TPTP Problem File: ITP149^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP149^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Poincare_Bendixson problem prob_755__19581512_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Poincare_Bendixson/prob_755__19581512_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 424 ( 195 unt; 71 typ; 0 def)
% Number of atoms : 935 ( 305 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2844 ( 77 ~; 18 |; 53 &;2418 @)
% ( 0 <=>; 278 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 5 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 64 usr; 12 con; 0-5 aty)
% Number of variables : 756 ( 58 ^; 688 !; 10 ?; 756 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:44:24.334
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Bounded____Linear____Function__Oblinfun_Itf__a_Mt__Real__Oreal_J,type,
bounde7994401a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (64)
thf(sy_c_Elementary__Metric__Spaces_Ocball_001t__Real__Oreal,type,
elemen1140313242l_real: real > real > set_real ).
thf(sy_c_Elementary__Metric__Spaces_Ocball_001tf__a,type,
elemen49976720ball_a: a > real > set_a ).
thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
uminus_uminus_a_a: ( a > a ) > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus773214379t_real: set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_a: a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_Initial__Value__Problem_Ointerval,type,
initia826609931terval: set_real > $o ).
thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001t__Real__Oreal,type,
inner_4346926r_real: real > real > real ).
thf(sy_c_Inner__Product_Oreal__inner__class_Oinner_001tf__a,type,
inner_1173012732nner_a: a > a > real ).
thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant_001tf__a,type,
auto_l630715367iant_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Line__Segment_Oclosed__segment_001tf__a,type,
line_c1152468841ment_a: a > a > set_a ).
thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
line_open_segment_a: a > a > set_a ).
thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped_001tf__a,type,
oDE_au1996717075pped_a: ( a > a ) > set_a > a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le861030508t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le1733505144t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
period720806154rbit_a: ( a > a ) > set_a > a > $o ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
period1305449585riod_a: ( a > a ) > set_a > a > real ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
period138238489rbit_a: ( a > a ) > set_a > a > $o ).
thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Orot_001tf__a,type,
poinca659159244_rot_a: a > a ).
thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection_001tf__a,type,
reacha1191408304tion_a: ( a > a ) > set_a > ( a > real ) > ( a > bounde7994401a_real ) > set_a > $o ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
real_V1934908667t_real: real > real > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001tf__a,type,
real_V1514887919dist_a: a > a > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or656347191t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Real__Oreal_J,type,
set_or1619725421t_real: set_real > set_real > set_set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2094724627_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or411607219Most_a: a > a > set_a ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X,type,
x: set_a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_d____,type,
d: real ).
thf(sy_v_f,type,
f: a > a ).
thf(sy_v_s____,type,
s: a ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_x____,type,
x2: a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (352)
thf(fact_0_d_I1_J,axiom,
ord_less_real @ zero_zero_real @ d ).
% d(1)
thf(fact_1__092_060open_062s_A_092_060in_062_Acball_Ax_Ad_092_060close_062,axiom,
member_a @ s @ ( elemen49976720ball_a @ x2 @ d ) ).
% \<open>s \<in> cball x d\<close>
thf(fact_2_seg_I1_J,axiom,
a2 != b ).
% seg(1)
thf(fact_3_nrm__dot,axiom,
! [X: a,Y: a] :
( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
= zero_zero_real ) ).
% nrm_dot
thf(fact_4__092_060open_062x_A_092_060in_062_AX_092_060close_062,axiom,
member_a @ x2 @ x ).
% \<open>x \<in> X\<close>
thf(fact_5__092_060open_062_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ s ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% \<open>\<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<close>
thf(fact_6__092_060open_062f_Ax_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060close_062,axiom,
( ( f @ x2 )
!= zero_zero_a ) ).
% \<open>f x \<noteq> (0::'a)\<close>
thf(fact_7_direction_I2_J,axiom,
ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% direction(2)
thf(fact_8_d_I2_J,axiom,
! [Y: a] :
( ( member_a @ Y @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
& ( member_a @ Y @ x )
& ( ( f @ Y )
!= zero_zero_a ) ) ) ).
% d(2)
thf(fact_9__092_060open_0620_A_060_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ x2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ).
% \<open>0 < f x \<bullet> rot (a - b)\<close>
thf(fact_10_dot__ortho,axiom,
! [X: a] :
( ( inner_1173012732nner_a @ X @ ( poinca659159244_rot_a @ X ) )
= zero_zero_real ) ).
% dot_ortho
thf(fact_11__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Acball_Ax_Ad_059_A_092_060forall_062x_092_060in_062cball_Ax_Ad_O_Af_As_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060le_062_Af_Ax_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S: a] :
( ( member_a @ S @ ( elemen49976720ball_a @ x2 @ d ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ ( elemen49976720ball_a @ x2 @ d ) )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ ( f @ S ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>s. \<lbrakk>s \<in> cball x d; \<forall>x\<in>cball x d. f s \<bullet> rot (a - b) \<le> f x \<bullet> rot (a - b)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_12__092_060open_062x_A_092_060in_062_A_123a_N_Nb_125_092_060close_062,axiom,
member_a @ x2 @ ( line_c1152468841ment_a @ a2 @ b ) ).
% \<open>x \<in> {a--b}\<close>
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_O_A_092_060lbrakk_0620_A_060_Ad_059_A_092_060And_062y_O_Ay_A_092_060in_062_Acball_Ax_Ad_A_092_060Longrightarrow_062_A0_A_060_Af_Ay_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_A_092_060and_062_Ay_A_092_060in_062_AX_A_092_060and_062_Af_Ay_A_092_060noteq_062_A_I0_058_058_Ha_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ~ ! [Y2: a] :
( ( member_a @ Y2 @ ( elemen49976720ball_a @ x2 @ D ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Y2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) )
& ( member_a @ Y2 @ x )
& ( ( f @ Y2 )
!= zero_zero_a ) ) ) ) ).
% \<open>\<And>thesis. (\<And>d. \<lbrakk>0 < d; \<And>y. y \<in> cball x d \<Longrightarrow> 0 < f y \<bullet> rot (a - b) \<and> y \<in> X \<and> f y \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_transversal,axiom,
poinca522724647ment_a @ f @ x @ a2 @ b ).
% transversal
thf(fact_15_inner__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( inner_4346926r_real @ X @ X ) )
= ( X != zero_zero_real ) ) ).
% inner_gt_zero_iff
thf(fact_16_inner__gt__zero__iff,axiom,
! [X: a] :
( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) )
= ( X != zero_zero_a ) ) ).
% inner_gt_zero_iff
thf(fact_17_centre__in__cball,axiom,
! [X: real,E: real] :
( ( member_real @ X @ ( elemen1140313242l_real @ X @ E ) )
= ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% centre_in_cball
thf(fact_18_centre__in__cball,axiom,
! [X: a,E: real] :
( ( member_a @ X @ ( elemen49976720ball_a @ X @ E ) )
= ( ord_less_eq_real @ zero_zero_real @ E ) ) ).
% centre_in_cball
thf(fact_19_all__zero__iff,axiom,
! [X: real] :
( ( ! [U: real] :
( ( inner_4346926r_real @ X @ U )
= zero_zero_real ) )
= ( X = zero_zero_real ) ) ).
% all_zero_iff
thf(fact_20_all__zero__iff,axiom,
! [X: a] :
( ( ! [U: a] :
( ( inner_1173012732nner_a @ X @ U )
= zero_zero_real ) )
= ( X = zero_zero_a ) ) ).
% all_zero_iff
thf(fact_21_inner__zero__left,axiom,
! [X: real] :
( ( inner_4346926r_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% inner_zero_left
thf(fact_22_inner__zero__left,axiom,
! [X: a] :
( ( inner_1173012732nner_a @ zero_zero_a @ X )
= zero_zero_real ) ).
% inner_zero_left
thf(fact_23_inner__zero__right,axiom,
! [X: real] :
( ( inner_4346926r_real @ X @ zero_zero_real )
= zero_zero_real ) ).
% inner_zero_right
thf(fact_24_inner__zero__right,axiom,
! [X: a] :
( ( inner_1173012732nner_a @ X @ zero_zero_a )
= zero_zero_real ) ).
% inner_zero_right
thf(fact_25_inner__eq__zero__iff,axiom,
! [X: real] :
( ( ( inner_4346926r_real @ X @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% inner_eq_zero_iff
thf(fact_26_inner__eq__zero__iff,axiom,
! [X: a] :
( ( ( inner_1173012732nner_a @ X @ X )
= zero_zero_real )
= ( X = zero_zero_a ) ) ).
% inner_eq_zero_iff
thf(fact_27_diff__gt__0__iff__gt,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_a @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_28_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_29_diff__ge__0__iff__ge,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_eq_a @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_30_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_31_rot__0,axiom,
( ( poinca659159244_rot_a @ zero_zero_a )
= zero_zero_a ) ).
% rot_0
thf(fact_32_rot__eq__0__iff,axiom,
! [X: a] :
( ( ( poinca659159244_rot_a @ X )
= zero_zero_a )
= ( X = zero_zero_a ) ) ).
% rot_eq_0_iff
thf(fact_33_seg_I3_J,axiom,
! [Z: a] :
( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% seg(3)
thf(fact_34_direction_I1_J,axiom,
member_a @ z @ ( line_c1152468841ment_a @ a2 @ b ) ).
% direction(1)
thf(fact_35_transversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% transversal_segment_reverse
thf(fact_36_transversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
= ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% transversal_segment_commute
thf(fact_37_in__segment__inner__rot2,axiom,
! [X: a,A: a,B: a,Y: a] :
( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
=> ( ( member_a @ Y @ ( line_c1152468841ment_a @ A @ B ) )
=> ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
= zero_zero_real ) ) ) ).
% in_segment_inner_rot2
thf(fact_38_in__segment__inner__rot,axiom,
! [X: a,A: a,B: a] :
( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
=> ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
= zero_zero_real ) ) ).
% in_segment_inner_rot
thf(fact_39_seg_I2_J,axiom,
ord_less_eq_set_a @ ( line_c1152468841ment_a @ a2 @ b ) @ x ).
% seg(2)
thf(fact_40_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_41_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_42_diff__zero,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_zero
thf(fact_43_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_44_diff__0__right,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_0_right
thf(fact_45_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_46_diff__self,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% diff_self
thf(fact_47_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_48_subset__cball,axiom,
! [D2: real,E: real,X: real] :
( ( ord_less_eq_real @ D2 @ E )
=> ( ord_less_eq_set_real @ ( elemen1140313242l_real @ X @ D2 ) @ ( elemen1140313242l_real @ X @ E ) ) ) ).
% subset_cball
thf(fact_49_subset__cball,axiom,
! [D2: real,E: real,X: a] :
( ( ord_less_eq_real @ D2 @ E )
=> ( ord_less_eq_set_a @ ( elemen49976720ball_a @ X @ D2 ) @ ( elemen49976720ball_a @ X @ E ) ) ) ).
% subset_cball
thf(fact_50_transversal__segment__def,axiom,
! [A: a,B: a] :
( ( poinca522724647ment_a @ f @ x @ A @ B )
= ( ( A != B )
& ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ A @ B ) @ x )
& ! [X3: a] :
( ( member_a @ X3 @ ( line_c1152468841ment_a @ A @ B ) )
=> ( ( inner_1173012732nner_a @ ( f @ X3 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) )
!= zero_zero_real ) ) ) ) ).
% transversal_segment_def
thf(fact_51_transversal__segmentE,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( ( X != Y )
=> ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
=> ~ ! [Z2: a] :
( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ( ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x )
=> ~ ! [Z2: a] :
( ( member_a @ Z2 @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% transversal_segmentE
thf(fact_52_transversal__segment__posD_I2_J,axiom,
! [X: a,Y: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
=> ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% transversal_segment_posD(2)
thf(fact_53_transversal__segment__negD_I2_J,axiom,
! [X: a,Y: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
=> ( ord_less_eq_set_a @ ( line_c1152468841ment_a @ X @ Y ) @ x ) ) ) ) ).
% transversal_segment_negD(2)
thf(fact_54_mem__Collect__eq,axiom,
! [A: set_real,P: set_real > $o] :
( ( member_set_real @ A @ ( collect_set_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_set_real] :
( ( collect_set_real
@ ^ [X3: set_real] : ( member_set_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_transversal__segment__sign__less,axiom,
! [W: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ W @ Y )
=> ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real )
=> ! [X2: a] :
( ( member_a @ X2 @ ( line_c1152468841ment_a @ W @ Y ) )
=> ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ W @ Y ) ) ) @ zero_zero_real ) ) ) ) ).
% transversal_segment_sign_less
thf(fact_63_transversal__segment__pos,axiom,
! [X: a,Y: a,W: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% transversal_segment_pos
thf(fact_64_transversal__segment__neg,axiom,
! [X: a,Y: a,W: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ W @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ W ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
=> ! [X2: a] :
( ( member_a @ X2 @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ X2 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% transversal_segment_neg
thf(fact_65_transversal__segment__posD_I1_J,axiom,
! [X: a,Y: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
=> ( X != Y ) ) ) ) ).
% transversal_segment_posD(1)
thf(fact_66_transversal__segment__posD_I3_J,axiom,
! [X: a,Y: a,Za: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) ) ) ) ) ) ).
% transversal_segment_posD(3)
thf(fact_67_transversal__segment__negD_I1_J,axiom,
! [X: a,Y: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
=> ( X != Y ) ) ) ) ).
% transversal_segment_negD(1)
thf(fact_68_transversal__segment__negD_I3_J,axiom,
! [X: a,Y: a,Za: a,Z: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( ( member_a @ Za @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Za ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real )
=> ( ( member_a @ Z @ ( line_c1152468841ment_a @ X @ Y ) )
=> ( ord_less_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) ) @ zero_zero_real ) ) ) ) ) ).
% transversal_segment_negD(3)
thf(fact_69__092_060open_062_092_060And_062z_O_Az_A_092_060in_062_A_123a_N_Nb_125_A_092_060Longrightarrow_062_A0_A_060_Af_Az_A_092_060bullet_062_Arot_A_Ia_A_N_Ab_J_092_060close_062,axiom,
! [Z: a] :
( ( member_a @ Z @ ( line_c1152468841ment_a @ a2 @ b ) )
=> ( ord_less_real @ zero_zero_real @ ( inner_1173012732nner_a @ ( f @ Z ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ).
% \<open>\<And>z. z \<in> {a--b} \<Longrightarrow> 0 < f z \<bullet> rot (a - b)\<close>
thf(fact_70_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
poinca522724647ment_a = poinca522724647ment_a ).
% c1_on_open_R2.transversal_segment.cong
thf(fact_71_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_72_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_73_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
= ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_74_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_75_diff__eq__diff__eq,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_76_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_77_inner__commute,axiom,
( inner_1173012732nner_a
= ( ^ [X3: a,Y3: a] : ( inner_1173012732nner_a @ Y3 @ X3 ) ) ) ).
% inner_commute
thf(fact_78_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: a,Z3: a] : Y4 = Z3 )
= ( ^ [A3: a,B2: a] :
( ( minus_minus_a @ A3 @ B2 )
= zero_zero_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_79_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : Y4 = Z3 )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_80_diff__eq__diff__less__eq,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D2 ) )
=> ( ( ord_less_eq_a @ A @ B )
= ( ord_less_eq_a @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_81_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_82_diff__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_83_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_84_diff__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_85_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_86_diff__mono,axiom,
! [A: a,B: a,D2: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ D2 @ C )
=> ( ord_less_eq_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_87_diff__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_88_diff__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_89_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_90_diff__strict__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_91_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_92_diff__eq__diff__less,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D2 ) )
=> ( ( ord_less_a @ A @ B )
= ( ord_less_a @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_93_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_94_diff__strict__mono,axiom,
! [A: a,B: a,D2: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ D2 @ C )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_95_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_96_inner__diff__right,axiom,
! [X: real,Y: real,Z: real] :
( ( inner_4346926r_real @ X @ ( minus_minus_real @ Y @ Z ) )
= ( minus_minus_real @ ( inner_4346926r_real @ X @ Y ) @ ( inner_4346926r_real @ X @ Z ) ) ) ).
% inner_diff_right
thf(fact_97_inner__diff__right,axiom,
! [X: a,Y: a,Z: a] :
( ( inner_1173012732nner_a @ X @ ( minus_minus_a @ Y @ Z ) )
= ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Y ) @ ( inner_1173012732nner_a @ X @ Z ) ) ) ).
% inner_diff_right
thf(fact_98_inner__diff__left,axiom,
! [X: real,Y: real,Z: real] :
( ( inner_4346926r_real @ ( minus_minus_real @ X @ Y ) @ Z )
= ( minus_minus_real @ ( inner_4346926r_real @ X @ Z ) @ ( inner_4346926r_real @ Y @ Z ) ) ) ).
% inner_diff_left
thf(fact_99_inner__diff__left,axiom,
! [X: a,Y: a,Z: a] :
( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ Y ) @ Z )
= ( minus_minus_real @ ( inner_1173012732nner_a @ X @ Z ) @ ( inner_1173012732nner_a @ Y @ Z ) ) ) ).
% inner_diff_left
thf(fact_100_mem__cball__leI,axiom,
! [X: real,Y: real,E: real,F: real] :
( ( member_real @ X @ ( elemen1140313242l_real @ Y @ E ) )
=> ( ( ord_less_eq_real @ E @ F )
=> ( member_real @ X @ ( elemen1140313242l_real @ Y @ F ) ) ) ) ).
% mem_cball_leI
thf(fact_101_mem__cball__leI,axiom,
! [X: a,Y: a,E: real,F: real] :
( ( member_a @ X @ ( elemen49976720ball_a @ Y @ E ) )
=> ( ( ord_less_eq_real @ E @ F )
=> ( member_a @ X @ ( elemen49976720ball_a @ Y @ F ) ) ) ) ).
% mem_cball_leI
thf(fact_102_le__iff__diff__le__0,axiom,
( ord_less_eq_a
= ( ^ [A3: a,B2: a] : ( ord_less_eq_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% le_iff_diff_le_0
thf(fact_103_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_104_less__iff__diff__less__0,axiom,
( ord_less_a
= ( ^ [A3: a,B2: a] : ( ord_less_a @ ( minus_minus_a @ A3 @ B2 ) @ zero_zero_a ) ) ) ).
% less_iff_diff_less_0
thf(fact_105_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_106_inner__ge__zero,axiom,
! [X: a] : ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ X @ X ) ) ).
% inner_ge_zero
thf(fact_107_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( ( period1305449585riod_a @ f @ x @ X )
= zero_zero_real ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_108_that,axiom,
! [D2: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( line_c1152468841ment_a @ a2 @ b ) )
=> ( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X4 @ Y5 ) @ D2 )
=> ( ord_less_eq_real @ B3 @ ( inner_1173012732nner_a @ ( f @ Y5 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) )
=> thesis ) ) ) ).
% that
thf(fact_109_inner__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( inner_4346926r_real @ A @ B ) ) ) ) ).
% inner_nonneg_nonneg
thf(fact_110_inner__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( inner_1173012732nner_a @ A @ B ) ) ) ) ).
% inner_nonneg_nonneg
thf(fact_111_in__closed__segment__iff__rot,axiom,
! [A: a,B: a,X: a] :
( ( A != B )
=> ( ( member_a @ X @ ( line_c1152468841ment_a @ A @ B ) )
= ( ( ( inner_1173012732nner_a @ ( minus_minus_a @ X @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) ) )
= zero_zero_real )
& ( member_real @ ( inner_1173012732nner_a @ X @ ( minus_minus_a @ B @ A ) ) @ ( set_or656347191t_real @ ( inner_1173012732nner_a @ A @ ( minus_minus_a @ B @ A ) ) @ ( inner_1173012732nner_a @ B @ ( minus_minus_a @ B @ A ) ) ) ) ) ) ) ).
% in_closed_segment_iff_rot
thf(fact_112_sectionD_I5_J,axiom,
! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
( ( reacha1191408304tion_a @ f @ x @ S2 @ Ds @ S3 )
=> ( ord_less_eq_set_a @ S3 @ x ) ) ).
% sectionD(5)
thf(fact_113_transversal__segment__exists,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
!= zero_zero_a )
=> ~ ! [A4: a,B4: a] :
( ( member_a @ X @ ( line_open_segment_a @ A4 @ B4 ) )
=> ~ ( poinca522724647ment_a @ f @ x @ A4 @ B4 ) ) ) ) ).
% transversal_segment_exists
thf(fact_114_invariant__iff__compl__invariant,axiom,
! [M: set_a] :
( ( auto_l630715367iant_a @ f @ x @ M )
= ( auto_l630715367iant_a @ f @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% invariant_iff_compl_invariant
thf(fact_115_cball__eq__cball__iff,axiom,
! [X: a,D2: real,Y: a,E: real] :
( ( ( elemen49976720ball_a @ X @ D2 )
= ( elemen49976720ball_a @ Y @ E ) )
= ( ( ( ord_less_real @ D2 @ zero_zero_real )
& ( ord_less_real @ E @ zero_zero_real ) )
| ( ( X = Y )
& ( D2 = E ) ) ) ) ).
% cball_eq_cball_iff
thf(fact_116_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B4: real,C2: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_117_mem__cball,axiom,
! [Y: real,X: real,E: real] :
( ( member_real @ Y @ ( elemen1140313242l_real @ X @ E ) )
= ( ord_less_eq_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% mem_cball
thf(fact_118_mem__cball,axiom,
! [Y: a,X: a,E: real] :
( ( member_a @ Y @ ( elemen49976720ball_a @ X @ E ) )
= ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% mem_cball
thf(fact_119_perfect__choose__dist,axiom,
! [R: real,X: a] :
( ( ord_less_real @ zero_zero_real @ R )
=> ? [A4: a] :
( ( A4 != X )
& ( ord_less_real @ ( real_V1514887919dist_a @ A4 @ X ) @ R ) ) ) ).
% perfect_choose_dist
thf(fact_120_interval__inner__leI_I2_J,axiom,
! [X: a,A: a,B: a,I: a] :
( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ I )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ X @ I ) @ ( inner_1173012732nner_a @ B @ I ) ) ) ) ).
% interval_inner_leI(2)
thf(fact_121_interval__inner__leI_I2_J,axiom,
! [X: real,A: real,B: real,I: real] :
( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ord_less_eq_real @ ( inner_4346926r_real @ X @ I ) @ ( inner_4346926r_real @ B @ I ) ) ) ) ).
% interval_inner_leI(2)
thf(fact_122_interval__inner__leI_I1_J,axiom,
! [X: a,A: a,B: a,I: a] :
( ( member_a @ X @ ( set_or411607219Most_a @ A @ B ) )
=> ( ( ord_less_eq_a @ zero_zero_a @ I )
=> ( ord_less_eq_real @ ( inner_1173012732nner_a @ A @ I ) @ ( inner_1173012732nner_a @ X @ I ) ) ) ) ).
% interval_inner_leI(1)
thf(fact_123_interval__inner__leI_I1_J,axiom,
! [X: real,A: real,B: real,I: real] :
( ( member_real @ X @ ( set_or656347191t_real @ A @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ I )
=> ( ord_less_eq_real @ ( inner_4346926r_real @ A @ I ) @ ( inner_4346926r_real @ X @ I ) ) ) ) ).
% interval_inner_leI(1)
thf(fact_124_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_125_invariant__def,axiom,
! [M: set_a] :
( ( auto_l630715367iant_a @ f @ x @ M )
= ( ! [X3: a] :
( ( member_a @ X3 @ M )
=> ( oDE_au1996717075pped_a @ f @ x @ X3 @ M ) ) ) ) ).
% invariant_def
thf(fact_126_periodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ f @ x @ X )
=> ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% periodic_orbit_period(1)
thf(fact_127_zero__less__dist__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) )
= ( X != Y ) ) ).
% zero_less_dist_iff
thf(fact_128_dist__le__zero__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real )
= ( X = Y ) ) ).
% dist_le_zero_iff
thf(fact_129_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D2: set_a] :
( ( ord_le318720350_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_130_atLeastatMost__subset__iff,axiom,
! [A: set_real,B: set_real,C: set_real,D2: set_real] :
( ( ord_le1733505144t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_real @ A @ B )
| ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_eq_set_real @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_131_atLeastatMost__subset__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or411607219Most_a @ A @ B ) @ ( set_or411607219Most_a @ C @ D2 ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( ( ord_less_eq_a @ C @ A )
& ( ord_less_eq_a @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_132_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_133_dist__self,axiom,
! [X: a] :
( ( real_V1514887919dist_a @ X @ X )
= zero_zero_real ) ).
% dist_self
thf(fact_134_atLeastAtMost__iff,axiom,
! [I: a,L: a,U2: a] :
( ( member_a @ I @ ( set_or411607219Most_a @ L @ U2 ) )
= ( ( ord_less_eq_a @ L @ I )
& ( ord_less_eq_a @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_135_atLeastAtMost__iff,axiom,
! [I: set_a,L: set_a,U2: set_a] :
( ( member_set_a @ I @ ( set_or2094724627_set_a @ L @ U2 ) )
= ( ( ord_less_eq_set_a @ L @ I )
& ( ord_less_eq_set_a @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_136_atLeastAtMost__iff,axiom,
! [I: set_real,L: set_real,U2: set_real] :
( ( member_set_real @ I @ ( set_or1619725421t_real @ L @ U2 ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_137_atLeastAtMost__iff,axiom,
! [I: real,L: real,U2: real] :
( ( member_real @ I @ ( set_or656347191t_real @ L @ U2 ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_138_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or2094724627_set_a @ L @ H )
= ( set_or2094724627_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_139_Icc__eq__Icc,axiom,
! [L: set_real,H: set_real,L2: set_real,H2: set_real] :
( ( ( set_or1619725421t_real @ L @ H )
= ( set_or1619725421t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_real @ L @ H )
& ~ ( ord_less_eq_set_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_140_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or656347191t_real @ L @ H )
= ( set_or656347191t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_141_dist__eq__0__iff,axiom,
! [X: a,Y: a] :
( ( ( real_V1514887919dist_a @ X @ Y )
= zero_zero_real )
= ( X = Y ) ) ).
% dist_eq_0_iff
thf(fact_142_dist__commute,axiom,
( real_V1514887919dist_a
= ( ^ [X3: a,Y3: a] : ( real_V1514887919dist_a @ Y3 @ X3 ) ) ) ).
% dist_commute
thf(fact_143_dist__commute__lessI,axiom,
! [Y: a,X: a,E: real] :
( ( ord_less_real @ ( real_V1514887919dist_a @ Y @ X ) @ E )
=> ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ E ) ) ).
% dist_commute_lessI
thf(fact_144_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D2: set_a] :
( ( ord_less_set_set_a @ ( set_or2094724627_set_a @ A @ B ) @ ( set_or2094724627_set_a @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D2 )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B @ D2 ) ) ) )
& ( ord_less_eq_set_a @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_145_atLeastatMost__psubset__iff,axiom,
! [A: set_real,B: set_real,C: set_real,D2: set_real] :
( ( ord_le861030508t_real @ ( set_or1619725421t_real @ A @ B ) @ ( set_or1619725421t_real @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_real @ A @ B )
| ( ( ord_less_eq_set_real @ C @ A )
& ( ord_less_eq_set_real @ B @ D2 )
& ( ( ord_less_set_real @ C @ A )
| ( ord_less_set_real @ B @ D2 ) ) ) )
& ( ord_less_eq_set_real @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_146_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D2 )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D2 ) ) ) )
& ( ord_less_eq_real @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_147_zero__le__dist,axiom,
! [X: a,Y: a] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ).
% zero_le_dist
thf(fact_148_dist__not__less__zero,axiom,
! [X: a,Y: a] :
~ ( ord_less_real @ ( real_V1514887919dist_a @ X @ Y ) @ zero_zero_real ) ).
% dist_not_less_zero
thf(fact_149_dist__pos__lt,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ord_less_real @ zero_zero_real @ ( real_V1514887919dist_a @ X @ Y ) ) ) ).
% dist_pos_lt
thf(fact_150_rev_Operiodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.periodic_orbit_period(1)
thf(fact_151_periodic__orbit__def,axiom,
! [X: a] :
( ( period138238489rbit_a @ f @ x @ X )
= ( ( period720806154rbit_a @ f @ x @ X )
& ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% periodic_orbit_def
thf(fact_152_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( uminus_uminus_a_a @ f @ X )
= zero_zero_a )
=> ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= zero_zero_real ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_153_closed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( ( period1305449585riod_a @ f @ x @ X )
= zero_zero_real )
=> ( ( f @ X )
= zero_zero_a ) ) ) ).
% closed_orbit_period_zero_fixed_point
thf(fact_154_closed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% closed_orbit_period_nonneg
thf(fact_155_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( member_a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_156_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_157_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_158_neg__equal__iff__equal,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_159_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_160_rev_Oclosed__orbit__eq__rev,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= ( period720806154rbit_a @ f @ x @ X ) ) ).
% rev.closed_orbit_eq_rev
thf(fact_161_rev_Oclosed__orbit__in__domain,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( member_a @ X @ x ) ) ).
% rev.closed_orbit_in_domain
thf(fact_162_rev_Oinvariant__eq__rev,axiom,
! [M: set_a] :
( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
= ( auto_l630715367iant_a @ f @ x @ M ) ) ).
% rev.invariant_eq_rev
thf(fact_163_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_164_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( uminus_uminus_a_a @ f @ X )
= zero_zero_a )
=> ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_165_rev_OsectionD_I5_J,axiom,
! [S2: a > real,Ds: a > bounde7994401a_real,S3: set_a] :
( ( reacha1191408304tion_a @ ( uminus_uminus_a_a @ f ) @ x @ S2 @ Ds @ S3 )
=> ( ord_less_eq_set_a @ S3 @ x ) ) ).
% rev.sectionD(5)
thf(fact_166_rev_Oinvariant__iff__compl__invariant,axiom,
! [M: set_a] :
( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
= ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ ( minus_minus_set_a @ x @ M ) ) ) ).
% rev.invariant_iff_compl_invariant
thf(fact_167_closed__orbit__periodic,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( ( f @ X )
!= zero_zero_a )
=> ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% closed_orbit_periodic
thf(fact_168_rev_Oinvariant__def,axiom,
! [M: set_a] :
( ( auto_l630715367iant_a @ ( uminus_uminus_a_a @ f ) @ x @ M )
= ( ! [X3: a] :
( ( member_a @ X3 @ M )
=> ( oDE_au1996717075pped_a @ ( uminus_uminus_a_a @ f ) @ x @ X3 @ M ) ) ) ) ).
% rev.invariant_def
thf(fact_169_rev_Oclosed__orbit__periodic,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( ( uminus_uminus_a_a @ f @ X )
!= zero_zero_a )
=> ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_periodic
thf(fact_170_add_Oinverse__neutral,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% add.inverse_neutral
thf(fact_171_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_172_neg__0__equal__iff__equal,axiom,
! [A: a] :
( ( zero_zero_a
= ( uminus_uminus_a @ A ) )
= ( zero_zero_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_173_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_174_neg__equal__0__iff__equal,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_175_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_176_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_177_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_178_neg__le__iff__le,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_179_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_180_neg__less__iff__less,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_181_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_182_minus__diff__eq,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
= ( minus_minus_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_183_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_184_inner__minus__left,axiom,
! [X: real,Y: real] :
( ( inner_4346926r_real @ ( uminus_uminus_real @ X ) @ Y )
= ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% inner_minus_left
thf(fact_185_inner__minus__left,axiom,
! [X: a,Y: a] :
( ( inner_1173012732nner_a @ ( uminus_uminus_a @ X ) @ Y )
= ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% inner_minus_left
thf(fact_186_inner__minus__right,axiom,
! [X: real,Y: real] :
( ( inner_4346926r_real @ X @ ( uminus_uminus_real @ Y ) )
= ( uminus_uminus_real @ ( inner_4346926r_real @ X @ Y ) ) ) ).
% inner_minus_right
thf(fact_187_inner__minus__right,axiom,
! [X: a,Y: a] :
( ( inner_1173012732nner_a @ X @ ( uminus_uminus_a @ Y ) )
= ( uminus_uminus_real @ ( inner_1173012732nner_a @ X @ Y ) ) ) ).
% inner_minus_right
thf(fact_188_rev_Oclosed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_period_nonneg
thf(fact_189_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= zero_zero_real )
=> ( ( uminus_uminus_a_a @ f @ X )
= zero_zero_a ) ) ) ).
% rev.closed_orbit_period_zero_fixed_point
thf(fact_190_rev_Operiodic__orbit__def,axiom,
! [X: a] :
( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
& ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.periodic_orbit_def
thf(fact_191_neg__0__le__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% neg_0_le_iff_le
thf(fact_192_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_193_neg__le__0__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_194_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_195_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_196_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_197_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_198_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_199_neg__0__less__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% neg_0_less_iff_less
thf(fact_200_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_201_neg__less__0__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_0_iff_less
thf(fact_202_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_203_diff__0,axiom,
! [A: a] :
( ( minus_minus_a @ zero_zero_a @ A )
= ( uminus_uminus_a @ A ) ) ).
% diff_0
thf(fact_204_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_205_le__imp__neg__le,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% le_imp_neg_le
thf(fact_206_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_207_minus__le__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_208_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_209_le__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_eq_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% le_minus_iff
thf(fact_210_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_211_minus__less__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_212_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_213_less__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% less_minus_iff
thf(fact_214_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_215_minus__diff__commute,axiom,
! [B: a,A: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
= ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_216_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_217_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_218_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_219_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_220_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_221_verit__minus__simplify_I3_J,axiom,
! [B: a] :
( ( minus_minus_a @ zero_zero_a @ B )
= ( uminus_uminus_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_222_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_223_psubsetI,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_224_psubsetI,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_real @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_225_uminus__apply,axiom,
( uminus_uminus_a_a
= ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% uminus_apply
thf(fact_226_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_227_compl__le__compl__iff,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ ( uminus773214379t_real @ Y ) )
= ( ord_less_eq_set_real @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_228_rot__diff__commute,axiom,
! [B: a,A: a] :
( ( poinca659159244_rot_a @ ( minus_minus_a @ B @ A ) )
= ( uminus_uminus_a @ ( poinca659159244_rot_a @ ( minus_minus_a @ A @ B ) ) ) ) ).
% rot_diff_commute
thf(fact_229_nrm__reverse,axiom,
! [A: a,X: a,Y: a] :
( ( inner_1173012732nner_a @ A @ ( poinca659159244_rot_a @ ( minus_minus_a @ X @ Y ) ) )
= ( inner_1173012732nner_a @ ( uminus_uminus_a @ A ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ Y @ X ) ) ) ) ).
% nrm_reverse
thf(fact_230_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_231_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_232_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
= ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_233_Compl__subset__Compl__iff,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ ( uminus773214379t_real @ A2 ) @ ( uminus773214379t_real @ B3 ) )
= ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_234_Compl__anti__mono,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_235_Compl__anti__mono,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ord_less_eq_set_real @ ( uminus773214379t_real @ B3 ) @ ( uminus773214379t_real @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_236_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_237_subset__antisym,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( ord_less_eq_set_real @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_238_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_239_subsetI,axiom,
! [A2: set_real,B3: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B3 ) )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_240_Diff__idemp,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_241_Diff__iff,axiom,
! [C: real,A2: set_real,B3: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_242_Diff__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_243_DiffI,axiom,
! [C: real,A2: set_real,B3: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B3 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_244_DiffI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_245_rot__rot,axiom,
! [V: a] :
( ( poinca659159244_rot_a @ ( poinca659159244_rot_a @ V ) )
= ( uminus_uminus_a @ V ) ) ).
% rot_rot
thf(fact_246_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_247_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_248_verit__negate__coefficient_I3_J,axiom,
! [A: a,B: a] :
( ( A = B )
=> ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_249_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_250_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_251_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X3: real] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z3: set_a] : Y4 = Z3 )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_253_set__eq__subset,axiom,
( ( ^ [Y4: set_real,Z3: set_real] : Y4 = Z3 )
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A5 @ B5 )
& ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_254_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_255_subset__trans,axiom,
! [A2: set_real,B3: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( ord_less_eq_set_real @ B3 @ C3 )
=> ( ord_less_eq_set_real @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_256_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_257_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_258_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_259_subset__refl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% subset_refl
thf(fact_260_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_261_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T: real] :
( ( member_real @ T @ A5 )
=> ( member_real @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_262_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_263_equalityD2,axiom,
! [A2: set_real,B3: set_real] :
( ( A2 = B3 )
=> ( ord_less_eq_set_real @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_264_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_265_equalityD1,axiom,
! [A2: set_real,B3: set_real] :
( ( A2 = B3 )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_266_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_267_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( member_real @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_268_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_269_equalityE,axiom,
! [A2: set_real,B3: set_real] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
=> ~ ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_270_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_271_subsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_272_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_273_in__mono,axiom,
! [A2: set_real,B3: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_274_DiffD2,axiom,
! [C: real,A2: set_real,B3: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
=> ~ ( member_real @ C @ B3 ) ) ).
% DiffD2
thf(fact_275_DiffD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_276_DiffD1,axiom,
! [C: real,A2: set_real,B3: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_277_DiffD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_278_DiffE,axiom,
! [C: real,A2: set_real,B3: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% DiffE
thf(fact_279_DiffE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_280_psubsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_281_psubsetD,axiom,
! [A2: set_real,B3: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_282_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_283_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_284_compl__le__swap2,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ X )
=> ( ord_less_eq_set_real @ ( uminus773214379t_real @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_285_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_286_compl__le__swap1,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ ( uminus773214379t_real @ X ) )
=> ( ord_less_eq_set_real @ X @ ( uminus773214379t_real @ Y ) ) ) ).
% compl_le_swap1
thf(fact_287_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_288_compl__mono,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ord_less_eq_set_real @ ( uminus773214379t_real @ Y ) @ ( uminus773214379t_real @ X ) ) ) ).
% compl_mono
thf(fact_289_verit__negate__coefficient_I2_J,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_290_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_291_fun__Compl__def,axiom,
( uminus_uminus_a_a
= ( ^ [A5: a > a,X3: a] : ( uminus_uminus_a @ ( A5 @ X3 ) ) ) ) ).
% fun_Compl_def
thf(fact_292_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_293_double__diff,axiom,
! [A2: set_real,B3: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( ord_less_eq_set_real @ B3 @ C3 )
=> ( ( minus_minus_set_real @ B3 @ ( minus_minus_set_real @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_294_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_295_Diff__subset,axiom,
! [A2: set_real,B3: set_real] : ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_296_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D4 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_297_Diff__mono,axiom,
! [A2: set_real,C3: set_real,D4: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A2 @ C3 )
=> ( ( ord_less_eq_set_real @ D4 @ B3 )
=> ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ ( minus_minus_set_real @ C3 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_298_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_299_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_set_real @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_300_subset__psubset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_301_subset__psubset__trans,axiom,
! [A2: set_real,B3: set_real,C3: set_real] :
( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ( ord_less_set_real @ B3 @ C3 )
=> ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_302_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_303_subset__not__subset__eq,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A5 @ B5 )
& ~ ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_304_psubset__subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_305_psubset__subset__trans,axiom,
! [A2: set_real,B3: set_real,C3: set_real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ( ord_less_eq_set_real @ B3 @ C3 )
=> ( ord_less_set_real @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_306_psubset__imp__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_307_psubset__imp__subset,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_308_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_309_psubset__eq,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_310_psubsetE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_311_psubsetE,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B3 )
=> ( ord_less_eq_set_real @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_312_psubset__imp__ex__mem,axiom,
! [A2: set_real,B3: set_real] :
( ( ord_less_set_real @ A2 @ B3 )
=> ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_313_psubset__imp__ex__mem,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_314_rev_Oopen__existence__ivl_H,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ~ ! [A4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.open_existence_ivl'
thf(fact_315_open__existence__ivl_H,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ~ ! [A4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ~ ( ord_less_eq_set_real @ ( set_or656347191t_real @ ( uminus_uminus_real @ A4 ) @ A4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% open_existence_ivl'
thf(fact_316_rev_Oopen__existence__ivl0,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ? [A4: real,B4: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 )
& ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.open_existence_ivl0
thf(fact_317_existence__ivl__zero,axiom,
! [X0: a] :
( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% existence_ivl_zero
thf(fact_318_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% general.mem_existence_ivl_iv_defined(2)
thf(fact_319_Compl__iff,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
= ( ~ ( member_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_320_Compl__iff,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
= ( ~ ( member_real @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_321_ComplI,axiom,
! [C: a,A2: set_a] :
( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% ComplI
thf(fact_322_ComplI,axiom,
! [C: real,A2: set_real] :
( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ ( uminus773214379t_real @ A2 ) ) ) ).
% ComplI
thf(fact_323_rev_Oexistence__ivl__zero,axiom,
! [X0: a] :
( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% rev.existence_ivl_zero
thf(fact_324_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% rev.general.mem_existence_ivl_iv_defined(2)
thf(fact_325_local_Oivl__subset__existence__ivl,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% local.ivl_subset_existence_ivl
thf(fact_326_local_Oivl__subset__existence__ivl_H,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% local.ivl_subset_existence_ivl'
thf(fact_327_open__existence__ivl0,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ? [A4: real,B4: real] :
( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B4 )
& ( ord_less_eq_set_real @ ( set_or656347191t_real @ A4 @ B4 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% open_existence_ivl0
thf(fact_328_local_Orev_Oivl__subset__existence__ivl,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ord_less_eq_set_real @ ( set_or656347191t_real @ zero_zero_real @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% local.rev.ivl_subset_existence_ivl
thf(fact_329_local_Orev_Oivl__subset__existence__ivl_H,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ord_less_eq_set_real @ ( set_or656347191t_real @ T2 @ zero_zero_real ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% local.rev.ivl_subset_existence_ivl'
thf(fact_330_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_331_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.rev.mem_existence_ivl_iv_defined(2)
thf(fact_332_ComplD,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
=> ~ ( member_a @ C @ A2 ) ) ).
% ComplD
thf(fact_333_ComplD,axiom,
! [C: real,A2: set_real] :
( ( member_real @ C @ ( uminus773214379t_real @ A2 ) )
=> ~ ( member_real @ C @ A2 ) ) ).
% ComplD
thf(fact_334_rev_Omvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% rev.mvar.interval_axioms
thf(fact_335_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_336_rev_Oclosed__orbit__global__existence,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= top_top_set_real ) ) ).
% rev.closed_orbit_global_existence
thf(fact_337_rev_Ofixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( uminus_uminus_a_a @ f @ X )
= zero_zero_a )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= top_top_set_real ) ) ) ).
% rev.fixpoint_sol(1)
thf(fact_338_interval__axioms,axiom,
initia826609931terval @ top_top_set_real ).
% interval_axioms
thf(fact_339_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_340_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_341_local_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T2 @ top_top_set_real ) ) ).
% local.mem_existence_ivl_subset
thf(fact_342_local_Oexistence__ivl__initial__time,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_initial_time
thf(fact_343_local_Orev_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ T2 @ top_top_set_real ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_344_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T2: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T0 @ top_top_set_real ) ) ).
% general.mem_existence_ivl_iv_defined(1)
thf(fact_345_general_Oexistence__ivl__initial__time,axiom,
! [T0: real,X0: a] :
( ( member_real @ T0 @ top_top_set_real )
=> ( ( member_a @ X0 @ x )
=> ( member_real @ ( minus_minus_real @ T0 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_initial_time
thf(fact_346_mem__existence__ivl__shift__autonomous2,axiom,
! [T2: real,S2: real,X: a] :
( ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( member_a @ X @ x )
=> ( ! [S: real,T3: real,X4: a] :
( ( member_a @ X4 @ x )
=> ( ( f @ X4 )
= ( f @ X4 ) ) )
=> ( ( top_top_set_real = top_top_set_real )
=> ( member_real @ ( minus_minus_real @ T2 @ S2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ) ) ).
% mem_existence_ivl_shift_autonomous2
thf(fact_347_general_Omem__existence__ivl__subset,axiom,
! [T2: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T2 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T2 @ top_top_set_real ) ) ).
% general.mem_existence_ivl_subset
thf(fact_348_local_Oexistence__ivl__subset,axiom,
! [X0: a] : ( ord_less_eq_set_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) @ top_top_set_real ) ).
% local.existence_ivl_subset
thf(fact_349_fixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( ( auto_l612940ivl0_a @ f @ x @ X )
= top_top_set_real ) ) ) ).
% fixpoint_sol(1)
thf(fact_350_closed__orbit__global__existence,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( auto_l612940ivl0_a @ f @ x @ X )
= top_top_set_real ) ) ).
% closed_orbit_global_existence
thf(fact_351_local_Orev_Oexistence__ivl__initial__time,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_initial_time
% Conjectures (1)
thf(conj_0,conjecture,
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ? [B7: real] :
( ( ord_less_real @ zero_zero_real @ B7 )
& ! [X4: a] :
( ( member_a @ X4 @ ( elemen49976720ball_a @ x2 @ D3 ) )
=> ( ord_less_eq_real @ B7 @ ( inner_1173012732nner_a @ ( f @ X4 ) @ ( poinca659159244_rot_a @ ( minus_minus_a @ a2 @ b ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------