TPTP Problem File: ITP149^2.p
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%------------------------------------------------------------------------------
% File : ITP149^2 : 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 : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 382 ( 86 unt; 62 typ; 0 def)
% Number of atoms : 869 ( 220 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 3838 ( 59 ~; 8 |; 37 &;3373 @)
% ( 0 <=>; 361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 60 usr; 8 con; 0-6 aty)
% Number of variables : 776 ( 39 ^; 680 !; 9 ?; 776 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:39.099
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
bounde2145540817linfun: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (58)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Inner__Product_Oreal__inner,type,
inner_real_inner:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
euclid925273238_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo890362671_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
execut510477386_space:
!>[A: $tType] : $o ).
thf(sy_c_Elementary__Metric__Spaces_Ocball,type,
elemen321786957_cball:
!>[A: $tType] : ( A > real > ( set @ A ) ) ).
thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
auto_l1112008849e_ivl0:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Initial__Value__Problem_Ointerval,type,
initia826609931terval: ( set @ real ) > $o ).
thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
inner_780170721_inner:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Invariance_Oauto__ll__on__open_Oinvariant,type,
auto_ll_on_invariant:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Line__Segment_Oclosed__segment,type,
line_closed_segment:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Line__Segment_Oopen__segment,type,
line_open_segment:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_ODE__Misc_Oauto__ll__on__open_Otrapped,type,
oDE_au1039603466rapped:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ A ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
period385816147_orbit:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
period1153813292period:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
period862636932_orbit:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Orot,type,
poinca1750768982en_rot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
poinca272511729egment:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
thf(sy_c_Reachability__Analysis_Oc1__on__open__euclidean_Osection,type,
reacha1084862253ection:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > ( A > real ) > ( A > ( bounde2145540817linfun @ A @ real ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V2000881966t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or331188842AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( 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 (255)
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 @ ( elemen321786957_cball @ 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_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_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 @ ( elemen321786957_cball @ a @ x2 @ d ) )
=> ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ s ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_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_780170721_inner @ a @ ( f @ z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ).
% direction(2)
thf(fact_8_d_I2_J,axiom,
! [Y2: a] :
( ( member @ a @ Y2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
& ( member @ a @ Y2 @ x )
& ( ( f @ Y2 )
!= ( 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_780170721_inner @ a @ ( f @ x2 ) @ ( poinca1750768982en_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_780170721_inner @ a @ X @ ( poinca1750768982en_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 @ ( elemen321786957_cball @ a @ x2 @ d ) )
=> ~ ! [X2: a] :
( ( member @ a @ X2 @ ( elemen321786957_cball @ a @ x2 @ d ) )
=> ( ord_less_eq @ real @ ( inner_780170721_inner @ a @ ( f @ S ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_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_closed_segment @ 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 )
=> ~ ! [Y3: a] :
( ( member @ a @ Y3 @ ( elemen321786957_cball @ a @ x2 @ D ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Y3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) )
& ( member @ a @ Y3 @ x )
& ( ( f @ Y3 )
!= ( 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,
poinca272511729egment @ a @ f @ x @ a2 @ b ).
% transversal
thf(fact_15_inner__gt__zero__iff,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% inner_gt_zero_iff
thf(fact_16_centre__in__cball,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,E: real] :
( ( member @ A @ X @ ( elemen321786957_cball @ A @ X @ E ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% centre_in_cball
thf(fact_17_all__zero__iff,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] :
( ( ! [U: A] :
( ( inner_780170721_inner @ A @ X @ U )
= ( zero_zero @ real ) ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% all_zero_iff
thf(fact_18_inner__zero__left,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] :
( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ real ) ) ) ).
% inner_zero_left
thf(fact_19_inner__zero__right,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] :
( ( inner_780170721_inner @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% inner_zero_right
thf(fact_20_inner__eq__zero__iff,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] :
( ( ( inner_780170721_inner @ A @ X @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% inner_eq_zero_iff
thf(fact_21_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
= ( ord_less @ A @ B @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_22_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
= ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_23_rot__0,axiom,
( ( poinca1750768982en_rot @ a @ ( zero_zero @ a ) )
= ( zero_zero @ a ) ) ).
% rot_0
thf(fact_24_rot__eq__0__iff,axiom,
! [X: a] :
( ( ( poinca1750768982en_rot @ a @ X )
= ( zero_zero @ a ) )
= ( X
= ( zero_zero @ a ) ) ) ).
% rot_eq_0_iff
thf(fact_25_seg_I3_J,axiom,
! [Z: a] :
( ( member @ a @ Z @ ( line_closed_segment @ a @ a2 @ b ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ).
% seg(3)
thf(fact_26_direction_I1_J,axiom,
member @ a @ z @ ( line_closed_segment @ a @ a2 @ b ) ).
% direction(1)
thf(fact_27_transversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% transversal_segment_reverse
thf(fact_28_transversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
= ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% transversal_segment_commute
thf(fact_29_in__segment__inner__rot2,axiom,
! [X: a,A2: a,B: a,Y: a] :
( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
=> ( ( member @ a @ Y @ ( line_closed_segment @ a @ A2 @ B ) )
=> ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ Y ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
= ( zero_zero @ real ) ) ) ) ).
% in_segment_inner_rot2
thf(fact_30_in__segment__inner__rot,axiom,
! [X: a,A2: a,B: a] :
( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
=> ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
= ( zero_zero @ real ) ) ) ).
% in_segment_inner_rot
thf(fact_31_seg_I2_J,axiom,
ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ a2 @ b ) @ x ).
% seg(2)
thf(fact_32_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_33_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_34_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_35_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_36_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_37_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_38_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_39_subset__cball,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [D2: real,E: real,X: A] :
( ( ord_less_eq @ real @ D2 @ E )
=> ( ord_less_eq @ ( set @ A ) @ ( elemen321786957_cball @ A @ X @ D2 ) @ ( elemen321786957_cball @ A @ X @ E ) ) ) ) ).
% subset_cball
thf(fact_40_transversal__segment__def,axiom,
! [A2: a,B: a] :
( ( poinca272511729egment @ a @ f @ x @ A2 @ B )
= ( ( A2 != B )
& ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ A2 @ B ) @ x )
& ! [X3: a] :
( ( member @ a @ X3 @ ( line_closed_segment @ a @ A2 @ B ) )
=> ( ( inner_780170721_inner @ a @ ( f @ X3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) )
!= ( zero_zero @ real ) ) ) ) ) ).
% transversal_segment_def
thf(fact_41_transversal__segmentE,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( ( X != Y )
=> ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
=> ~ ! [Z2: a] :
( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ( ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x )
=> ~ ! [Z2: a] :
( ( member @ a @ Z2 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ) ) ) ) ) ).
% transversal_segmentE
thf(fact_42_transversal__segment__posD_I2_J,axiom,
! [X: a,Y: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
=> ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% transversal_segment_posD(2)
thf(fact_43_transversal__segment__negD_I2_J,axiom,
! [X: a,Y: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
=> ( ord_less_eq @ ( set @ a ) @ ( line_closed_segment @ a @ X @ Y ) @ x ) ) ) ) ).
% transversal_segment_negD(2)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_transversal__segment__sign__less,axiom,
! [W: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ W @ Y )
=> ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) )
=> ! [X2: a] :
( ( member @ a @ X2 @ ( line_closed_segment @ a @ W @ Y ) )
=> ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ W @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ).
% transversal_segment_sign_less
thf(fact_49_transversal__segment__pos,axiom,
! [X: a,Y: a,W: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
=> ! [X2: a] :
( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% transversal_segment_pos
thf(fact_50_transversal__segment__neg,axiom,
! [X: a,Y: a,W: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ W @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ W ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
=> ! [X2: a] :
( ( member @ a @ X2 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ X2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% transversal_segment_neg
thf(fact_51_transversal__segment__posD_I1_J,axiom,
! [X: a,Y: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
=> ( X != Y ) ) ) ) ).
% transversal_segment_posD(1)
thf(fact_52_transversal__segment__posD_I3_J,axiom,
! [X: a,Y: a,Za: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) ) ) ) ) ) ).
% transversal_segment_posD(3)
thf(fact_53_transversal__segment__negD_I1_J,axiom,
! [X: a,Y: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
=> ( X != Y ) ) ) ) ).
% transversal_segment_negD(1)
thf(fact_54_transversal__segment__negD_I3_J,axiom,
! [X: a,Y: a,Za: a,Z3: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( ( member @ a @ Za @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Za ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) )
=> ( ( member @ a @ Z3 @ ( line_closed_segment @ a @ X @ Y ) )
=> ( ord_less @ real @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% transversal_segment_negD(3)
thf(fact_55__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,
! [Z3: a] :
( ( member @ a @ Z3 @ ( line_closed_segment @ a @ a2 @ b ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ a @ ( f @ Z3 ) @ ( poinca1750768982en_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_56_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ( poinca272511729egment @ A )
= ( poinca272511729egment @ A ) ) ) ).
% c1_on_open_R2.transversal_segment.cong
thf(fact_57_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_58_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_59_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_60_inner__commute,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ( ( inner_780170721_inner @ A )
= ( ^ [X3: A,Y4: A] : ( inner_780170721_inner @ A @ Y4 @ X3 ) ) ) ) ).
% inner_commute
thf(fact_61_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_62_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_63_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_64_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_65_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_66_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_67_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_68_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% diff_right_mono
thf(fact_69_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% diff_left_mono
thf(fact_70_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,D2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ D2 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_71_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_72_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% diff_strict_left_mono
thf(fact_73_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B )
= ( ord_less @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_74_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A,D2: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ D2 @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_75_inner__diff__right,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( inner_780170721_inner @ A @ X @ ( minus_minus @ A @ Y @ Z3 ) )
= ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Y ) @ ( inner_780170721_inner @ A @ X @ Z3 ) ) ) ) ).
% inner_diff_right
thf(fact_76_inner__diff__left,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( inner_780170721_inner @ A @ ( minus_minus @ A @ X @ Y ) @ Z3 )
= ( minus_minus @ real @ ( inner_780170721_inner @ A @ X @ Z3 ) @ ( inner_780170721_inner @ A @ Y @ Z3 ) ) ) ) ).
% inner_diff_left
thf(fact_77_mem__cball__leI,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A,E: real,F: real] :
( ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ E ) )
=> ( ( ord_less_eq @ real @ E @ F )
=> ( member @ A @ X @ ( elemen321786957_cball @ A @ Y @ F ) ) ) ) ) ).
% mem_cball_leI
thf(fact_78_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_79_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_80_inner__ge__zero,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ X @ X ) ) ) ).
% inner_ge_zero
thf(fact_81_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( period1153813292period @ a @ f @ x @ X )
= ( zero_zero @ real ) ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_82_that,axiom,
! [D3: real,B4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
=> ( ! [X4: a,Y6: a] :
( ( member @ a @ X4 @ ( line_closed_segment @ a @ a2 @ b ) )
=> ( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ a @ X4 @ Y6 ) @ D3 )
=> ( ord_less_eq @ real @ B4 @ ( inner_780170721_inner @ a @ ( f @ Y6 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) )
=> thesis ) ) ) ).
% that
thf(fact_83_inner__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere890947078_space @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( inner_780170721_inner @ A @ A2 @ B ) ) ) ) ) ).
% inner_nonneg_nonneg
thf(fact_84_in__closed__segment__iff__rot,axiom,
! [A2: a,B: a,X: a] :
( ( A2 != B )
=> ( ( member @ a @ X @ ( line_closed_segment @ a @ A2 @ B ) )
= ( ( ( inner_780170721_inner @ a @ ( minus_minus @ a @ X @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) ) )
= ( zero_zero @ real ) )
& ( member @ real @ ( inner_780170721_inner @ a @ X @ ( minus_minus @ a @ B @ A2 ) ) @ ( set_or331188842AtMost @ real @ ( inner_780170721_inner @ a @ A2 @ ( minus_minus @ a @ B @ A2 ) ) @ ( inner_780170721_inner @ a @ B @ ( minus_minus @ a @ B @ A2 ) ) ) ) ) ) ) ).
% in_closed_segment_iff_rot
thf(fact_85_sectionD_I5_J,axiom,
! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
( ( reacha1084862253ection @ a @ f @ x @ S2 @ Ds @ S3 )
=> ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% sectionD(5)
thf(fact_86_transversal__segment__exists,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
!= ( zero_zero @ a ) )
=> ~ ! [A5: a,B5: a] :
( ( member @ a @ X @ ( line_open_segment @ a @ A5 @ B5 ) )
=> ~ ( poinca272511729egment @ a @ f @ x @ A5 @ B5 ) ) ) ) ).
% transversal_segment_exists
thf(fact_87_invariant__iff__compl__invariant,axiom,
! [M2: set @ a] :
( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
= ( auto_ll_on_invariant @ a @ f @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% invariant_iff_compl_invariant
thf(fact_88_cball__eq__cball__iff,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [X: A,D2: real,Y: A,E: real] :
( ( ( elemen321786957_cball @ A @ X @ D2 )
= ( elemen321786957_cball @ 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_89_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B )
=> ( ! [A5: real,B5: real,C2: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C2 )
=> ( ( ord_less_eq @ real @ A5 @ B5 )
=> ( ( ord_less_eq @ real @ B5 @ C2 )
=> ( P @ A5 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq @ real @ A5 @ X4 )
& ( ord_less_eq @ real @ X4 @ B5 )
& ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_90_mem__cball,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [Y: A,X: A,E: real] :
( ( member @ A @ Y @ ( elemen321786957_cball @ A @ X @ E ) )
= ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% mem_cball
thf(fact_91_perfect__choose__dist,axiom,
! [A: $tType] :
( ( ( real_V2090557954_space @ A )
& ( topolo890362671_space @ A ) )
=> ! [R: real,X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [A5: A] :
( ( A5 != X )
& ( ord_less @ real @ ( real_V2000881966t_dist @ A @ A5 @ X ) @ R ) ) ) ) ).
% perfect_choose_dist
thf(fact_92_interval__inner__leI_I2_J,axiom,
! [A: $tType] :
( ( ordere890947078_space @ A )
=> ! [X: A,A2: A,B: A,I: A] :
( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
=> ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ X @ I ) @ ( inner_780170721_inner @ A @ B @ I ) ) ) ) ) ).
% interval_inner_leI(2)
thf(fact_93_interval__inner__leI_I1_J,axiom,
! [A: $tType] :
( ( ordere890947078_space @ A )
=> ! [X: A,A2: A,B: A,I: A] :
( ( member @ A @ X @ ( set_or331188842AtMost @ A @ A2 @ B ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ I )
=> ( ord_less_eq @ real @ ( inner_780170721_inner @ A @ A2 @ I ) @ ( inner_780170721_inner @ A @ X @ I ) ) ) ) ) ).
% interval_inner_leI(1)
thf(fact_94_eucl__less__le__not__le,axiom,
! [A: $tType] :
( ( ordere890947078_space @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_95_invariant__def,axiom,
! [M2: set @ a] :
( ( auto_ll_on_invariant @ a @ f @ x @ M2 )
= ( ! [X3: a] :
( ( member @ a @ X3 @ M2 )
=> ( oDE_au1039603466rapped @ a @ f @ x @ X3 @ M2 ) ) ) ) ).
% invariant_def
thf(fact_96_periodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ f @ x @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% periodic_orbit_period(1)
thf(fact_97_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_98_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_99_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A,C: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B )
| ( ( ord_less_eq @ A @ C @ A2 )
& ( ord_less_eq @ A @ B @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_100_dist__self,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A] :
( ( real_V2000881966t_dist @ A @ X @ X )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_101_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U2 ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_102_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L2: A,H2: A] :
( ( ( set_or331188842AtMost @ A @ L @ H )
= ( set_or331188842AtMost @ A @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_103_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
( ( ( real_V2000881966t_dist @ A @ X @ Y )
= ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_eq_0_iff
thf(fact_104_dist__commute,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ( ( real_V2000881966t_dist @ A )
= ( ^ [X3: A,Y4: A] : ( real_V2000881966t_dist @ A @ Y4 @ X3 ) ) ) ) ).
% dist_commute
thf(fact_105_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [Y: A,X: A,E: real] :
( ( ord_less @ real @ ( real_V2000881966t_dist @ A @ Y @ X ) @ E )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ E ) ) ) ).
% dist_commute_lessI
thf(fact_106_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A,C: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B ) @ ( set_or331188842AtMost @ A @ C @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B )
| ( ( ord_less_eq @ A @ C @ A2 )
& ( ord_less_eq @ A @ B @ D2 )
& ( ( ord_less @ A @ C @ A2 )
| ( ord_less @ A @ B @ D2 ) ) ) )
& ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_107_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_108_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V2000881966t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_109_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_110_rev_Operiodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.periodic_orbit_period(1)
thf(fact_111_periodic__orbit__def,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ f @ x @ X )
= ( ( period385816147_orbit @ a @ f @ x @ X )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% periodic_orbit_def
thf(fact_112_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 ) )
=> ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( zero_zero @ real ) ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_113_closed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( ( period1153813292period @ a @ f @ x @ X )
= ( zero_zero @ real ) )
=> ( ( f @ X )
= ( zero_zero @ a ) ) ) ) ).
% closed_orbit_period_zero_fixed_point
thf(fact_114_closed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% closed_orbit_period_nonneg
thf(fact_115_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( member @ a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_116_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_117_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B ) )
= ( A2 = B ) ) ) ).
% neg_equal_iff_equal
thf(fact_118_rev_Oclosed__orbit__eq__rev,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% rev.closed_orbit_eq_rev
thf(fact_119_rev_Oclosed__orbit__in__domain,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( member @ a @ X @ x ) ) ).
% rev.closed_orbit_in_domain
thf(fact_120_rev_Oinvariant__eq__rev,axiom,
! [M2: set @ a] :
( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
= ( auto_ll_on_invariant @ a @ f @ x @ M2 ) ) ).
% rev.invariant_eq_rev
thf(fact_121_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_122_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 ) )
=> ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_123_rev_OsectionD_I5_J,axiom,
! [S2: a > real,Ds: a > ( bounde2145540817linfun @ a @ real ),S3: set @ a] :
( ( reacha1084862253ection @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ S2 @ Ds @ S3 )
=> ( ord_less_eq @ ( set @ a ) @ S3 @ x ) ) ).
% rev.sectionD(5)
thf(fact_124_rev_Oinvariant__iff__compl__invariant,axiom,
! [M2: set @ a] :
( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
= ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( minus_minus @ ( set @ a ) @ x @ M2 ) ) ) ).
% rev.invariant_iff_compl_invariant
thf(fact_125_closed__orbit__periodic,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( ( f @ X )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% closed_orbit_periodic
thf(fact_126_rev_Oinvariant__def,axiom,
! [M2: set @ a] :
( ( auto_ll_on_invariant @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ M2 )
= ( ! [X3: a] :
( ( member @ a @ X3 @ M2 )
=> ( oDE_au1039603466rapped @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X3 @ M2 ) ) ) ) ).
% rev.invariant_def
thf(fact_127_rev_Oclosed__orbit__periodic,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_periodic
thf(fact_128_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_129_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_130_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_131_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_132_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_133_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% neg_le_iff_le
thf(fact_134_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% neg_less_iff_less
thf(fact_135_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B ) )
= ( minus_minus @ A @ B @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_136_inner__minus__left,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A,Y: A] :
( ( inner_780170721_inner @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% inner_minus_left
thf(fact_137_inner__minus__right,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X: A,Y: A] :
( ( inner_780170721_inner @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( uminus_uminus @ real @ ( inner_780170721_inner @ A @ X @ Y ) ) ) ) ).
% inner_minus_right
thf(fact_138_rev_Oclosed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_period_nonneg
thf(fact_139_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( ( period1153813292period @ 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_140_rev_Operiodic__orbit__def,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.periodic_orbit_def
thf(fact_141_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_142_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_143_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_144_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_145_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_146_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_147_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_148_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_149_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_150_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_151_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_152_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B ) )
= ( ord_less_eq @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_153_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B )
= ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_154_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B ) )
= ( ord_less @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_155_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B ) ) ) ).
% minus_diff_commute
thf(fact_156_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( uminus_uminus @ A @ B ) )
= ( B
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_157_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( ( uminus_uminus @ A @ A2 )
= B )
= ( ( uminus_uminus @ A @ B )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_158_verit__minus__simplify_I3_J,axiom,
! [B2: $tType] :
( ( group_add @ B2 )
=> ! [B: B2] :
( ( minus_minus @ B2 @ ( zero_zero @ B2 ) @ B )
= ( uminus_uminus @ B2 @ B ) ) ) ).
% verit_minus_simplify(3)
thf(fact_159_psubsetI,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( A3 != B6 )
=> ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ).
% psubsetI
thf(fact_160_uminus__apply,axiom,
! [B2: $tType,A: $tType] :
( ( uminus @ B2 )
=> ( ( uminus_uminus @ ( A > B2 ) )
= ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_161_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_162_rot__diff__commute,axiom,
! [B: a,A2: a] :
( ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ B @ A2 ) )
= ( uminus_uminus @ a @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ A2 @ B ) ) ) ) ).
% rot_diff_commute
thf(fact_163_nrm__reverse,axiom,
! [A2: a,X: a,Y: a] :
( ( inner_780170721_inner @ a @ A2 @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ X @ Y ) ) )
= ( inner_780170721_inner @ a @ ( uminus_uminus @ a @ A2 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ Y @ X ) ) ) ) ).
% nrm_reverse
thf(fact_164_verit__minus__simplify_I4_J,axiom,
! [B2: $tType] :
( ( group_add @ B2 )
=> ! [B: B2] :
( ( uminus_uminus @ B2 @ ( uminus_uminus @ B2 @ B ) )
= B ) ) ).
% verit_minus_simplify(4)
thf(fact_165_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_166_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_167_minus__apply,axiom,
! [B2: $tType,A: $tType] :
( ( minus @ B2 )
=> ( ( minus_minus @ ( A > B2 ) )
= ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% minus_apply
thf(fact_168_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_169_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_170_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% subset_antisym
thf(fact_171_subsetI,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% subsetI
thf(fact_172_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ B6 )
= ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ).
% Diff_idemp
thf(fact_173_Diff__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
= ( ( member @ A @ C @ A3 )
& ~ ( member @ A @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_174_DiffI,axiom,
! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ~ ( member @ A @ C @ B6 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).
% DiffI
thf(fact_175_rot__rot,axiom,
! [V: a] :
( ( poinca1750768982en_rot @ a @ ( poinca1750768982en_rot @ a @ V ) )
= ( uminus_uminus @ a @ V ) ) ).
% rot_rot
thf(fact_176_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( A2 = B )
| ~ ( ord_less_eq @ A @ A2 @ B )
| ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_177_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_178_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( A2 = B )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_179_fun__diff__def,axiom,
! [B2: $tType,A: $tType] :
( ( minus @ B2 )
=> ( ( minus_minus @ ( A > B2 ) )
= ( ^ [A6: A > B2,B7: A > B2,X3: A] : ( minus_minus @ B2 @ ( A6 @ X3 ) @ ( B7 @ X3 ) ) ) ) ) ).
% fun_diff_def
thf(fact_180_Collect__mono__iff,axiom,
! [A: $tType,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_181_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4 )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_182_subset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_183_Collect__mono,axiom,
! [A: $tType,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_184_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_185_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_186_equalityD2,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).
% equalityD2
thf(fact_187_equalityD1,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% equalityD1
thf(fact_188_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_189_equalityE,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( A3 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% equalityE
thf(fact_190_subsetD,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B6 ) ) ) ).
% subsetD
thf(fact_191_in__mono,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_192_DiffD2,axiom,
! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ~ ( member @ A @ C @ B6 ) ) ).
% DiffD2
thf(fact_193_DiffD1,axiom,
! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ( member @ A @ C @ A3 ) ) ).
% DiffD1
thf(fact_194_DiffE,axiom,
! [A: $tType,C: A,A3: set @ A,B6: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B6 ) ) ) ).
% DiffE
thf(fact_195_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_196_psubsetD,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_197_verit__comp__simplify1_I3_J,axiom,
! [B2: $tType] :
( ( linorder @ B2 )
=> ! [B8: B2,A7: B2] :
( ( ~ ( ord_less_eq @ B2 @ B8 @ A7 ) )
= ( ord_less @ B2 @ A7 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_198_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_199_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_200_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_201_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_202_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_203_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_204_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_205_fun__Compl__def,axiom,
! [B2: $tType,A: $tType] :
( ( uminus @ B2 )
=> ( ( uminus_uminus @ ( A > B2 ) )
= ( ^ [A6: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A6 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_206_double__diff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B6 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_207_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ A3 ) ).
% Diff_subset
thf(fact_208_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,D5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D5 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D5 ) ) ) ) ).
% Diff_mono
thf(fact_209_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_210_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_211_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_212_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_213_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_214_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_215_psubsetE,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% psubsetE
thf(fact_216_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_217_rev_Oopen__existence__ivl_H,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ~ ! [A5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
=> ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.open_existence_ivl'
thf(fact_218_open__existence__ivl_H,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ~ ! [A5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A5 )
=> ~ ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( uminus_uminus @ real @ A5 ) @ A5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% open_existence_ivl'
thf(fact_219_rev_Oopen__existence__ivl0,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ? [A5: real,B5: real] :
( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.open_existence_ivl0
thf(fact_220_existence__ivl0__cong,axiom,
! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
( ( x = Y7 )
=> ( ! [X4: a,T2: B2] :
( ( member @ a @ X4 @ Y7 )
=> ( ( f @ X4 )
= ( G @ X4 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% existence_ivl0_cong
thf(fact_221_existence__ivl__zero,axiom,
! [X0: a] :
( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% existence_ivl_zero
thf(fact_222_rev_Oexistence__ivl0__cong,axiom,
! [B2: $tType,Y7: set @ a,G: a > a,X0: a] :
( ( x = Y7 )
=> ( ! [X4: a,T2: B2] :
( ( member @ a @ X4 @ Y7 )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X4 )
= ( G @ X4 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y7 @ X0 ) ) ) ) ).
% rev.existence_ivl0_cong
thf(fact_223_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T3: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% general.mem_existence_ivl_iv_defined(2)
thf(fact_224_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B6: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( A3 = B6 ) ) ).
% Compl_eq_Compl_iff
thf(fact_225_Compl__iff,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_226_ComplI,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_227_rev_Oexistence__ivl__zero,axiom,
! [X0: a] :
( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% rev.existence_ivl_zero
thf(fact_228_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T3: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% rev.general.mem_existence_ivl_iv_defined(2)
thf(fact_229_local_Oivl__subset__existence__ivl,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% local.ivl_subset_existence_ivl
thf(fact_230_local_Oivl__subset__existence__ivl_H,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% local.ivl_subset_existence_ivl'
thf(fact_231_open__existence__ivl0,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ? [A5: real,B5: real] :
( ( ord_less @ real @ A5 @ ( zero_zero @ real ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ A5 @ B5 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% open_existence_ivl0
thf(fact_232_local_Orev_Oivl__subset__existence__ivl,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% local.rev.ivl_subset_existence_ivl
thf(fact_233_local_Orev_Oivl__subset__existence__ivl_H,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ord_less_eq @ ( set @ real ) @ ( set_or331188842AtMost @ real @ T3 @ ( zero_zero @ real ) ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% local.rev.ivl_subset_existence_ivl'
thf(fact_234_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_235_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% local.rev.mem_existence_ivl_iv_defined(2)
thf(fact_236_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_237_ComplD,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C @ A3 ) ) ).
% ComplD
thf(fact_238_rev_Omvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% rev.mvar.interval_axioms
thf(fact_239_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_240_rev_Oclosed__orbit__global__existence,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ).
% rev.closed_orbit_global_existence
thf(fact_241_rev_Ofixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% rev.fixpoint_sol(1)
thf(fact_242_interval__axioms,axiom,
initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% interval_axioms
thf(fact_243_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_244_local_Omem__existence__ivl__subset,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% local.mem_existence_ivl_subset
thf(fact_245_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_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_initial_time
thf(fact_246_local_Orev_Omem__existence__ivl__subset,axiom,
! [T3: real,X0: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_247_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T3: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) ) ) ).
% general.mem_existence_ivl_iv_defined(1)
thf(fact_248_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_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_initial_time
thf(fact_249_mem__existence__ivl__shift__autonomous2,axiom,
! [T3: real,S2: real,X: a] :
( ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( member @ a @ X @ x )
=> ( ! [S: real,T2: real,X4: a] :
( ( member @ a @ X4 @ x )
=> ( ( f @ X4 )
= ( f @ X4 ) ) )
=> ( ( ( top_top @ ( set @ real ) )
= ( top_top @ ( set @ real ) ) )
=> ( member @ real @ ( minus_minus @ real @ T3 @ S2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ) ) ).
% mem_existence_ivl_shift_autonomous2
thf(fact_250_general_Omem__existence__ivl__subset,axiom,
! [T3: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T3 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T3 @ ( top_top @ ( set @ real ) ) ) ) ).
% general.mem_existence_ivl_subset
thf(fact_251_local_Oexistence__ivl__subset,axiom,
! [X0: a] : ( ord_less_eq @ ( set @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) @ ( top_top @ ( set @ real ) ) ) ).
% local.existence_ivl_subset
thf(fact_252_fixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% fixpoint_sol(1)
thf(fact_253_closed__orbit__global__existence,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ).
% closed_orbit_global_existence
thf(fact_254_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_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_initial_time
% Subclasses (17)
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( type @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( zero @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ominus,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( minus @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ouminus,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( uminus @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oord,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ord @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Oorder,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( order @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( preorder @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ab_group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Inner__Product_Oreal__inner,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( inner_real_inner @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordered_ab_group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( cancel1352612707id_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( cancel146912293up_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Euclidean__Space_Oeuclidean__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( euclid925273238_space @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( real_V2090557954_space @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Topological__Spaces_Operfect__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( topolo890362671_space @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere890947078_space @ A ) ) ).
% Type constructors (46)
thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
countable @ $o ).
thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( countable @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Countable_Ocountable_2,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( countable @ A9 ) )
=> ( countable @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
finite_finite @ $o ).
thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( real_V2090557954_space @ A9 ) )
=> ( real_V2090557954_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 )
=> ( boolean_algebra @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A8: $tType,A9: $tType] :
( ( uminus @ A9 )
=> ( uminus @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 )
=> ( minus @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_10,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A8: $tType] :
( ( zero @ A8 )
=> ( zero @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_15,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_16,axiom,
minus @ $o ).
thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
execut510477386_space @ real ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
ordere890947078_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo890362671_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_17,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
euclid925273238_space @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
inner_real_inner @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_20,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_21,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_22,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_23,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_24,axiom,
zero @ real ).
% Free types (1)
thf(tfree_0,hypothesis,
execut510477386_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ? [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
& ! [X4: a] :
( ( member @ a @ X4 @ ( elemen321786957_cball @ a @ x2 @ D4 ) )
=> ( ord_less_eq @ real @ B9 @ ( inner_780170721_inner @ a @ ( f @ X4 ) @ ( poinca1750768982en_rot @ a @ ( minus_minus @ a @ a2 @ b ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------