 EST 2026
% 0.30/1.71 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.71 %%% Logic specification used:
% 0.30/1.71 %%% thf(spec, logic, $$dhol).
% 0.30/1.71
% 0.30/1.71 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2DTF2468.p
% See solution above
% 0.30/1.71 ------------------------
% 0.30/1.75 ---- Cleaned THF ---
% 0.30/1.76 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/1.76 set_real: $tType ).
% 0.30/1.76
% 0.30/1.76 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.30/1.76 set_a: $tType ).
% 0.30/1.76
% 0.30/1.76 thf(ty_n_t__Real__Oreal,type,
% 0.30/1.76 real: $tType ).
% 0.30/1.76
% 0.30/1.76 thf(ty_n_tf__a,type,
% 0.30/1.76 a: $tType ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Elementary__Metric__Spaces_Oball_001t__Real__Oreal,type,
% 0.30/1.76 elemen341675809l_real: real > real > set_real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Elementary__Metric__Spaces_Oball_001tf__a,type,
% 0.30/1.76 elemen154694473ball_a: a > real > set_a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
% 0.30/1.76 auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Flow_Oauto__ll__on__open_Oflow0_001tf__a,type,
% 0.30/1.76 auto_ll_on_flow0_a: ( a > a ) > set_a > a > real > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 0.30/1.76 abs_abs_real: real > real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Oabs__class_Oabs_001tf__a,type,
% 0.30/1.76 abs_abs_a: a > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 0.30/1.76 one_one_real: real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.30/1.76 plus_plus_real: real > real > real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
% 0.30/1.76 plus_plus_a: a > a > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
% 0.30/1.76 uminus_uminus_a_a: ( a > a ) > a > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.30/1.76 uminus_uminus_real: real > real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
% 0.30/1.76 uminus_uminus_a: a > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.30/1.76 zero_zero_real: real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
% 0.30/1.76 zero_zero_a: a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_If_001t__Real__Oreal,type,
% 0.30/1.76 if_real: $o > real > real > real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Initial__Value__Problem_Ointerval,type,
% 0.30/1.76 initia826609931terval: set_real > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point_001tf__a,type,
% 0.30/1.76 limit_1865182413oint_a: ( a > a ) > set_a > a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point_001tf__a,type,
% 0.30/1.76 limit_94460170oint_a: ( a > a ) > set_a > a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
% 0.30/1.76 line_open_segment_a: a > a > set_a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/1.76 bot_bot_set_real: set_real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.30/1.76 bot_bot_set_a: set_a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.30/1.76 ord_less_real: real > real > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
% 0.30/1.76 ord_less_a: a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.30/1.76 ord_less_eq_real: real > real > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.30/1.76 ord_less_eq_a: a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/1.76 top_top_set_real: set_real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
% 0.30/1.76 period720806154rbit_a: ( a > a ) > set_a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
% 0.30/1.76 period1305449585riod_a: ( a > a ) > set_a > a > real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
% 0.30/1.76 period138238489rbit_a: ( a > a ) > set_a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
% 0.30/1.76 poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.30/1.76 collect_real: ( real > $o ) > set_real ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.30/1.76 collect_a: ( a > $o ) > set_a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001t__Real__Oreal,type,
% 0.30/1.76 topolo1710226732a_real: set_a > ( a > real ) > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/1.76 member_real: real > set_real > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_c_member_001tf__a,type,
% 0.30/1.76 member_a: a > set_a > $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_X,type,
% 0.30/1.76 x: set_a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_a,type,
% 0.30/1.76 a2: a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_b,type,
% 0.30/1.76 b: a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_f,type,
% 0.30/1.76 f: a > a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_p,type,
% 0.30/1.76 p: a ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_thesisa____,type,
% 0.30/1.76 thesisa: $o ).
% 0.30/1.76
% 0.30/1.76 thf(sy_v_x,type,
% 0.30/1.76 x2: a ).
% 0.30/1.76
% 0.30/1.76 thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
% 0.30/1.76 ! [X: a,A: a,B: a,Y: a] :
% 0.30/1.76 ( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
% 0.30/1.76 => ( ( member_a @ Y @ ( line_open_segment_a @ A @ B ) )
% 0.30/1.76 => ( ( X = Y )
% 0.30/1.76 | ( member_a @ Y @ ( line_open_segment_a @ X @ B ) )
% 0.30/1.76 | ( member_a @ Y @ ( line_open_segment_a @ A @ X ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062,axiom,
% 0.30/1.76 member_a @ p @ x ).
% 0.30/1.76
% 0.30/1.76 thf(fact_2_assms_I4_J,axiom,
% 0.30/1.76 member_a @ p @ ( line_open_segment_a @ a2 @ b ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/1.76 ~ ! [D: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/1.76 => ! [T: a > real] :
% 0.30/1.76 ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ D ) @ T )
% 0.30/1.76 => ( ! [Y2: a] :
% 0.30/1.76 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ p @ D ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.30/1.76 => ( ! [Y2: a] :
% 0.30/1.76 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ p @ D ) )
% 0.30/1.76 => ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ one_one_real ) )
% 0.30/1.76 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ D ) @ T )
% 0.30/1.76 => ( ( T @ p )
% 0.30/1.76 != zero_zero_real ) ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_4_that,axiom,
% 0.30/1.76 ! [Delta: real,Tau: a > real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ Delta )
% 0.30/1.76 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta ) @ Tau )
% 0.30/1.76 => ( ( ( Tau @ p )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 => ( ! [Y3: a] :
% 0.30/1.76 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta ) )
% 0.30/1.76 => ( ord_less_real @ ( abs_abs_real @ ( Tau @ Y3 ) ) @ one_one_real ) )
% 0.30/1.76 => ( ! [Y3: a] :
% 0.30/1.76 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y3 @ ( Tau @ Y3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.30/1.76 => thesisa ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_5_assms_I1_J,axiom,
% 0.30/1.76 poinca522724647ment_a @ f @ x @ a2 @ b ).
% 0.30/1.76
% 0.30/1.76 thf(fact_6_assms_I3_J,axiom,
% 0.30/1.76 limit_94460170oint_a @ f @ x @ x2 @ p ).
% 0.30/1.76
% 0.30/1.76 thf(fact_7_centre__in__ball,axiom,
% 0.30/1.76 ! [X: real,E: real] :
% 0.30/1.76 ( ( member_real @ X @ ( elemen341675809l_real @ X @ E ) )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ E ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_8_centre__in__ball,axiom,
% 0.30/1.76 ! [X: a,E: real] :
% 0.30/1.76 ( ( member_a @ X @ ( elemen154694473ball_a @ X @ E ) )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ E ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_9_zero__less__abs__iff,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_a @ zero_zero_a @ ( abs_abs_a @ A ) )
% 0.30/1.76 = ( A != zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_10_zero__less__abs__iff,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.30/1.76 = ( A != zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_11_cross__time__continuous,axiom,
% 0.30/1.76 ! [A: a,B: a,X: a,E: real] :
% 0.30/1.76 ( ( poinca522724647ment_a @ f @ x @ A @ B )
% 0.30/1.76 => ( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
% 0.30/1.76 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/1.76 => ~ ! [D: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/1.76 => ! [T: a > real] :
% 0.30/1.76 ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
% 0.30/1.76 => ( ! [Y2: a] :
% 0.30/1.76 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment_a @ A @ B ) ) )
% 0.30/1.76 => ( ! [Y2: a] :
% 0.30/1.76 ( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
% 0.30/1.76 => ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ E ) )
% 0.30/1.76 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
% 0.30/1.76 => ( ( T @ X )
% 0.30/1.76 != zero_zero_real ) ) ) ) ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_12_abs__1,axiom,
% 0.30/1.76 ( ( abs_abs_real @ one_one_real )
% 0.30/1.76 = one_one_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_13_abs__0,axiom,
% 0.30/1.76 ( ( abs_abs_real @ zero_zero_real )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_14_abs__0__eq,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( zero_zero_real
% 0.30/1.76 = ( abs_abs_real @ A ) )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_15_abs__0__eq,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( zero_zero_a
% 0.30/1.76 = ( abs_abs_a @ A ) )
% 0.30/1.76 = ( A = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_16_abs__eq__0,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ( abs_abs_real @ A )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_17_abs__eq__0,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ( abs_abs_a @ A )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 = ( A = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_18_abs__zero,axiom,
% 0.30/1.76 ( ( abs_abs_real @ zero_zero_real )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_19_abs__zero,axiom,
% 0.30/1.76 ( ( abs_abs_a @ zero_zero_a )
% 0.30/1.76 = zero_zero_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_20_fixpoint__sol_I2_J,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
% 0.30/1.76 = X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_21_gt__one__absI,axiom,
% 0.30/1.76 ! [K: real] :
% 0.30/1.76 ( ( ord_less_real @ ( abs_abs_real @ K ) @ one_one_real )
% 0.30/1.76 => ( ord_less_real @ K @ one_one_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_22_transversal__segment__reverse,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.30/1.76 => ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_23_transversal__segment__commute,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( poinca522724647ment_a @ f @ x @ X @ Y )
% 0.30/1.76 = ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_24_abs__idempotent,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/1.76 = ( abs_abs_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_25_abs__idempotent,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( abs_abs_a @ ( abs_abs_a @ A ) )
% 0.30/1.76 = ( abs_abs_a @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_26_abs__abs,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/1.76 = ( abs_abs_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_27_transversal__segment__exists,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ~ ! [A2: a,B2: a] :
% 0.30/1.76 ( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
% 0.30/1.76 => ~ ( poinca522724647ment_a @ f @ x @ A2 @ B2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_28_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
% 0.30/1.76 poinca522724647ment_a = poinca522724647ment_a ).
% 0.30/1.76
% 0.30/1.76 thf(fact_29_zero__reorient,axiom,
% 0.30/1.76 ! [X: real] :
% 0.30/1.76 ( ( zero_zero_real = X )
% 0.30/1.76 = ( X = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_30_zero__reorient,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( zero_zero_a = X )
% 0.30/1.76 = ( X = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_31_linorder__neqE__linordered__idom,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( X != Y )
% 0.30/1.76 => ( ~ ( ord_less_real @ X @ Y )
% 0.30/1.76 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_32_one__reorient,axiom,
% 0.30/1.76 ! [X: real] :
% 0.30/1.76 ( ( one_one_real = X )
% 0.30/1.76 = ( X = one_one_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_33_zero__neq__one,axiom,
% 0.30/1.76 zero_zero_real != one_one_real ).
% 0.30/1.76
% 0.30/1.76 thf(fact_34_abs__eq__0__iff,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ( abs_abs_real @ A )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_35_abs__one,axiom,
% 0.30/1.76 ( ( abs_abs_real @ one_one_real )
% 0.30/1.76 = one_one_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_36_not__one__less__zero,axiom,
% 0.30/1.76 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_37_zero__less__one,axiom,
% 0.30/1.76 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/1.76
% 0.30/1.76 thf(fact_38_abs__not__less__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_39_abs__not__less__zero,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ~ ( ord_less_a @ ( abs_abs_a @ A ) @ zero_zero_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_40_abs__of__pos,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( abs_abs_real @ A )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_41_abs__of__pos,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( abs_abs_a @ A )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_42_periodic__orbit__imp__flow0__regular,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( period138238489rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ( f @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) )
% 0.30/1.76 != zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_43_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( period1305449585riod_a @ f @ x @ X )
% 0.30/1.76 = zero_zero_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_44_local_Oflow__initial__time__if,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = X0 ) )
% 0.30/1.76 & ( ~ ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_45_rev_Otransversal__segment__exists,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ~ ! [A2: a,B2: a] :
% 0.30/1.76 ( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
% 0.30/1.76 => ~ ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A2 @ B2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_46_rev_Ofixpoint__sol_I2_J,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 )
% 0.30/1.76 = X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_47_mem__Collect__eq,axiom,
% 0.30/1.76 ! [A: a,P: a > $o] :
% 0.30/1.76 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.30/1.76 = ( P @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_48_mem__Collect__eq,axiom,
% 0.30/1.76 ! [A: real,P: real > $o] :
% 0.30/1.76 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.30/1.76 = ( P @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_49_Collect__mem__eq,axiom,
% 0.30/1.76 ! [A3: set_a] :
% 0.30/1.76 ( ( collect_a
% 0.30/1.76 @ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
% 0.30/1.76 = A3 ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_50_Collect__mem__eq,axiom,
% 0.30/1.76 ! [A3: set_real] :
% 0.30/1.76 ( ( collect_real
% 0.30/1.76 @ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
% 0.30/1.76 = A3 ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_51_Collect__cong,axiom,
% 0.30/1.76 ! [P: real > $o,Q: real > $o] :
% 0.30/1.76 ( ! [X3: real] :
% 0.30/1.76 ( ( P @ X3 )
% 0.30/1.76 = ( Q @ X3 ) )
% 0.30/1.76 => ( ( collect_real @ P )
% 0.30/1.76 = ( collect_real @ Q ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_52_Collect__cong,axiom,
% 0.30/1.76 ! [P: a > $o,Q: a > $o] :
% 0.30/1.76 ( ! [X3: a] :
% 0.30/1.76 ( ( P @ X3 )
% 0.30/1.76 = ( Q @ X3 ) )
% 0.30/1.76 => ( ( collect_a @ P )
% 0.30/1.76 = ( collect_a @ Q ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_53_local_Oflow__undefined0,axiom,
% 0.30/1.76 ! [T2: real,X: a] :
% 0.30/1.76 ( ~ ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
% 0.30/1.76 = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_54_rev__transversal__segment,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A @ B )
% 0.30/1.76 = ( poinca522724647ment_a @ f @ x @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_55_rev_Otransversal__segment__reverse,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
% 0.30/1.76 => ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_56_rev_Otransversal__segment__commute,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
% 0.30/1.76 = ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_57_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_58_closed__orbit__in__domain,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( member_a @ X @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_59_add_Oinverse__inverse,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_60_add_Oinverse__inverse,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_61_neg__equal__iff__equal,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ( uminus_uminus_a @ A )
% 0.30/1.76 = ( uminus_uminus_a @ B ) )
% 0.30/1.76 = ( A = B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_62_neg__equal__iff__equal,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ( uminus_uminus_real @ A )
% 0.30/1.76 = ( uminus_uminus_real @ B ) )
% 0.30/1.76 = ( A = B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_63_existence__ivl__zero,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_64_local_Omem__existence__ivl__subset,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_65_rev_Oclosed__orbit__eq__rev,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = ( period720806154rbit_a @ f @ x @ X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_66_rev_Oclosed__orbit__in__domain,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( member_a @ X @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_67_closed__orbit__flow0,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( period720806154rbit_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_68_rev_Oexistence__ivl__zero,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_69_local_Oexistence__ivl__initial__time,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_70_local_Orev_Omem__existence__ivl__subset,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ T2 @ top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_71_fixpoint__sol_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.30/1.76 = top_top_set_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_72_rev_Oclosed__orbit__flow0,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_73_closed__orbit__global__existence,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ f @ x @ X )
% 0.30/1.76 = top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_74_closed__orbitI,axiom,
% 0.30/1.76 ! [T2: real,T3: real,Y: a] :
% 0.30/1.76 ( ( T2 != T3 )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.30/1.76 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
% 0.30/1.76 => ( period720806154rbit_a @ f @ x @ Y ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_75_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_76_periodic__orbit__period_I2_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( period1305449585riod_a @ f @ x @ X ) )
% 0.30/1.76 = X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_77_closed__orbit__periodic,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ( ( f @ X )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_78_local_Orev_Oexistence__ivl__initial__time,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_79_rev_Ofixpoint__sol_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = top_top_set_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_80_local_Orev_Oflow__undefined0,axiom,
% 0.30/1.76 ! [T2: real,X: a] :
% 0.30/1.76 ( ~ ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 )
% 0.30/1.76 = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_81_closed__orbit__def,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 = ( ? [X2: real] :
% 0.30/1.76 ( ( member_real @ X2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.30/1.76 & ( X2 != zero_zero_real )
% 0.30/1.76 & ( ( auto_ll_on_flow0_a @ f @ x @ X @ X2 )
% 0.30/1.76 = X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_82_rev_Oclosed__orbit__global__existence,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_83_rev_Oclosed__orbitI,axiom,
% 0.30/1.76 ! [T2: real,T3: real,Y: a] :
% 0.30/1.76 ( ( T2 != T3 )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.30/1.76 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T3 ) )
% 0.30/1.76 => ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_84_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( member_a @ X @ x )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 => ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = zero_zero_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_85_rev_Operiodic__orbit__imp__flow0__regular,axiom,
% 0.30/1.76 ! [X: a,T2: real] :
% 0.30/1.76 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ( uminus_uminus_a_a @ f @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T2 ) )
% 0.30/1.76 != zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_86_periodic__orbitI,axiom,
% 0.30/1.76 ! [T2: real,T3: real,Y: a] :
% 0.30/1.76 ( ( T2 != T3 )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.30/1.76 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
% 0.30/1.76 => ( ( ( f @ Y )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ( period138238489rbit_a @ f @ x @ Y ) ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_87_periodic__orbit__period_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_88_closed__orbit__period__zero__fixed__point,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ( ( period1305449585riod_a @ f @ x @ X )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 => ( ( f @ X )
% 0.30/1.76 = zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_89_rev_Operiodic__orbit__period_I2_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.30/1.76 = X ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_90_rev_Oclosed__orbit__periodic,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_91_add_Oinverse__neutral,axiom,
% 0.30/1.76 ( ( uminus_uminus_a @ zero_zero_a )
% 0.30/1.76 = zero_zero_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_92_add_Oinverse__neutral,axiom,
% 0.30/1.76 ( ( uminus_uminus_real @ zero_zero_real )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_93_neg__0__equal__iff__equal,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( zero_zero_a
% 0.30/1.76 = ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( zero_zero_a = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_94_neg__0__equal__iff__equal,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( zero_zero_real
% 0.30/1.76 = ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( zero_zero_real = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_95_neg__equal__0__iff__equal,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ( uminus_uminus_a @ A )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 = ( A = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_96_neg__equal__0__iff__equal,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ( uminus_uminus_real @ A )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_97_equal__neg__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_98_neg__equal__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ( uminus_uminus_real @ A )
% 0.30/1.76 = A )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_99_local_Orev_Oflow__initial__time__if,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = X0 ) )
% 0.30/1.76 & ( ~ ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_100_neg__less__iff__less,axiom,
% 0.30/1.76 ! [B: a,A: a] :
% 0.30/1.76 ( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( ord_less_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_101_neg__less__iff__less,axiom,
% 0.30/1.76 ! [B: real,A: real] :
% 0.30/1.76 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_102_rev_Oclosed__orbit__def,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = ( ? [X2: real] :
% 0.30/1.76 ( ( member_real @ X2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.30/1.76 & ( X2 != zero_zero_real )
% 0.30/1.76 & ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ X2 )
% 0.30/1.76 = X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_103_abs__minus,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( abs_abs_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_104_abs__minus__cancel,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( abs_abs_a @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( abs_abs_a @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_105_abs__minus__cancel,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( abs_abs_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_106_rev_Operiodic__orbitI,axiom,
% 0.30/1.76 ! [T2: real,T3: real,Y: a] :
% 0.30/1.76 ( ( T2 != T3 )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.30/1.76 => ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T3 ) )
% 0.30/1.76 => ( ( ( uminus_uminus_a_a @ f @ Y )
% 0.30/1.76 != zero_zero_a )
% 0.30/1.76 => ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_107_rev_Operiodic__orbit__period_I1_J,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_108_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 => ( ( uminus_uminus_a_a @ f @ X )
% 0.30/1.76 = zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_109_periodic__orbit__def,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ f @ x @ X )
% 0.30/1.76 = ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_110_rev_Operiodic__orbit__def,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 = ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 & ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_111_less__neg__neg,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_112_neg__less__pos,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_113_neg__0__less__iff__less,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_114_neg__0__less__iff__less,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_115_neg__less__0__iff__less,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.30/1.76 = ( ord_less_a @ zero_zero_a @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_116_neg__less__0__iff__less,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_117_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ X0 @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_118_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ X0 @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_119_flow0__defined,axiom,
% 0.30/1.76 ! [Xa: real,X0: a] :
% 0.30/1.76 ( ( member_real @ Xa @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_120_local_Oflow__in__domain,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_121_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_122_local_Oexistence__ivl__initial__time__iff,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 = ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_123_local_Oflow__initial__time,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = X0 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_124_rev_Oflow0__defined,axiom,
% 0.30/1.76 ! [Xa: real,X0: a] :
% 0.30/1.76 ( ( member_real @ Xa @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ Xa ) @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_125_local_Orev_Oflow__in__domain,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ T2 ) @ x ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_126_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
% 0.30/1.76 ! [T2: real,X0: a] :
% 0.30/1.76 ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_127_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 = ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 & ( member_a @ X0 @ x ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_128_local_Orev_Oflow__initial__time,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ zero_zero_real )
% 0.30/1.76 = X0 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_129_equation__minus__iff,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( uminus_uminus_a @ B ) )
% 0.30/1.76 = ( B
% 0.30/1.76 = ( uminus_uminus_a @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_130_equation__minus__iff,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( uminus_uminus_real @ B ) )
% 0.30/1.76 = ( B
% 0.30/1.76 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_131_minus__equation__iff,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ( uminus_uminus_a @ A )
% 0.30/1.76 = B )
% 0.30/1.76 = ( ( uminus_uminus_a @ B )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_132_minus__equation__iff,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ( uminus_uminus_real @ A )
% 0.30/1.76 = B )
% 0.30/1.76 = ( ( uminus_uminus_real @ B )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_133_less__minus__iff,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
% 0.30/1.76 = ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_134_less__minus__iff,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/1.76 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_135_minus__less__iff,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
% 0.30/1.76 = ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_136_minus__less__iff,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/1.76 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_137_abs__eq__iff,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( ( abs_abs_real @ X )
% 0.30/1.76 = ( abs_abs_real @ Y ) )
% 0.30/1.76 = ( ( X = Y )
% 0.30/1.76 | ( X
% 0.30/1.76 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_138_abs__less__iff,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.30/1.76 = ( ( ord_less_real @ A @ B )
% 0.30/1.76 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_139_abs__if,axiom,
% 0.30/1.76 ( abs_abs_real
% 0.30/1.76 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_140_abs__of__neg,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( abs_abs_a @ A )
% 0.30/1.76 = ( uminus_uminus_a @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_141_abs__of__neg,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( abs_abs_real @ A )
% 0.30/1.76 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_142__092_060alpha_062__limit__point__eq__rev,axiom,
% 0.30/1.76 ! [X: a,P2: a] :
% 0.30/1.76 ( ( limit_1865182413oint_a @ f @ x @ X @ P2 )
% 0.30/1.76 = ( limit_94460170oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_143_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
% 0.30/1.76 ! [X: a,P2: a] :
% 0.30/1.76 ( ( limit_1865182413oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 )
% 0.30/1.76 = ( limit_94460170oint_a @ f @ x @ X @ P2 ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_144_rev_Omvar_Ointerval__axioms,axiom,
% 0.30/1.76 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_145_mvar_Ointerval__axioms,axiom,
% 0.30/1.76 ! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_146_abs__neg__one,axiom,
% 0.30/1.76 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/1.76 = one_one_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_147_rev_Oclosed__orbitE,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ~ ! [T4: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/1.76 => ~ ! [T5: real] :
% 0.30/1.76 ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( plus_plus_real @ T5 @ T4 ) )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T5 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_148_local_Orev_Oexistence__ivl__notempty,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.30/1.76 != bot_bot_set_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_149_rev_Oclosed__orbit__period__nonneg,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
% 0.30/1.76 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_150_rev_Ovec__simps_I11_J,axiom,
% 0.30/1.76 ! [Ae: a,Be: a,Cd: a] :
% 0.30/1.76 ( ( plus_plus_a @ ( plus_plus_a @ Ae @ Be ) @ Cd )
% 0.30/1.76 = ( plus_plus_a @ Ae @ ( plus_plus_a @ Be @ Cd ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_151_rev_Ovec__simps_I11_J,axiom,
% 0.30/1.76 ! [Ae: real,Be: real,Cd: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( plus_plus_real @ Ae @ Be ) @ Cd )
% 0.30/1.76 = ( plus_plus_real @ Ae @ ( plus_plus_real @ Be @ Cd ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_152_rev_Ovec__simps_I12_J,axiom,
% 0.30/1.76 ( plus_plus_a
% 0.30/1.76 = ( ^ [Ag: a,Bg: a] : ( plus_plus_a @ Bg @ Ag ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_153_rev_Ovec__simps_I12_J,axiom,
% 0.30/1.76 ( plus_plus_real
% 0.30/1.76 = ( ^ [Ag: real,Bg: real] : ( plus_plus_real @ Bg @ Ag ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_154_rev_Ovec__simps_I13_J,axiom,
% 0.30/1.76 ! [Bg2: a,Ag2: a,Cf: a] :
% 0.30/1.76 ( ( plus_plus_a @ Bg2 @ ( plus_plus_a @ Ag2 @ Cf ) )
% 0.30/1.76 = ( plus_plus_a @ Ag2 @ ( plus_plus_a @ Bg2 @ Cf ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_155_rev_Ovec__simps_I13_J,axiom,
% 0.30/1.76 ! [Bg2: real,Ag2: real,Cf: real] :
% 0.30/1.76 ( ( plus_plus_real @ Bg2 @ ( plus_plus_real @ Ag2 @ Cf ) )
% 0.30/1.76 = ( plus_plus_real @ Ag2 @ ( plus_plus_real @ Bg2 @ Cf ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_156_interval__axioms,axiom,
% 0.30/1.76 initia826609931terval @ top_top_set_real ).
% 0.30/1.76
% 0.30/1.76 thf(fact_157_add__right__cancel,axiom,
% 0.30/1.76 ! [B: a,A: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ B @ A )
% 0.30/1.76 = ( plus_plus_a @ C @ A ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_158_add__right__cancel,axiom,
% 0.30/1.76 ! [B: real,A: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ B @ A )
% 0.30/1.76 = ( plus_plus_real @ C @ A ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_159_add__left__cancel,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ A @ B )
% 0.30/1.76 = ( plus_plus_a @ A @ C ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_160_add__left__cancel,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ A @ B )
% 0.30/1.76 = ( plus_plus_real @ A @ C ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_161_rev__eq__flow,axiom,
% 0.30/1.76 ! [Y: a,T2: real] :
% 0.30/1.76 ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ f @ x @ Y @ ( uminus_uminus_real @ T2 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_162_rev_Orev__eq__flow,axiom,
% 0.30/1.76 ! [Y: a,T2: real] :
% 0.30/1.76 ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ ( uminus_uminus_real @ T2 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_163_recurrence__time__flip__sign_I1_J,axiom,
% 0.30/1.76 ! [T6: real,X: a] :
% 0.30/1.76 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T6 )
% 0.30/1.76 = X )
% 0.30/1.76 => ( member_real @ ( uminus_uminus_real @ T6 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_164_recurrence__time__flip__sign_I2_J,axiom,
% 0.30/1.76 ! [T6: real,X: a] :
% 0.30/1.76 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T6 )
% 0.30/1.76 = X )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( uminus_uminus_real @ T6 ) )
% 0.30/1.76 = X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_165_rev_Orecurrence__time__flip__sign_I1_J,axiom,
% 0.30/1.76 ! [T6: real,X: a] :
% 0.30/1.76 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T6 )
% 0.30/1.76 = X )
% 0.30/1.76 => ( member_real @ ( uminus_uminus_real @ T6 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_166_rev_Orecurrence__time__flip__sign_I2_J,axiom,
% 0.30/1.76 ! [T6: real,X: a] :
% 0.30/1.76 ( ( member_real @ T6 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
% 0.30/1.76 => ( ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ T6 )
% 0.30/1.76 = X )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( uminus_uminus_real @ T6 ) )
% 0.30/1.76 = X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_167_local_Oflow__trans,axiom,
% 0.30/1.76 ! [S: real,X0: a,T2: real] :
% 0.30/1.76 ( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ f @ x @ X0 @ ( plus_plus_real @ S @ T2 ) )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_168_local_Oexistence__ivl__trans_H,axiom,
% 0.30/1.76 ! [T2: real,S: real,X0: a] :
% 0.30/1.76 ( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_169_local_Oexistence__ivl__trans,axiom,
% 0.30/1.76 ! [S: real,X0: a,T2: real] :
% 0.30/1.76 ( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
% 0.30/1.76 => ( member_real @ ( plus_plus_real @ S @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_170_local_Orev_Oflow__trans,axiom,
% 0.30/1.76 ! [S: real,X0: a,T2: real] :
% 0.30/1.76 ( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) ) )
% 0.30/1.76 => ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ ( plus_plus_real @ S @ T2 ) )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_171_local_Orev_Oexistence__ivl__trans_H,axiom,
% 0.30/1.76 ! [T2: real,S: real,X0: a] :
% 0.30/1.76 ( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ T2 ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_172_local_Orev_Oexistence__ivl__trans,axiom,
% 0.30/1.76 ! [S: real,X0: a,T2: real] :
% 0.30/1.76 ( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
% 0.30/1.76 => ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 @ S ) ) )
% 0.30/1.76 => ( member_real @ ( plus_plus_real @ S @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_173_local_Oexistence__ivl__notempty,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.30/1.76 != bot_bot_set_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_174_closed__orbit__period__nonneg,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_175_neg__le__iff__le,axiom,
% 0.30/1.76 ! [B: a,A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_176_neg__le__iff__le,axiom,
% 0.30/1.76 ! [B: real,A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_177_add__cancel__right__right,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = ( B = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_178_add__cancel__right__right,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( plus_plus_a @ A @ B ) )
% 0.30/1.76 = ( B = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_179_add__cancel__right__left,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( plus_plus_real @ B @ A ) )
% 0.30/1.76 = ( B = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_180_add__cancel__right__left,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( A
% 0.30/1.76 = ( plus_plus_a @ B @ A ) )
% 0.30/1.76 = ( B = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_181_add__cancel__left__right,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ A @ B )
% 0.30/1.76 = A )
% 0.30/1.76 = ( B = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_182_add__cancel__left__right,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ A @ B )
% 0.30/1.76 = A )
% 0.30/1.76 = ( B = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_183_add__cancel__left__left,axiom,
% 0.30/1.76 ! [B: real,A: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ B @ A )
% 0.30/1.76 = A )
% 0.30/1.76 = ( B = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_184_add__cancel__left__left,axiom,
% 0.30/1.76 ! [B: a,A: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ B @ A )
% 0.30/1.76 = A )
% 0.30/1.76 = ( B = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_185_double__zero__sym,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( zero_zero_real
% 0.30/1.76 = ( plus_plus_real @ A @ A ) )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_186_linordered__ab__group__add__class_Odouble__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ A @ A )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_187_add_Oright__neutral,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_188_add_Oright__neutral,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_189_add_Oleft__neutral,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_190_add_Oleft__neutral,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_191_add__le__cancel__right,axiom,
% 0.30/1.76 ! [A: a,C: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.30/1.76 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_192_add__le__cancel__right,axiom,
% 0.30/1.76 ! [A: real,C: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/1.76 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_193_add__le__cancel__left,axiom,
% 0.30/1.76 ! [C: a,A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.30/1.76 = ( ord_less_eq_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_194_add__le__cancel__left,axiom,
% 0.30/1.76 ! [C: real,A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/1.76 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_195_add__less__cancel__right,axiom,
% 0.30/1.76 ! [A: real,C: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/1.76 = ( ord_less_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_196_add__less__cancel__right,axiom,
% 0.30/1.76 ! [A: a,C: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.30/1.76 = ( ord_less_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_197_add__less__cancel__left,axiom,
% 0.30/1.76 ! [C: real,A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/1.76 = ( ord_less_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_198_add__less__cancel__left,axiom,
% 0.30/1.76 ! [C: a,A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.30/1.76 = ( ord_less_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_199_closed__orbitE,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( period720806154rbit_a @ f @ x @ X )
% 0.30/1.76 => ~ ! [T4: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/1.76 => ~ ! [T5: real] :
% 0.30/1.76 ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( plus_plus_real @ T5 @ T4 ) )
% 0.30/1.76 = ( auto_ll_on_flow0_a @ f @ x @ X @ T5 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_200_minus__add__distrib,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
% 0.30/1.76 = ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_201_minus__add__distrib,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_202_minus__add__cancel,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
% 0.30/1.76 = B ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_203_minus__add__cancel,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = B ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_204_add__minus__cancel,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
% 0.30/1.76 = B ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_205_add__minus__cancel,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.30/1.76 = B ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_206_abs__add__abs,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( abs_abs_a @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) )
% 0.30/1.76 = ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_207_abs__add__abs,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.30/1.76 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_208_neg__0__le__iff__le,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_209_neg__0__le__iff__le,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_210_neg__le__0__iff__le,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
% 0.30/1.76 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_211_neg__le__0__iff__le,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_212_less__eq__neg__nonpos,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_213_neg__less__eq__nonneg,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_214_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_215_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/1.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_216_le__add__same__cancel2,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.30/1.76 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_217_le__add__same__cancel2,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_218_le__add__same__cancel1,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.30/1.76 = ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_219_le__add__same__cancel1,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_220_add__le__same__cancel2,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.30/1.76 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_221_add__le__same__cancel2,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/1.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_222_add__le__same__cancel1,axiom,
% 0.30/1.76 ! [B: a,A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.30/1.76 = ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_223_add__le__same__cancel1,axiom,
% 0.30/1.76 ! [B: real,A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/1.76 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_224_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_225_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/1.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_226_less__add__same__cancel2,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_227_less__add__same__cancel2,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
% 0.30/1.76 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_228_less__add__same__cancel1,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_229_less__add__same__cancel1,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
% 0.30/1.76 = ( ord_less_a @ zero_zero_a @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_230_add__less__same__cancel2,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/1.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_231_add__less__same__cancel2,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
% 0.30/1.76 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_232_add__less__same__cancel1,axiom,
% 0.30/1.76 ! [B: real,A: real] :
% 0.30/1.76 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/1.76 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_233_add__less__same__cancel1,axiom,
% 0.30/1.76 ! [B: a,A: a] :
% 0.30/1.76 ( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
% 0.30/1.76 = ( ord_less_a @ A @ zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_234_add_Oleft__inverse,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
% 0.30/1.76 = zero_zero_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_235_add_Oleft__inverse,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_236_add_Oright__inverse,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ A @ ( uminus_uminus_a @ A ) )
% 0.30/1.76 = zero_zero_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_237_add_Oright__inverse,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_238_abs__le__zero__iff,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ zero_zero_a )
% 0.30/1.76 = ( A = zero_zero_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_239_abs__le__zero__iff,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.30/1.76 = ( A = zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_240_abs__le__self__iff,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ A )
% 0.30/1.76 = ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_241_abs__le__self__iff,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.30/1.76 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_242_abs__of__nonneg,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( abs_abs_a @ A )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_243_abs__of__nonneg,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( abs_abs_real @ A )
% 0.30/1.76 = A ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_244_ball__trivial,axiom,
% 0.30/1.76 ! [X: a] :
% 0.30/1.76 ( ( elemen154694473ball_a @ X @ zero_zero_real )
% 0.30/1.76 = bot_bot_set_a ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_245_ball__trivial,axiom,
% 0.30/1.76 ! [X: real] :
% 0.30/1.76 ( ( elemen341675809l_real @ X @ zero_zero_real )
% 0.30/1.76 = bot_bot_set_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_246_add__neg__numeral__special_I7_J,axiom,
% 0.30/1.76 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_247_add__neg__numeral__special_I8_J,axiom,
% 0.30/1.76 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/1.76 = zero_zero_real ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_248_abs__of__nonpos,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( abs_abs_a @ A )
% 0.30/1.76 = ( uminus_uminus_a @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_249_abs__of__nonpos,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( abs_abs_real @ A )
% 0.30/1.76 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_250_ball__eq__empty,axiom,
% 0.30/1.76 ! [X: a,E: real] :
% 0.30/1.76 ( ( ( elemen154694473ball_a @ X @ E )
% 0.30/1.76 = bot_bot_set_a )
% 0.30/1.76 = ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_251_ball__eq__empty,axiom,
% 0.30/1.76 ! [X: real,E: real] :
% 0.30/1.76 ( ( ( elemen341675809l_real @ X @ E )
% 0.30/1.76 = bot_bot_set_real )
% 0.30/1.76 = ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_252_local_Oexistence__ivl__undefined,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ~ ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_253_local_Orev_Oexistence__ivl__undefined,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ~ ( member_a @ X0 @ x )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_254_local_Oexistence__ivl__empty__iff,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real )
% 0.30/1.76 = ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 | ~ ( member_a @ X0 @ x ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_255_local_Oexistence__ivl__empty1,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ f @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_256_local_Orev_Oexistence__ivl__empty__iff,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real )
% 0.30/1.76 = ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 | ~ ( member_a @ X0 @ x ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_257_local_Orev_Oexistence__ivl__empty1,axiom,
% 0.30/1.76 ! [X0: a] :
% 0.30/1.76 ( ~ ( member_real @ zero_zero_real @ top_top_set_real )
% 0.30/1.76 => ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
% 0.30/1.76 = bot_bot_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_258_is__num__normalize_I8_J,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/1.76 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_259_abs__triangle__ineq,axiom,
% 0.30/1.76 ! [A: a,B: a] : ( ord_less_eq_a @ ( abs_abs_a @ ( plus_plus_a @ A @ B ) ) @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_260_abs__triangle__ineq,axiom,
% 0.30/1.76 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_261_le__numeral__extra_I3_J,axiom,
% 0.30/1.76 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/1.76
% 0.30/1.76 thf(fact_262_le__numeral__extra_I4_J,axiom,
% 0.30/1.76 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.30/1.76
% 0.30/1.76 thf(fact_263_is__num__normalize_I1_J,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/1.76 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_264_add__le__imp__le__right,axiom,
% 0.30/1.76 ! [A: a,C: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
% 0.30/1.76 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_265_add__le__imp__le__right,axiom,
% 0.30/1.76 ! [A: real,C: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/1.76 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_266_add__le__imp__le__left,axiom,
% 0.30/1.76 ! [C: a,A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
% 0.30/1.76 => ( ord_less_eq_a @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_267_add__le__imp__le__left,axiom,
% 0.30/1.76 ! [C: real,A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/1.76 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_268_add__less__le__mono,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a,D2: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ B )
% 0.30/1.76 => ( ( ord_less_eq_a @ C @ D2 )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_269_add__less__le__mono,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real,D2: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ B )
% 0.30/1.76 => ( ( ord_less_eq_real @ C @ D2 )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_270_add__le__less__mono,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a,D2: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ B )
% 0.30/1.76 => ( ( ord_less_a @ C @ D2 )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_271_add__le__less__mono,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real,D2: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ B )
% 0.30/1.76 => ( ( ord_less_real @ C @ D2 )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_272_add__nonpos__eq__0__iff,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ X @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_eq_a @ Y @ zero_zero_a )
% 0.30/1.76 => ( ( ( plus_plus_a @ X @ Y )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 = ( ( X = zero_zero_a )
% 0.30/1.76 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_273_add__nonpos__eq__0__iff,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.30/1.76 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( ( X = zero_zero_real )
% 0.30/1.76 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_274_add__nonneg__eq__0__iff,axiom,
% 0.30/1.76 ! [X: a,Y: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ X )
% 0.30/1.76 => ( ( ord_less_eq_a @ zero_zero_a @ Y )
% 0.30/1.76 => ( ( ( plus_plus_a @ X @ Y )
% 0.30/1.76 = zero_zero_a )
% 0.30/1.76 = ( ( X = zero_zero_a )
% 0.30/1.76 & ( Y = zero_zero_a ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_275_add__nonneg__eq__0__iff,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/1.76 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/1.76 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 = ( ( X = zero_zero_real )
% 0.30/1.76 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_276_add__nonpos__nonpos,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_eq_a @ B @ zero_zero_a )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_277_add__nonpos__nonpos,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_278_add__nonneg__nonneg,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.30/1.76 => ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_279_add__nonneg__nonneg,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/1.76 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_280_add__increasing2,axiom,
% 0.30/1.76 ! [C: a,B: a,A: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ C )
% 0.30/1.76 => ( ( ord_less_eq_a @ B @ A )
% 0.30/1.76 => ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_281_add__increasing2,axiom,
% 0.30/1.76 ! [C: real,B: real,A: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/1.76 => ( ( ord_less_eq_real @ B @ A )
% 0.30/1.76 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_282_add__decreasing2,axiom,
% 0.30/1.76 ! [C: a,A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ C @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_eq_a @ A @ B )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_283_add__decreasing2,axiom,
% 0.30/1.76 ! [C: real,A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_eq_real @ A @ B )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_284_add__right__mono,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ B )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_285_add__right__mono,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ B )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_286_add__increasing,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_eq_a @ B @ C )
% 0.30/1.76 => ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_287_add__increasing,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_eq_real @ B @ C )
% 0.30/1.76 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_288_add__decreasing,axiom,
% 0.30/1.76 ! [A: a,C: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_eq_a @ C @ B )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_289_add__decreasing,axiom,
% 0.30/1.76 ! [A: real,C: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_eq_real @ C @ B )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_290_add__left__mono,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ B )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_291_add__left__mono,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ B )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_292_add__right__imp__eq,axiom,
% 0.30/1.76 ! [B: a,A: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ B @ A )
% 0.30/1.76 = ( plus_plus_a @ C @ A ) )
% 0.30/1.76 => ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_293_add__right__imp__eq,axiom,
% 0.30/1.76 ! [B: real,A: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ B @ A )
% 0.30/1.76 = ( plus_plus_real @ C @ A ) )
% 0.30/1.76 => ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_294_add__left__imp__eq,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ A @ B )
% 0.30/1.76 = ( plus_plus_a @ A @ C ) )
% 0.30/1.76 => ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_295_add__left__imp__eq,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ A @ B )
% 0.30/1.76 = ( plus_plus_real @ A @ C ) )
% 0.30/1.76 => ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_296_add__mono,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a,D2: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ B )
% 0.30/1.76 => ( ( ord_less_eq_a @ C @ D2 )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_297_add__mono,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real,D2: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ B )
% 0.30/1.76 => ( ( ord_less_eq_real @ C @ D2 )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_298_add_Oleft__commute,axiom,
% 0.30/1.76 ! [B: a,A: a,C: a] :
% 0.30/1.76 ( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
% 0.30/1.76 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_299_add_Oleft__commute,axiom,
% 0.30/1.76 ! [B: real,A: real,C: real] :
% 0.30/1.76 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.30/1.76 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_300_add_Ocommute,axiom,
% 0.30/1.76 ( plus_plus_a
% 0.30/1.76 = ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_301_add_Ocommute,axiom,
% 0.30/1.76 ( plus_plus_real
% 0.30/1.76 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_302_add_Oright__cancel,axiom,
% 0.30/1.76 ! [B: a,A: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ B @ A )
% 0.30/1.76 = ( plus_plus_a @ C @ A ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_303_add_Oright__cancel,axiom,
% 0.30/1.76 ! [B: real,A: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ B @ A )
% 0.30/1.76 = ( plus_plus_real @ C @ A ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_304_add_Oleft__cancel,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ( plus_plus_a @ A @ B )
% 0.30/1.76 = ( plus_plus_a @ A @ C ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_305_add_Oleft__cancel,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ( plus_plus_real @ A @ B )
% 0.30/1.76 = ( plus_plus_real @ A @ C ) )
% 0.30/1.76 = ( B = C ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_306_add_Oassoc,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.30/1.76 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_307_add_Oassoc,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/1.76 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_308_group__cancel_Oadd2,axiom,
% 0.30/1.76 ! [B4: a,K: a,B: a,A: a] :
% 0.30/1.76 ( ( B4
% 0.30/1.76 = ( plus_plus_a @ K @ B ) )
% 0.30/1.76 => ( ( plus_plus_a @ A @ B4 )
% 0.30/1.76 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_309_group__cancel_Oadd2,axiom,
% 0.30/1.76 ! [B4: real,K: real,B: real,A: real] :
% 0.30/1.76 ( ( B4
% 0.30/1.76 = ( plus_plus_real @ K @ B ) )
% 0.30/1.76 => ( ( plus_plus_real @ A @ B4 )
% 0.30/1.76 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_310_group__cancel_Oadd1,axiom,
% 0.30/1.76 ! [A3: a,K: a,A: a,B: a] :
% 0.30/1.76 ( ( A3
% 0.30/1.76 = ( plus_plus_a @ K @ A ) )
% 0.30/1.76 => ( ( plus_plus_a @ A3 @ B )
% 0.30/1.76 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_311_group__cancel_Oadd1,axiom,
% 0.30/1.76 ! [A3: real,K: real,A: real,B: real] :
% 0.30/1.76 ( ( A3
% 0.30/1.76 = ( plus_plus_real @ K @ A ) )
% 0.30/1.76 => ( ( plus_plus_real @ A3 @ B )
% 0.30/1.76 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_312_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( ord_less_eq_a @ I @ J )
% 0.30/1.76 & ( ord_less_eq_a @ K @ L ) )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_313_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/1.76 & ( ord_less_eq_real @ K @ L ) )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_314_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( I = J )
% 0.30/1.76 & ( ord_less_eq_a @ K @ L ) )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_315_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( I = J )
% 0.30/1.76 & ( ord_less_eq_real @ K @ L ) )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_316_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( ord_less_eq_a @ I @ J )
% 0.30/1.76 & ( K = L ) )
% 0.30/1.76 => ( ord_less_eq_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_317_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/1.76 & ( K = L ) )
% 0.30/1.76 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_318_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( I = J )
% 0.30/1.76 & ( K = L ) )
% 0.30/1.76 => ( ( plus_plus_a @ I @ K )
% 0.30/1.76 = ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_319_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( I = J )
% 0.30/1.76 & ( K = L ) )
% 0.30/1.76 => ( ( plus_plus_real @ I @ K )
% 0.30/1.76 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_320_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( ord_less_a @ I @ J )
% 0.30/1.76 & ( ord_less_eq_a @ K @ L ) )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_321_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( ord_less_real @ I @ J )
% 0.30/1.76 & ( ord_less_eq_real @ K @ L ) )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_322_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/1.76 ! [I: a,J: a,K: a,L: a] :
% 0.30/1.76 ( ( ( ord_less_eq_a @ I @ J )
% 0.30/1.76 & ( ord_less_a @ K @ L ) )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ I @ K ) @ ( plus_plus_a @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_323_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/1.76 ! [I: real,J: real,K: real,L: real] :
% 0.30/1.76 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/1.76 & ( ord_less_real @ K @ L ) )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.30/1.76 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/1.76 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_326_add__strict__increasing2,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_a @ B @ C )
% 0.30/1.76 => ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_327_add__strict__increasing2,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_real @ B @ C )
% 0.30/1.76 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_328_add__strict__increasing,axiom,
% 0.30/1.76 ! [A: a,B: a,C: a] :
% 0.30/1.76 ( ( ord_less_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_eq_a @ B @ C )
% 0.30/1.76 => ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_329_add__strict__increasing,axiom,
% 0.30/1.76 ! [A: real,B: real,C: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_eq_real @ B @ C )
% 0.30/1.76 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_330_add__pos__nonneg,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_eq_a @ zero_zero_a @ B )
% 0.30/1.76 => ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_331_add__pos__nonneg,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/1.76 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_332_add__nonpos__neg,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_a @ B @ zero_zero_a )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_333_add__nonpos__neg,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_334_add__nonneg__pos,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_eq_a @ zero_zero_a @ A )
% 0.30/1.76 => ( ( ord_less_a @ zero_zero_a @ B )
% 0.30/1.76 => ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_335_add__nonneg__pos,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/1.76 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/1.76 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_336_add__neg__nonpos,axiom,
% 0.30/1.76 ! [A: a,B: a] :
% 0.30/1.76 ( ( ord_less_a @ A @ zero_zero_a )
% 0.30/1.76 => ( ( ord_less_eq_a @ B @ zero_zero_a )
% 0.30/1.76 => ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_337_add__neg__nonpos,axiom,
% 0.30/1.76 ! [A: real,B: real] :
% 0.30/1.76 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/1.76 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/1.76 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_338_Elementary__Metric__Spaces_Oball__empty,axiom,
% 0.30/1.76 ! [E: real,X: a] :
% 0.30/1.76 ( ( ord_less_eq_real @ E @ zero_zero_real )
% 0.30/1.76 => ( ( elemen154694473ball_a @ X @ E )
% 0.30/1.76 = bot_bot_set_a ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_339_Elementary__Metric__Spaces_Oball__empty,axiom,
% 0.30/1.76 ! [E: real,X: real] :
% 0.30/1.76 ( ( ord_less_eq_real @ E @ zero_zero_real )
% 0.30/1.76 => ( ( elemen341675809l_real @ X @ E )
% 0.30/1.76 = bot_bot_set_real ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_340_add_Ogroup__left__neutral,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_341_add_Ogroup__left__neutral,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_342_add_Ocomm__neutral,axiom,
% 0.30/1.76 ! [A: real] :
% 0.30/1.76 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(fact_343_add_Ocomm__neutral,axiom,
% 0.30/1.76 ! [A: a] :
% 0.30/1.76 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.30/1.76 = A ) ).
% 0.30/1.76
% 0.30/1.76 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.30/1.76 ! [P: $o] :
% 0.30/1.76 ( ( P = $true )
% 0.30/1.76 | ( P = $false ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( if_real @ $false @ X @ Y )
% 0.30/1.76 = Y ) ).
% 0.30/1.76
% 0.30/1.76 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.30/1.76 ! [X: real,Y: real] :
% 0.30/1.76 ( ( if_real @ $true @ X @ Y )
% 0.30/1.76 = X ) ).
% 0.30/1.76
% 0.30/1.76 thf(conj_0,hypothesis,
% 0.30/1.76 ! [Delta2: real,Tau2: a > real] :
% 0.30/1.76 ( ( ord_less_real @ zero_zero_real @ Delta2 )
% 0.30/1.76 => ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta2 ) @ Tau2 )
% 0.30/1.76 => ( ( ( Tau2 @ p )
% 0.30/1.76 = zero_zero_real )
% 0.30/1.76 => ( ! [Y3: a] :
% 0.30/1.76 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta2 ) )
% 0.30/1.76 => ( ord_less_real @ ( abs_abs_real @ ( Tau2 @ Y3 ) ) @ one_one_real ) )
% 0.30/1.76 => ( ! [Y3: a] :
% 0.30/1.76 ( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta2 ) )
% 0.30/1.76 => ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y3 @ ( Tau2 @ Y3 ) ) @ ( line_open_segment_a @ a2 @ b ) ) )
% 0.30/1.76 => thesisa ) ) ) ) ) ).
% 0.30/1.76
% 0.30/1.76 thf(conj_1,conjecture,
% 0.30/1.76 thesisa ).
% 0.30/1.76 ------------------------
% 3.40/1.80 % (2582)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/2Mi)
% 3.40/1.80 % (2585)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/18Mi)
% 3.40/1.80 % (2583)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_2468 for (2999ds/2Mi)
% 3.40/1.80 % (2580)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_2468 for (2999ds/4Mi)
% 3.40/1.80 % (2581)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_2468 for (2999ds/27Mi)
% 3.40/1.80 % (2579)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/183Mi)
% 3.40/1.80 % (2584)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_2468 for (2999ds/275Mi)
% 3.40/1.80 % (2582)Instruction limit reached!
% 3.40/1.80 % (2582)------------------------------
% 3.40/1.80 % (2582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.80 % (2582)Termination reason: Unknown
% 3.40/1.80 % (2582)Termination phase: shuffling
% 3.40/1.80
% 3.40/1.80 % (2582)Memory used [KB]: 1279
% 3.40/1.80 % (2582)Time elapsed: 0.003 s
% 3.40/1.80 % (2582)Instructions burned: 2 (million)
% 3.40/1.80 % (2582)------------------------------
% 3.40/1.80 % (2582)------------------------------
% 3.40/1.80 % (2583)Instruction limit reached!
% 3.40/1.80 % (2583)------------------------------
% 3.40/1.80 % (2583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.80 % (2583)Termination reason: Unknown
% 3.40/1.80 % (2583)Termination phase: shuffling
% 3.40/1.80
% 3.40/1.80 % (2583)Memory used [KB]: 1279
% 3.40/1.80 % (2583)Time elapsed: 0.003 s
% 3.40/1.80 % (2583)Instructions burned: 2 (million)
% 3.40/1.80 % (2583)------------------------------
% 3.40/1.80 % (2583)------------------------------
% 3.40/1.81 % (2580)Instruction limit reached!
% 3.40/1.81 % (2580)------------------------------
% 3.40/1.81 % (2580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.81 % (2580)Termination reason: Unknown
% 3.40/1.81 % (2580)Termination phase: shuffling
% 3.40/1.81
% 3.40/1.81 % (2580)Memory used [KB]: 1407
% 3.40/1.81 % (2580)Time elapsed: 0.005 s
% 3.40/1.81 % (2580)Instructions burned: 5 (million)
% 3.40/1.81 % (2580)------------------------------
% 3.40/1.81 % (2580)------------------------------
% 3.40/1.81 % (2585)Instruction limit reached!
% 3.40/1.81 % (2585)------------------------------
% 3.40/1.81 % (2585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.81 % (2585)Termination reason: Unknown
% 3.40/1.81 % (2585)Termination phase: shuffling
% 3.40/1.81
% 3.40/1.81 % (2585)Memory used [KB]: 1535
% 3.40/1.81 % (2585)Time elapsed: 0.012 s
% 3.40/1.81 % (2585)Instructions burned: 19 (million)
% 3.40/1.81 % (2585)------------------------------
% 3.40/1.81 % (2585)------------------------------
% 3.40/1.82 % (2581)Instruction limit reached!
% 3.40/1.82 % (2581)------------------------------
% 3.40/1.82 % (2581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.82 % (2581)Termination reason: Unknown
% 3.40/1.82 % (2581)Termination phase: Property scanning
% 3.40/1.82
% 3.40/1.82 % (2581)Memory used [KB]: 1535
% 3.40/1.82 % (2581)Time elapsed: 0.017 s
% 3.40/1.82 % (2581)Instructions burned: 28 (million)
% 3.40/1.82 % (2581)------------------------------
% 3.40/1.82 % (2581)------------------------------
% 3.40/1.82 % (2586)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/3Mi)
% 3.40/1.82 % (2587)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_2468 for (2999ds/37Mi)
% 3.40/1.82 % (2588)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_2468 for (2999ds/15Mi)
% 3.40/1.82 % (2586)Instruction limit reached!
% 3.40/1.82 % (2586)------------------------------
% 3.40/1.82 % (2586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.82 % (2586)Termination reason: Unknown
% 3.40/1.82 % (2586)Termination phase: shuffling
% 3.40/1.82
% 3.40/1.82 % (2586)Memory used [KB]: 1279
% 3.40/1.82 % (2586)Time elapsed: 0.004 s
% 3.40/1.82 % (2586)Instructions burned: 3 (million)
% 3.40/1.82 % (2586)------------------------------
% 3.40/1.82 % (2586)------------------------------
% 3.40/1.83 % (2589)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_2468 for (2999ds/3Mi)
% 3.40/1.83 % (2588)Instruction limit reached!
% 3.40/1.83 % (2588)------------------------------
% 3.40/1.83 % (2588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.83 % (2588)Termination reason: Unknown
% 3.40/1.83 % (2588)Termination phase: shuffling
% 3.40/1.83
% 3.40/1.83 % (2588)Memory used [KB]: 1535
% 3.40/1.83 % (2588)Time elapsed: 0.010 s
% 3.40/1.83 % (2588)Instructions burned: 16 (million)
% 3.40/1.83 % (2588)------------------------------
% 3.40/1.83 % (2588)------------------------------
% 3.40/1.83 % (2589)Instruction limit reached!
% 3.40/1.83 % (2589)------------------------------
% 3.40/1.83 % (2589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.83 % (2589)Termination reason: Unknown
% 3.40/1.83 % (2589)Termination phase: shuffling
% 3.40/1.83
% 3.40/1.83 % (2589)Memory used [KB]: 1279
% 3.40/1.83 % (2589)Time elapsed: 0.003 s
% 3.40/1.83 % (2589)Instructions burned: 3 (million)
% 3.40/1.83 % (2589)------------------------------
% 3.40/1.83 % (2589)------------------------------
% 3.40/1.84 % (2590)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_2468 for (2999ds/1041Mi)
% 3.40/1.84 % (2587)Instruction limit reached!
% 3.40/1.84 % (2587)------------------------------
% 3.40/1.84 % (2587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.84 % (2587)Termination reason: Unknown
% 3.40/1.84 % (2587)Termination phase: shuffling
% 3.40/1.84
% 3.40/1.84 % (2587)Memory used [KB]: 2046
% 3.40/1.84 % (2587)Time elapsed: 0.021 s
% 3.40/1.84 % (2587)Instructions burned: 37 (million)
% 3.40/1.84 % (2587)------------------------------
% 3.40/1.84 % (2587)------------------------------
% 3.40/1.84 % (2591)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/7Mi)
% 3.40/1.85 % (2591)Instruction limit reached!
% 3.40/1.85 % (2591)------------------------------
% 3.40/1.85 % (2591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.85 % (2591)Termination reason: Unknown
% 3.40/1.85 % (2591)Termination phase: shuffling
% 3.40/1.85
% 3.40/1.85 % (2591)Memory used [KB]: 1407
% 3.40/1.85 % (2591)Time elapsed: 0.006 s
% 3.40/1.85 % (2591)Instructions burned: 8 (million)
% 3.40/1.85 % (2591)------------------------------
% 3.40/1.85 % (2591)------------------------------
% 3.40/1.85 % (2592)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/16Mi)
% 3.40/1.85 % (2593)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_2468 for (2999ds/3Mi)
% 3.40/1.85 % (2593)Instruction limit reached!
% 3.40/1.85 % (2593)------------------------------
% 3.40/1.85 % (2593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.85 % (2593)Termination reason: Unknown
% 3.40/1.85 % (2593)Termination phase: shuffling
% 3.40/1.85
% 3.40/1.85 % (2593)Memory used [KB]: 1279
% 3.40/1.85 % (2593)Time elapsed: 0.004 s
% 3.40/1.85 % (2593)Instructions burned: 4 (million)
% 3.40/1.85 % (2593)------------------------------
% 3.40/1.85 % (2593)------------------------------
% 3.40/1.86 % (2592)Instruction limit reached!
% 3.40/1.86 % (2592)------------------------------
% 3.40/1.86 % (2592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.86 % (2592)Termination reason: Unknown
% 3.40/1.86 % (2592)Termination phase: shuffling
% 3.40/1.86
% 3.40/1.86 % (2592)Memory used [KB]: 1535
% 3.40/1.86 % (2592)Time elapsed: 0.010 s
% 3.40/1.86 % (2592)Instructions burned: 16 (million)
% 3.40/1.86 % (2592)------------------------------
% 3.40/1.86 % (2592)------------------------------
% 3.40/1.86 % (2594)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_2468 for (2999ds/3Mi)
% 3.40/1.86 % (2594)Instruction limit reached!
% 3.40/1.86 % (2594)------------------------------
% 3.40/1.86 % (2594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.86 % (2594)Termination reason: Unknown
% 3.40/1.86 % (2594)Termination phase: shuffling
% 3.40/1.86
% 3.40/1.86 % (2594)Memory used [KB]: 1279
% 3.40/1.86 % (2594)Time elapsed: 0.004 s
% 3.40/1.86 % (2594)Instructions burned: 3 (million)
% 3.40/1.86 % (2594)------------------------------
% 3.40/1.86 % (2594)------------------------------
% 3.40/1.87 % (2595)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/7Mi)
% 3.40/1.87 % (2595)Instruction limit reached!
% 3.40/1.87 % (2595)------------------------------
% 3.40/1.87 % (2595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.87 % (2595)Termination reason: Unknown
% 3.40/1.87 % (2595)Termination phase: shuffling
% 3.40/1.87
% 3.40/1.87 % (2595)Memory used [KB]: 1407
% 3.40/1.87 % (2595)Time elapsed: 0.006 s
% 3.40/1.87 % (2595)Instructions burned: 8 (million)
% 3.40/1.87 % (2595)------------------------------
% 3.40/1.87 % (2595)------------------------------
% 3.40/1.87 % (2596)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/3Mi)
% 3.40/1.87 % (2596)Instruction limit reached!
% 3.40/1.87 % (2596)------------------------------
% 3.40/1.87 % (2596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.87 % (2596)Termination reason: Unknown
% 3.40/1.87 % (2596)Termination phase: shuffling
% 3.40/1.87
% 3.40/1.87 % (2596)Memory used [KB]: 1279
% 3.40/1.87 % (2596)Time elapsed: 0.004 s
% 3.40/1.87 % (2596)Instructions burned: 3 (million)
% 3.40/1.87 % (2596)------------------------------
% 3.40/1.87 % (2596)------------------------------
% 3.40/1.88 % (2597)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/4Mi)
% 3.40/1.88 % (2598)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2468 for (2999ds/18Mi)
% 3.40/1.88 % (2597)Instruction limit reached!
% 3.40/1.88 % (2597)------------------------------
% 3.40/1.88 % (2597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.88 % (2597)Termination reason: Unknown
% 3.40/1.88 % (2597)Termination phase: shuffling
% 3.40/1.88
% 3.40/1.88 % (2597)Memory used [KB]: 1279
% 3.40/1.88 % (2597)Time elapsed: 0.004 s
% 3.40/1.88 % (2597)Instructions burned: 5 (million)
% 3.40/1.88 % (2597)------------------------------
% 3.40/1.88 % (2597)------------------------------
% 3.40/1.89 % (2598)Instruction limit reached!
% 3.40/1.89 % (2598)------------------------------
% 3.40/1.89 % (2598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.89 % (2598)Termination reason: Unknown
% 3.40/1.89 % (2598)Termination phase: shuffling
% 3.40/1.89
% 3.40/1.89 % (2598)Memory used [KB]: 1535
% 3.40/1.89 % (2598)Time elapsed: 0.012 s
% 3.40/1.89 % (2598)Instructions burned: 18 (million)
% 3.40/1.89 % (2598)------------------------------
% 3.40/1.89 % (2598)------------------------------
% 3.40/1.89 % (2599)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2468 for (2998ds/710Mi)
% 3.40/1.89 % (2600)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2468 for (2998ds/6Mi)
% 3.40/1.90 % (2600)Instruction limit reached!
% 3.40/1.90 % (2600)------------------------------
% 3.40/1.90 % (2600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.90 % (2600)Termination reason: Unknown
% 3.40/1.90 % (2600)Termination phase: shuffling
% 3.40/1.90
% 3.40/1.90 % (2600)Memory used [KB]: 1407
% 3.40/1.90 % (2600)Time elapsed: 0.005 s
% 3.40/1.90 % (2600)Instructions burned: 6 (million)
% 3.40/1.90 % (2600)------------------------------
% 3.40/1.90 % (2600)------------------------------
% 3.40/1.90 % (2601)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_2468 for (2998ds/902Mi)
% 3.40/1.90 % (2579)Instruction limit reached!
% 3.40/1.90 % (2579)------------------------------
% 3.40/1.90 % (2579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.90 % (2579)Termination reason: Unknown
% 3.40/1.90 % (2579)Termination phase: Saturation
% 3.40/1.90
% 3.40/1.90 % (2579)Memory used [KB]: 6908
% 3.40/1.90 % (2579)Time elapsed: 0.101 s
% 3.40/1.90 % (2579)Instructions burned: 183 (million)
% 3.40/1.90 % (2579)------------------------------
% 3.40/1.90 % (2579)------------------------------
% 3.40/1.91 % (2602)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_2468 for (2998ds/21Mi)
% 3.40/1.91 % (2584)First to succeed.
% 3.40/1.92 % (2604)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2468 for (2998ds/5Mi)
% 3.40/1.92 % (2584)Refutation found. Thanks to Tanya!
% 3.40/1.92 % SZS status Theorem for DTF2THF_2468
% 3.40/1.92 % SZS output start Proof for DTF2THF_2468
% 3.40/1.92 thf(type_def_5, type, real: $tType).
% 3.40/1.92 thf(type_def_6, type, set_real: $tType).
% 3.40/1.92 thf(type_def_8, type, a: $tType).
% 3.40/1.92 thf(type_def_9, type, set_a: $tType).
% 3.40/1.92 thf(func_def_0, type, set_real: $tType).
% 3.40/1.92 thf(func_def_1, type, set_a: $tType).
% 3.40/1.92 thf(func_def_2, type, real: $tType).
% 3.40/1.92 thf(func_def_3, type, a: $tType).
% 3.40/1.92 thf(func_def_4, type, elemen341675809l_real: real > real > set_real).
% 3.40/1.92 thf(func_def_5, type, elemen154694473ball_a: a > real > set_a).
% 3.40/1.92 thf(func_def_6, type, auto_l612940ivl0_a: (a > a) > set_a > a > set_real).
% 3.40/1.92 thf(func_def_7, type, auto_ll_on_flow0_a: (a > a) > set_a > a > real > a).
% 3.40/1.92 thf(func_def_8, type, abs_abs_real: real > real).
% 3.40/1.92 thf(func_def_9, type, abs_abs_a: a > a).
% 3.40/1.92 thf(func_def_10, type, one_one_real: real).
% 3.40/1.92 thf(func_def_11, type, plus_plus_real: real > real > real).
% 3.40/1.92 thf(func_def_12, type, plus_plus_a: a > a > a).
% 3.40/1.92 thf(func_def_13, type, uminus_uminus_a_a: (a > a) > a > a).
% 3.40/1.92 thf(func_def_14, type, uminus_uminus_real: real > real).
% 3.40/1.92 thf(func_def_15, type, uminus_uminus_a: a > a).
% 3.40/1.92 thf(func_def_16, type, zero_zero_real: real).
% 3.40/1.92 thf(func_def_17, type, zero_zero_a: a).
% 3.40/1.92 thf(func_def_18, type, if_real: $o > real > real > real).
% 3.40/1.92 thf(func_def_19, type, initia826609931terval: set_real > $o).
% 3.40/1.92 thf(func_def_20, type, limit_1865182413oint_a: (a > a) > set_a > a > a > $o).
% 3.40/1.92 thf(func_def_21, type, limit_94460170oint_a: (a > a) > set_a > a > a > $o).
% 3.40/1.92 thf(func_def_22, type, line_open_segment_a: a > a > set_a).
% 3.40/1.92 thf(func_def_23, type, bot_bot_set_real: set_real).
% 3.40/1.92 thf(func_def_24, type, bot_bot_set_a: set_a).
% 3.40/1.92 thf(func_def_25, type, ord_less_real: real > real > $o).
% 3.40/1.92 thf(func_def_26, type, ord_less_a: a > a > $o).
% 3.40/1.92 thf(func_def_27, type, ord_less_eq_real: real > real > $o).
% 3.40/1.92 thf(func_def_28, type, ord_less_eq_a: a > a > $o).
% 3.40/1.92 thf(func_def_29, type, top_top_set_real: set_real).
% 3.40/1.92 thf(func_def_30, type, period720806154rbit_a: (a > a) > set_a > a > $o).
% 3.40/1.92 thf(func_def_31, type, period1305449585riod_a: (a > a) > set_a > a > real).
% 3.40/1.92 thf(func_def_32, type, period138238489rbit_a: (a > a) > set_a > a > $o).
% 3.40/1.92 thf(func_def_33, type, poinca522724647ment_a: (a > a) > set_a > a > a > $o).
% 3.40/1.92 thf(func_def_34, type, collect_real: (real > $o) > set_real).
% 3.40/1.92 thf(func_def_35, type, collect_a: (a > $o) > set_a).
% 3.40/1.92 thf(func_def_36, type, topolo1710226732a_real: set_a > (a > real) > $o).
% 3.40/1.92 thf(func_def_37, type, member_real: real > set_real > $o).
% 3.40/1.92 thf(func_def_38, type, member_a: a > set_a > $o).
% 3.40/1.92 thf(func_def_39, type, x: set_a).
% 3.40/1.92 thf(func_def_40, type, a2: a).
% 3.40/1.92 thf(func_def_41, type, b: a).
% 3.40/1.92 thf(func_def_42, type, f: a > a).
% 3.40/1.92 thf(func_def_43, type, p: a).
% 3.40/1.92 thf(func_def_45, type, x2: a).
% 3.40/1.92 thf(func_def_52, type, sK0: real).
% 3.40/1.92 thf(func_def_53, type, sK1: a > real).
% 3.40/1.92 thf(func_def_54, type, sK2: (a > real) > real > a).
% 3.40/1.92 thf(func_def_55, type, sK3: (a > real) > real > a).
% 3.40/1.92 thf(func_def_56, type, sK4: (a > real) > real > a).
% 3.40/1.92 thf(func_def_57, type, sK5: (a > real) > real > a).
% 3.40/1.92 thf(func_def_58, type, sK6: real > a > a > a > real).
% 3.40/1.92 thf(func_def_59, type, sK7: real > a > a > a > a > real).
% 3.40/1.92 thf(func_def_60, type, sK8: a > a).
% 3.40/1.92 thf(func_def_61, type, sK9: a > a).
% 3.40/1.92 thf(func_def_62, type, sK10: a > a).
% 3.40/1.92 thf(func_def_63, type, sK11: a > a).
% 3.40/1.92 thf(func_def_65, type, ph13: !>[X0: $tType]:(X0)).
% 3.40/1.92 thf(f1623,plain,(
% 3.40/1.92 $false),
% 3.40/1.92 inference(avatar_sat_refutation,[status(thm)],[f1409,f1410,f1414,f1418,f1422,f1622])).
% 3.40/1.92 thf(f1622,plain,(
% 3.40/1.92 ~spl12_1 | ~spl12_3 | ~spl12_4 | ~spl12_5),
% 3.40/1.92 inference(avatar_contradiction_clause,[status(thm)],[f1621])).
% 3.40/1.92 thf(f1621,plain,(
% 3.40/1.92 $false | (~spl12_1 | ~spl12_3 | ~spl12_4 | ~spl12_5)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1620,f1473])).
% 3.40/1.92 thf(f1473,plain,(
% 3.40/1.92 ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | (~spl12_1 | ~spl12_4)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1472,f1261])).
% 3.40/1.92 thf(f1261,plain,(
% 3.40/1.92 ((ord_less_real @ (sK1 @ p) @ sK0) = $true)),
% 3.40/1.92 inference(definition_unfolding,[status(thm)],[f1054,f1059])).
% 3.40/1.92 thf(f1059,plain,(
% 3.40/1.92 (zero_zero_real = (sK1 @ p))),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f980])).
% 3.40/1.92 thf(f980,plain,(
% 3.40/1.92 ((zero_zero_real = (sK1 @ p)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (sK1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ((member_a @ X2 @ (elemen154694473ball_a @ p @ sK0)) != $true)) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ sK0))) | ((ord_less_real @ (abs_abs_real @ (sK1 @ X3)) @ one_one_real) = $true))) & ($true = (ord_less_real @ zero_zero_real @ sK0))),
% 3.40/1.92 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f887,f979,f978])).
% 3.40/1.92 thf(f978,plain,(
% 3.40/1.92 ? [X0 : real] : (? [X1 : a > real] : ((zero_zero_real = (X1 @ p)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ($true != (member_a @ X2 @ (elemen154694473ball_a @ p @ X0)))) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) | ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)))) & ((ord_less_real @ zero_zero_real @ X0) = $true)) => (? [X1 : a > real] : ((zero_zero_real = (X1 @ p)) & ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X1) = $true) & ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X1) = $true) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ((member_a @ X2 @ (elemen154694473ball_a @ p @ sK0)) != $true)) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ sK0))) | ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)))) & ($true = (ord_less_real @ zero_zero_real @ sK0)))),
% 3.40/1.92 introduced(definition,[],[choice_axiom])).
% 3.40/1.92 thf(f979,plain,(
% 3.40/1.92 ? [X1 : a > real] : ((zero_zero_real = (X1 @ p)) & ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X1) = $true) & ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X1) = $true) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ((member_a @ X2 @ (elemen154694473ball_a @ p @ sK0)) != $true)) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ sK0))) | ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)))) => ((zero_zero_real = (sK1 @ p)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (sK1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ((member_a @ X2 @ (elemen154694473ball_a @ p @ sK0)) != $true)) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ sK0))) | ((ord_less_real @ (abs_abs_real @ (sK1 @ X3)) @ one_one_real) = $true)))),
% 3.40/1.92 introduced(definition,[],[choice_axiom])).
% 3.40/1.92 thf(f887,plain,(
% 3.40/1.92 ? [X0 : real] : (? [X1 : a > real] : ((zero_zero_real = (X1 @ p)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ($true != (member_a @ X2 @ (elemen154694473ball_a @ p @ X0)))) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) | ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)))) & ((ord_less_real @ zero_zero_real @ X0) = $true))),
% 3.40/1.92 inference(flattening,[status(thm)],[f886])).
% 3.40/1.92 thf(f886,plain,(
% 3.40/1.92 ? [X0 : real] : (? [X1 : a > real] : (((((zero_zero_real = (X1 @ p)) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1))) & ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) | ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)))) & ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ($true != (member_a @ X2 @ (elemen154694473ball_a @ p @ X0))))) & ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1))) & ((ord_less_real @ zero_zero_real @ X0) = $true))),
% 3.40/1.92 inference(ennf_transformation,[status(thm)],[f429])).
% 3.40/1.92 thf(f429,plain,(
% 3.40/1.92 ~! [X0 : real] : (((ord_less_real @ zero_zero_real @ X0) = $true) => ! [X1 : a > real] : (($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) => (! [X2 : a] : (($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0))) => ((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true)) => (! [X3 : a] : (($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) => ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real))) => (($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) => (zero_zero_real != (X1 @ p)))))))),
% 3.40/1.92 inference(fool_elimination,[status(thm)],[f428])).
% 3.40/1.92 thf(f428,plain,(
% 3.40/1.92 ~! [X0 : real] : ((ord_less_real @ zero_zero_real @ X0) => ! [X1 : a > real] : ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1) => (! [X2 : a] : ((member_a @ X2 @ (elemen154694473ball_a @ p @ X0)) => (member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b))) => (! [X3 : a] : ((member_a @ X3 @ (elemen154694473ball_a @ p @ X0)) => (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real)) => ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1) => (zero_zero_real != (X1 @ p)))))))),
% 3.40/1.92 inference(rectify,[status(thm)],[f4])).
% 3.40/1.92 thf(f4,axiom,(
% 3.40/1.92 ~! [X4 : real] : ((ord_less_real @ zero_zero_real @ X4) => ! [X5 : a > real] : ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X4) @ X5) => (! [X6 : a] : ((member_a @ X6 @ (elemen154694473ball_a @ p @ X4)) => (member_a @ (auto_ll_on_flow0_a @ f @ x @ X6 @ (X5 @ X6)) @ (line_open_segment_a @ a2 @ b))) => (! [X6 : a] : ((member_a @ X6 @ (elemen154694473ball_a @ p @ X4)) => (ord_less_real @ (abs_abs_real @ (X5 @ X6)) @ one_one_real)) => ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X4) @ X5) => (zero_zero_real != (X5 @ p)))))))),
% 3.40/1.92 file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062)).
% 3.40/1.92 thf(f1054,plain,(
% 3.40/1.92 ($true = (ord_less_real @ zero_zero_real @ sK0))),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f980])).
% 3.40/1.92 thf(f1472,plain,(
% 3.40/1.92 ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | (~spl12_1 | ~spl12_4)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1471,f1058])).
% 3.40/1.92 thf(f1058,plain,(
% 3.40/1.92 ($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1))),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f980])).
% 3.40/1.92 thf(f1471,plain,(
% 3.40/1.92 ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | (~spl12_1 | ~spl12_4)),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1470])).
% 3.40/1.92 thf(f1470,plain,(
% 3.40/1.92 ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ($true != $true) | ((sK1 @ p) != (sK1 @ p)) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | (~spl12_1 | ~spl12_4)),
% 3.40/1.92 inference(duplicate_literal_removal,[status(thm)],[f1468])).
% 3.40/1.92 thf(f1468,plain,(
% 3.40/1.92 ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ($true != $true) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ((sK1 @ p) != (sK1 @ p)) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | (~spl12_1 | ~spl12_4)),
% 3.40/1.92 inference(superposition,[status(thm)],[f1445,f1404])).
% 3.40/1.92 thf(f1404,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((X1 @ p) != (sK1 @ p))) ) | ~spl12_1),
% 3.40/1.92 inference(avatar_component_clause,[status(thm)],[f1403])).
% 3.40/1.92 thf(f1403,definition,(
% 3.40/1.92 spl12_1 <=> ! [X0 : real,X1 : a > real] : (($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((X1 @ p) != (sK1 @ p)))),
% 3.40/1.92 introduced(definition,[new_symbols(naming,[spl12_1])],[avatar_definition])).
% 3.40/1.92 thf(f1445,plain,(
% 3.40/1.92 ( ! [X0 : real] : (($true != (member_a @ (sK2 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ sK0))) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ sK1)) | ($true != (ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ X0))) @ one_one_real)) | ($true != (ord_less_real @ (sK1 @ p) @ X0))) ) | ~spl12_4),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1441])).
% 3.40/1.92 thf(f1441,plain,(
% 3.40/1.92 ( ! [X0 : real] : (((sK1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ X0))) @ one_one_real)) | ($true != (member_a @ (sK2 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ sK0))) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ sK1)) | ($true != $true)) ) | ~spl12_4),
% 3.40/1.92 inference(superposition,[status(thm)],[f1417,f1056])).
% 3.40/1.92 thf(f1056,plain,(
% 3.40/1.92 ( ! [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (sK1 @ X2)) @ (line_open_segment_a @ a2 @ b)) = $true) | ((member_a @ X2 @ (elemen154694473ball_a @ p @ sK0)) != $true)) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f980])).
% 3.40/1.92 thf(f1417,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((X1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1))) ) | ~spl12_4),
% 3.40/1.92 inference(avatar_component_clause,[status(thm)],[f1416])).
% 3.40/1.92 thf(f1416,definition,(
% 3.40/1.92 spl12_4 <=> ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | ((X1 @ p) != (sK1 @ p)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)))),
% 3.40/1.92 introduced(definition,[new_symbols(naming,[spl12_4])],[avatar_definition])).
% 3.40/1.92 thf(f1620,plain,(
% 3.40/1.92 ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) = $true) | (~spl12_1 | ~spl12_3 | ~spl12_4 | ~spl12_5)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1618,f1058])).
% 3.40/1.92 thf(f1618,plain,(
% 3.40/1.92 ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) = $true) | (~spl12_1 | ~spl12_3 | ~spl12_4 | ~spl12_5)),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1616])).
% 3.40/1.92 thf(f1616,plain,(
% 3.40/1.92 ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ($true != $true) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) = $true) | ((sK1 @ p) != (sK1 @ p)) | (~spl12_1 | ~spl12_3 | ~spl12_4 | ~spl12_5)),
% 3.40/1.92 inference(superposition,[status(thm)],[f1478,f1432])).
% 3.40/1.92 thf(f1432,plain,(
% 3.40/1.92 ( ! [X0 : a > real] : (((member_a @ (sK2 @ X0 @ sK0) @ (elemen154694473ball_a @ p @ sK0)) = $true) | ((X0 @ p) != (sK1 @ p)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X0)) | ($true = (ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ X0 @ sK0))) @ one_one_real))) ) | ~spl12_5),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1431,f1261])).
% 3.40/1.92 thf(f1431,plain,(
% 3.40/1.92 ( ! [X0 : a > real] : (((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ((member_a @ (sK2 @ X0 @ sK0) @ (elemen154694473ball_a @ p @ sK0)) = $true) | ((X0 @ p) != (sK1 @ p)) | ($true = (ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ X0 @ sK0))) @ one_one_real)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X0))) ) | ~spl12_5),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1430])).
% 3.40/1.92 thf(f1430,plain,(
% 3.40/1.92 ( ! [X0 : a > real] : (((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ((member_a @ (sK2 @ X0 @ sK0) @ (elemen154694473ball_a @ p @ sK0)) = $true) | ($true = (ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ X0 @ sK0))) @ one_one_real)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ X0)) | ((X0 @ p) != (sK1 @ p)) | ($true != $true)) ) | ~spl12_5),
% 3.40/1.92 inference(superposition,[status(thm)],[f1055,f1421])).
% 3.40/1.92 thf(f1421,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ((X1 @ p) != (sK1 @ p))) ) | ~spl12_5),
% 3.40/1.92 inference(avatar_component_clause,[status(thm)],[f1420])).
% 3.40/1.92 thf(f1420,definition,(
% 3.40/1.92 spl12_5 <=> ! [X0 : real,X1 : a > real] : (($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) | ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ((X1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)))),
% 3.40/1.92 introduced(definition,[new_symbols(naming,[spl12_5])],[avatar_definition])).
% 3.40/1.92 thf(f1055,plain,(
% 3.40/1.92 ( ! [X3 : a] : (($true != (member_a @ X3 @ (elemen154694473ball_a @ p @ sK0))) | ((ord_less_real @ (abs_abs_real @ (sK1 @ X3)) @ one_one_real) = $true)) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f980])).
% 3.40/1.92 thf(f1478,plain,(
% 3.40/1.92 ($true != (member_a @ (sK2 @ sK1 @ sK0) @ (elemen154694473ball_a @ p @ sK0))) | (~spl12_1 | ~spl12_3 | ~spl12_4)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1477,f1058])).
% 3.40/1.92 thf(f1477,plain,(
% 3.40/1.92 ($true != (member_a @ (sK2 @ sK1 @ sK0) @ (elemen154694473ball_a @ p @ sK0))) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | (~spl12_1 | ~spl12_3 | ~spl12_4)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1476,f1261])).
% 3.40/1.92 thf(f1476,plain,(
% 3.40/1.92 ($true != (member_a @ (sK2 @ sK1 @ sK0) @ (elemen154694473ball_a @ p @ sK0))) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | (~spl12_1 | ~spl12_3 | ~spl12_4)),
% 3.40/1.92 inference(subsumption_resolution,[status(thm)],[f1475,f1473])).
% 3.40/1.92 thf(f1475,plain,(
% 3.40/1.92 ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) = $true) | ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ($true != (member_a @ (sK2 @ sK1 @ sK0) @ (elemen154694473ball_a @ p @ sK0))) | ~spl12_3),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1474])).
% 3.40/1.92 thf(f1474,plain,(
% 3.40/1.92 ((ord_less_real @ (sK1 @ p) @ sK0) != $true) | ((ord_less_real @ (abs_abs_real @ (sK1 @ (sK3 @ sK1 @ sK0))) @ one_one_real) = $true) | ($true != (member_a @ (sK2 @ sK1 @ sK0) @ (elemen154694473ball_a @ p @ sK0))) | ($true != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ sK0) @ sK1)) | ~spl12_3),
% 3.40/1.92 inference(superposition,[status(thm)],[f1055,f1455])).
% 3.40/1.92 thf(f1455,plain,(
% 3.40/1.92 ( ! [X0 : real] : (((member_a @ (sK3 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ sK1)) | ($true != (member_a @ (sK2 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ sK0))) | ($true != (ord_less_real @ (sK1 @ p) @ X0))) ) | ~spl12_3),
% 3.40/1.92 inference(trivial_inequality_removal,[status(thm)],[f1451])).
% 3.40/1.92 thf(f1451,plain,(
% 3.40/1.92 ( ! [X0 : real] : (($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ sK1)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((member_a @ (sK3 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ($true != $true) | ((sK1 @ p) != (sK1 @ p)) | ($true != (member_a @ (sK2 @ sK1 @ X0) @ (elemen154694473ball_a @ p @ sK0)))) ) | ~spl12_3),
% 3.40/1.92 inference(superposition,[status(thm)],[f1413,f1056])).
% 3.40/1.92 thf(f1413,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((X1 @ p) != (sK1 @ p))) ) | ~spl12_3),
% 3.40/1.92 inference(avatar_component_clause,[status(thm)],[f1412])).
% 3.40/1.92 thf(f1412,definition,(
% 3.40/1.92 spl12_3 <=> ! [X0 : real,X1 : a > real] : (($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ((X1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)))),
% 3.40/1.92 introduced(definition,[new_symbols(naming,[spl12_3])],[avatar_definition])).
% 3.40/1.92 thf(f1422,plain,(
% 3.40/1.92 spl12_2 | spl12_5),
% 3.40/1.92 inference(avatar_split_clause,[status(thm)],[f1278,f1420,f1406])).
% 3.40/1.92 thf(f1406,definition,(
% 3.40/1.92 spl12_2 <=> (thesisa = $true)),
% 3.40/1.92 introduced(definition,[new_symbols(naming,[spl12_2])],[avatar_definition])).
% 3.40/1.92 thf(f1278,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((X1 @ p) != (sK1 @ p)) | ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | (thesisa = $true) | ($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)))) )),
% 3.40/1.92 inference(definition_unfolding,[status(thm)],[f1085,f1059,f1059])).
% 3.40/1.92 thf(f1085,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)))) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f990])).
% 3.40/1.92 thf(f990,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : ((((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) & ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true)) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | (($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real))))),
% 3.40/1.92 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f987,f989,f988])).
% 3.40/1.92 thf(f988,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : (? [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) != $true) & ($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0)))) => (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) & ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true)))),
% 3.40/1.92 introduced(definition,[],[choice_axiom])).
% 3.40/1.92 thf(f989,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : (? [X3 : a] : (($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real))) => (($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real))))),
% 3.40/1.92 introduced(definition,[],[choice_axiom])).
% 3.40/1.92 thf(f987,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : (? [X2 : a] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ X2 @ (X1 @ X2)) @ (line_open_segment_a @ a2 @ b)) != $true) & ($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0)))) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ? [X3 : a] : (($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ X3)) @ one_one_real))))),
% 3.40/1.92 inference(rectify,[status(thm)],[f870])).
% 3.40/1.92 thf(f870,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : (? [X3 : a] : (($true != (member_a @ (auto_ll_on_flow0_a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment_a @ a2 @ b))) & ($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0)))) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ? [X2 : a] : (($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ X2)) @ one_one_real))))),
% 3.40/1.92 inference(flattening,[status(thm)],[f869])).
% 3.40/1.92 thf(f869,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : ((((((thesisa = $true) | ? [X3 : a] : (($true != (member_a @ (auto_ll_on_flow0_a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment_a @ a2 @ b))) & ($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))))) | ? [X2 : a] : (($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0))) & ($true != (ord_less_real @ (abs_abs_real @ (X1 @ X2)) @ one_one_real)))) | (zero_zero_real != (X1 @ p))) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1))) | ((ord_less_real @ zero_zero_real @ X0) != $true))),
% 3.40/1.92 inference(ennf_transformation,[status(thm)],[f397])).
% 3.40/1.92 thf(f397,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : (((ord_less_real @ zero_zero_real @ X0) = $true) => (($true = (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) => ((zero_zero_real = (X1 @ p)) => (! [X2 : a] : (($true = (member_a @ X2 @ (elemen154694473ball_a @ p @ X0))) => ($true = (ord_less_real @ (abs_abs_real @ (X1 @ X2)) @ one_one_real))) => (! [X3 : a] : (($true = (member_a @ X3 @ (elemen154694473ball_a @ p @ X0))) => ($true = (member_a @ (auto_ll_on_flow0_a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment_a @ a2 @ b)))) => (thesisa = $true))))))),
% 3.40/1.92 inference(fool_elimination,[status(thm)],[f396])).
% 3.40/1.92 thf(f396,plain,(
% 3.40/1.92 ! [X0 : real,X1 : a > real] : ((ord_less_real @ zero_zero_real @ X0) => ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1) => ((zero_zero_real = (X1 @ p)) => (! [X2 : a] : ((member_a @ X2 @ (elemen154694473ball_a @ p @ X0)) => (ord_less_real @ (abs_abs_real @ (X1 @ X2)) @ one_one_real)) => (! [X3 : a] : ((member_a @ X3 @ (elemen154694473ball_a @ p @ X0)) => (member_a @ (auto_ll_on_flow0_a @ f @ x @ X3 @ (X1 @ X3)) @ (line_open_segment_a @ a2 @ b))) => thesisa)))))),
% 3.40/1.92 inference(rectify,[status(thm)],[f348])).
% 3.40/1.92 thf(f348,axiom,(
% 3.40/1.92 ! [X44 : real,X45 : a > real] : ((ord_less_real @ zero_zero_real @ X44) => ((topolo1710226732a_real @ (elemen154694473ball_a @ p @ X44) @ X45) => ((zero_zero_real = (X45 @ p)) => (! [X9 : a] : ((member_a @ X9 @ (elemen154694473ball_a @ p @ X44)) => (ord_less_real @ (abs_abs_real @ (X45 @ X9)) @ one_one_real)) => (! [X9 : a] : ((member_a @ X9 @ (elemen154694473ball_a @ p @ X44)) => (member_a @ (auto_ll_on_flow0_a @ f @ x @ X9 @ (X45 @ X9)) @ (line_open_segment_a @ a2 @ b))) => thesisa)))))),
% 3.40/1.92 file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',conj_0)).
% 3.40/1.92 thf(f1418,plain,(
% 3.40/1.92 spl12_4 | spl12_2),
% 3.40/1.92 inference(avatar_split_clause,[status(thm)],[f1277,f1406,f1416])).
% 3.40/1.92 thf(f1277,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | ((X1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | (thesisa = $true)) )),
% 3.40/1.92 inference(definition_unfolding,[status(thm)],[f1086,f1059,f1059])).
% 3.40/1.92 thf(f1086,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real))) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f990])).
% 3.40/1.92 thf(f1414,plain,(
% 3.40/1.92 spl12_3 | spl12_2),
% 3.40/1.92 inference(avatar_split_clause,[status(thm)],[f1276,f1406,f1412])).
% 3.40/1.92 thf(f1276,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0))) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((X1 @ p) != (sK1 @ p)) | ((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (thesisa = $true)) )),
% 3.40/1.92 inference(definition_unfolding,[status(thm)],[f1087,f1059,f1059])).
% 3.40/1.92 thf(f1087,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (auto_ll_on_flow0_a @ f @ x @ (sK2 @ X1 @ X0) @ (X1 @ (sK2 @ X1 @ X0))) @ (line_open_segment_a @ a2 @ b)) != $true) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ($true = (member_a @ (sK3 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)))) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f990])).
% 3.40/1.92 thf(f1410,plain,(
% 3.40/1.92 ~spl12_2),
% 3.40/1.92 inference(avatar_split_clause,[status(thm)],[f1113,f1406])).
% 3.40/1.92 thf(f1113,plain,(
% 3.40/1.92 (thesisa != $true)),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f791])).
% 3.40/1.92 thf(f791,plain,(
% 3.40/1.92 (thesisa != $true)),
% 3.40/1.92 inference(flattening,[status(thm)],[f631])).
% 3.40/1.92 thf(f631,plain,(
% 3.40/1.92 ~(thesisa = $true)),
% 3.40/1.92 inference(fool_elimination,[status(thm)],[f630])).
% 3.40/1.92 thf(f630,plain,(
% 3.40/1.92 ~thesisa),
% 3.40/1.92 inference(rectify,[status(thm)],[f350])).
% 3.40/1.92 thf(f350,negated_conjecture,(
% 3.40/1.92 ~thesisa),
% 3.40/1.92 inference(negated_conjecture,[status(cth)],[f349])).
% 3.40/1.92 thf(f349,conjecture,(
% 3.40/1.92 thesisa),
% 3.40/1.92 file('/export/starexec/sandbox/tmp/tmp.pd4Y4akRnK/DTF2THF_2468.p',conj_1)).
% 3.40/1.92 thf(f1409,plain,(
% 3.40/1.92 spl12_1 | spl12_2),
% 3.40/1.92 inference(avatar_split_clause,[status(thm)],[f1279,f1406,f1403])).
% 3.40/1.92 thf(f1279,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : ((thesisa = $true) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real)) | ((X1 @ p) != (sK1 @ p)) | ($true != (ord_less_real @ (sK1 @ p) @ X0)) | ((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1))) )),
% 3.40/1.92 inference(definition_unfolding,[status(thm)],[f1084,f1059,f1059])).
% 3.40/1.92 thf(f1084,plain,(
% 3.40/1.92 ( ! [X0 : real,X1 : a > real] : (((member_a @ (sK2 @ X1 @ X0) @ (elemen154694473ball_a @ p @ X0)) = $true) | ((ord_less_real @ zero_zero_real @ X0) != $true) | (thesisa = $true) | ($true != (topolo1710226732a_real @ (elemen154694473ball_a @ p @ X0) @ X1)) | (zero_zero_real != (X1 @ p)) | ($true != (ord_less_real @ (abs_abs_real @ (X1 @ (sK3 @ X1 @ X0))) @ one_one_real))) )),
% 3.40/1.92 inference(cnf_transformation,[status(thm)],[f990])).
% 3.40/1.92 % SZS output end Proof for DTF2THF_2468
% 3.40/1.92 % (2584)------------------------------
% 3.40/1.92 % (2584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.92 % (2584)Termination reason: Refutation
% 3.40/1.92
% 3.40/1.92 % (2584)Memory used [KB]: 6908
% 3.40/1.92 % (2584)Time elapsed: 0.115 s
% 3.40/1.92 % (2584)Instructions burned: 193 (million)
% 3.40/1.92 % (2584)------------------------------
% 3.40/1.92 % (2584)------------------------------
% 3.40/1.92 % (2578)Success in time 0.144 s
%------------------------------------------------------------------------------