TPTP Problem File: ITP148^2.p
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%------------------------------------------------------------------------------
% File : ITP148^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Poincare_Bendixson problem prob_301__19573322_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_301__19573322_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 330 ( 89 unt; 45 typ; 0 def)
% Number of atoms : 813 ( 349 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 4129 ( 35 ~; 2 |; 64 &;3664 @)
% ( 0 <=>; 364 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 423 ( 423 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 41 usr; 0 con; 1-8 aty)
% Number of variables : 1184 ( 32 ^;1083 !; 19 ?;1184 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:43.127
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
bounde2145540817linfun: $tType > $tType > $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (39)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Inner__Product_Oreal__inner,type,
inner_real_inner:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Space_Oeuclidean__space,type,
euclid925273238_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
execut510477386_space:
!>[A: $tType] : $o ).
thf(sy_c_Affine_Oaff__dim,type,
aff_dim:
!>[A: $tType] : ( ( set @ A ) > int ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Inner__Product_Oreal__inner__class_Oinner,type,
inner_780170721_inner:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Linear__Algebra_Oadjoint,type,
linear_adjoint:
!>[A: $tType,B: $tType] : ( ( A > B ) > B > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Path__Connected_Oarc,type,
path_arc:
!>[A: $tType] : ( ( real > A ) > $o ) ).
thf(sy_c_Path__Connected_Opath,type,
path_path:
!>[A: $tType] : ( ( real > A ) > $o ) ).
thf(sy_c_Path__Connected_Opathfinish,type,
path_pathfinish:
!>[A: $tType] : ( ( real > A ) > A ) ).
thf(sy_c_Path__Connected_Opathstart,type,
path_pathstart:
!>[A: $tType] : ( ( real > A ) > A ) ).
thf(sy_c_Path__Connected_Orectpath,type,
path_rectpath: complex > complex > real > complex ).
thf(sy_c_Path__Connected_Osimple__path,type,
path_simple_path:
!>[A: $tType] : ( ( real > A ) > $o ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2,type,
poinca1076805489pen_R2:
!>[A: $tType] : ( ( A > A ) > ( A > ( bounde2145540817linfun @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Ocomplex__of,type,
poinca1531302022lex_of:
!>[A: $tType] : ( A > complex ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Oreal__of,type,
poinca179318102eal_of:
!>[A: $tType] : ( complex > A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V1632203528linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
real_V1072672427axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
iso_tu2011167877assist:
!>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Starlike_Ocoplanar,type,
coplanar:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_c,type,
c: real > a ).
% Relevant facts (255)
thf(fact_0_assms_I2_J,axiom,
( ( path_pathfinish @ a @ c )
= ( path_pathstart @ a @ c ) ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
path_simple_path @ a @ c ).
% assms(1)
thf(fact_2_a1,axiom,
path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ).
% a1
thf(fact_3_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F: B > A,G: C > B,X: C] : ( F @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_4_pathfinish__compose,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo503727757_space @ B )
& ( topolo503727757_space @ A ) )
=> ! [F2: B > A,P: real > B] :
( ( path_pathfinish @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
= ( F2 @ ( path_pathfinish @ B @ P ) ) ) ) ).
% pathfinish_compose
thf(fact_5_pathstart__compose,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo503727757_space @ B )
& ( topolo503727757_space @ A ) )
=> ! [F2: B > A,P: real > B] :
( ( path_pathstart @ A @ ( comp @ B @ A @ real @ F2 @ P ) )
= ( F2 @ ( path_pathstart @ B @ P ) ) ) ) ).
% pathstart_compose
thf(fact_6_complex__of__bounded__linear,axiom,
real_V1632203528linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% complex_of_bounded_linear
thf(fact_7_complex__of__linear,axiom,
real_Vector_linear @ a @ complex @ ( poinca1531302022lex_of @ a ) ).
% complex_of_linear
thf(fact_8_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F: B > C,G: A > B,X: A] : ( F @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_9_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_10_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_11_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_12_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X2: C,F3: D > A,G3: E > D,X3: E] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
= ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_13_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_14_pathstart__linear__image__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A )
& ( topolo503727757_space @ A )
& ( real_V1076094709vector @ B )
& ( topolo503727757_space @ B ) )
=> ! [F2: A > B,G2: real > A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( path_pathstart @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
= ( F2 @ ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% pathstart_linear_image_eq
thf(fact_15_pathfinish__linear__image,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A )
& ( topolo503727757_space @ A )
& ( real_V1076094709vector @ B )
& ( topolo503727757_space @ B ) )
=> ! [F2: A > B,G2: real > A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( path_pathfinish @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
= ( F2 @ ( path_pathfinish @ A @ G2 ) ) ) ) ) ).
% pathfinish_linear_image
thf(fact_16_bounded__linear_Olinear,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ! [F2: A > B] :
( ( real_V1632203528linear @ A @ B @ F2 )
=> ( real_Vector_linear @ A @ B @ F2 ) ) ) ).
% bounded_linear.linear
thf(fact_17_linear__conv__bounded__linear,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( real_V55928688vector @ B ) )
=> ( ( real_Vector_linear @ A @ B )
= ( real_V1632203528linear @ A @ B ) ) ) ).
% linear_conv_bounded_linear
thf(fact_18_linear__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A )
& ( real_V1076094709vector @ B )
& ( real_V1076094709vector @ C ) )
=> ! [F2: A > B,G2: B > C] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_linear @ B @ C @ G2 )
=> ( real_Vector_linear @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) ) ) ) ) ).
% linear_compose
thf(fact_19__092_060open_062_092_060And_062g_O_Ainj_Acomplex__of_A_092_060Longrightarrow_062_Asimple__path_A_Icomplex__of_A_092_060circ_062_Ag_J_A_061_Asimple__path_Ag_092_060close_062,axiom,
! [G2: real > a] :
( ( inj_on @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) )
=> ( ( path_simple_path @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ G2 ) )
= ( path_simple_path @ a @ G2 ) ) ) ).
% \<open>\<And>g. inj complex_of \<Longrightarrow> simple_path (complex_of \<circ> g) = simple_path g\<close>
thf(fact_20_pathfinish__rectpath,axiom,
! [A1: complex,A3: complex] :
( ( path_pathfinish @ complex @ ( path_rectpath @ A1 @ A3 ) )
= A1 ) ).
% pathfinish_rectpath
thf(fact_21_pathstart__rectpath,axiom,
! [A1: complex,A3: complex] :
( ( path_pathstart @ complex @ ( path_rectpath @ A1 @ A3 ) )
= A1 ) ).
% pathstart_rectpath
thf(fact_22_bounded__linear_Obounded__linear,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ! [F2: A > B] :
( ( real_V1632203528linear @ A @ B @ F2 )
=> ( real_V1632203528linear @ A @ B @ F2 ) ) ) ).
% bounded_linear.bounded_linear
thf(fact_23_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F2: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F2 @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_24_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_25_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F2: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H )
= ( comp @ B @ D @ A @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_26_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_27_injD,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ).
% injD
thf(fact_28_injI,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ! [X4: A,Y2: A] :
( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_29_inj__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ).
% inj_eq
thf(fact_30_inj__def,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X: A,Y3: A] :
( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ).
% inj_def
thf(fact_31_inj__onD,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_32_inj__onI,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B] :
( ! [X4: A,Y2: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
=> ( X4 = Y2 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% inj_onI
thf(fact_33_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F: A > B,A5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_34_inj__on__cong,axiom,
! [B: $tType,A: $tType,A4: set @ A,F2: A > B,G2: A > B] :
( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ( F2 @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( inj_on @ A @ B @ F2 @ A4 )
= ( inj_on @ A @ B @ G2 @ A4 ) ) ) ).
% inj_on_cong
thf(fact_35_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_36_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( X2 != Y )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_37_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A4: set @ A,G2: B > A,F2: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( G2 @ ( F2 @ X4 ) )
= X4 ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% inj_on_inverseI
thf(fact_38_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: C > A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ C @ A @ G2 @ ( top_top @ ( set @ C ) ) )
=> ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F2 @ G2 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% inj_compose
thf(fact_39_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,F2: A > C,A4: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F2 ) @ A4 )
=> ( inj_on @ A @ C @ F2 @ A4 ) ) ).
% inj_on_imageI2
thf(fact_40_simple__path__linear__image__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F2: A > B,G2: real > A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( path_simple_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
= ( path_simple_path @ A @ G2 ) ) ) ) ) ).
% simple_path_linear_image_eq
thf(fact_41_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F2 ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% fun.inj_map
thf(fact_42_real__of__bounded__linear,axiom,
real_V1632203528linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% real_of_bounded_linear
thf(fact_43_real__of__linear,axiom,
real_Vector_linear @ complex @ a @ ( poinca179318102eal_of @ a ) ).
% real_of_linear
thf(fact_44_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G2 @ X4 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_50_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_51_bounded__linear_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_V1072672427axioms @ A @ B @ F2 )
=> ( real_V1632203528linear @ A @ B @ F2 ) ) ) ) ).
% bounded_linear.intro
thf(fact_52_bounded__linear__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ( ( real_V1632203528linear @ A @ B )
= ( ^ [F: A > B] :
( ( real_Vector_linear @ A @ B @ F )
& ( real_V1072672427axioms @ A @ B @ F ) ) ) ) ) ).
% bounded_linear_def
thf(fact_53_arc__linear__image__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F2: A > B,G2: real > A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( path_arc @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
= ( path_arc @ A @ G2 ) ) ) ) ) ).
% arc_linear_image_eq
thf(fact_54_real__of__inj,axiom,
inj_on @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) ).
% real_of_inj
thf(fact_55_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_56_arc__imp__simple__path,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_arc @ A @ G2 )
=> ( path_simple_path @ A @ G2 ) ) ) ).
% arc_imp_simple_path
thf(fact_57_arc__distinct__ends,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_arc @ A @ G2 )
=> ( ( path_pathfinish @ A @ G2 )
!= ( path_pathstart @ A @ G2 ) ) ) ) ).
% arc_distinct_ends
thf(fact_58_bounded__linear_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ! [F2: A > B] :
( ( real_V1632203528linear @ A @ B @ F2 )
=> ( real_V1072672427axioms @ A @ B @ F2 ) ) ) ).
% bounded_linear.axioms(2)
thf(fact_59_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_60_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_61_simple__path__imp__arc,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_simple_path @ A @ G2 )
=> ( ( ( path_pathfinish @ A @ G2 )
!= ( path_pathstart @ A @ G2 ) )
=> ( path_arc @ A @ G2 ) ) ) ) ).
% simple_path_imp_arc
thf(fact_62_simple__path__eq__arc,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( ( path_pathfinish @ A @ G2 )
!= ( path_pathstart @ A @ G2 ) )
=> ( ( path_simple_path @ A @ G2 )
= ( path_arc @ A @ G2 ) ) ) ) ).
% simple_path_eq_arc
thf(fact_63_simple__path__cases,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_simple_path @ A @ G2 )
=> ( ( path_arc @ A @ G2 )
| ( ( path_pathfinish @ A @ G2 )
= ( path_pathstart @ A @ G2 ) ) ) ) ) ).
% simple_path_cases
thf(fact_64_arc__simple__path,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( ( path_arc @ A )
= ( ^ [G: real > A] :
( ( path_simple_path @ A @ G )
& ( ( path_pathfinish @ A @ G )
!= ( path_pathstart @ A @ G ) ) ) ) ) ) ).
% arc_simple_path
thf(fact_65_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_66_complex__of__real__of,axiom,
( ( comp @ a @ complex @ complex @ ( poinca1531302022lex_of @ a ) @ ( poinca179318102eal_of @ a ) )
= ( id @ complex ) ) ).
% complex_of_real_of
thf(fact_67_real__of__complex__of,axiom,
( ( comp @ complex @ a @ a @ ( poinca179318102eal_of @ a ) @ ( poinca1531302022lex_of @ a ) )
= ( id @ a ) ) ).
% real_of_complex_of
thf(fact_68_real__of__bij,axiom,
bij_betw @ complex @ a @ ( poinca179318102eal_of @ a ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ a ) ) ).
% real_of_bij
thf(fact_69_complex__of__bij,axiom,
bij_betw @ a @ complex @ ( poinca1531302022lex_of @ a ) @ ( top_top @ ( set @ a ) ) @ ( top_top @ ( set @ complex ) ) ).
% complex_of_bij
thf(fact_70_linear__injective__left__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A )
& ( real_V1076094709vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ? [G4: B > A] :
( ( real_Vector_linear @ B @ A @ G4 )
& ( ( comp @ B @ A @ A @ G4 @ F2 )
= ( id @ A ) ) ) ) ) ) ).
% linear_injective_left_inverse
thf(fact_71_path__linear__image__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F2: A > B,G2: real > A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( path_path @ B @ ( comp @ A @ B @ real @ F2 @ G2 ) )
= ( path_path @ A @ G2 ) ) ) ) ) ).
% path_linear_image_eq
thf(fact_72_linear__inj__iff__eq__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A )
& ( real_V1076094709vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X: A] :
( ( ( F2 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_inj_iff_eq_0
thf(fact_73_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X2: C,N: D > A,H: C > D,F2: A > E] :
( ( ( M @ ( G2 @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X2 )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_74_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_apply
thf(fact_75_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_76_path__rectpath,axiom,
! [A2: complex,B2: complex] : ( path_path @ complex @ ( path_rectpath @ A2 @ B2 ) ) ).
% path_rectpath
thf(fact_77_comp__id,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ A @ B @ A @ F2 @ ( id @ A ) )
= F2 ) ).
% comp_id
thf(fact_78_id__comp,axiom,
! [B: $tType,A: $tType,G2: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
= G2 ) ).
% id_comp
thf(fact_79_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T )
= T ) ).
% fun.map_id
thf(fact_80_bij__betw__id,axiom,
! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% bij_betw_id
thf(fact_81_bij__id,axiom,
! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% bij_id
thf(fact_82_DEADID_Oin__rel,axiom,
! [B: $tType] :
( ( ^ [Y4: B,Z: B] : Y4 = Z )
= ( ^ [A7: B,B3: B] :
? [Z2: B] :
( ( member @ B @ Z2 @ ( top_top @ ( set @ B ) ) )
& ( ( id @ B @ Z2 )
= A7 )
& ( ( id @ B @ Z2 )
= B3 ) ) ) ) ).
% DEADID.in_rel
thf(fact_83_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P2: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P2 )
= P2 ) ).
% top_conj(1)
thf(fact_84_top__conj_I2_J,axiom,
! [A: $tType,P2: $o,X2: A] :
( ( P2
& ( top_top @ ( A > $o ) @ X2 ) )
= P2 ) ).
% top_conj(2)
thf(fact_85_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_86_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_def
thf(fact_87_bij__betwE,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ B @ ( F2 @ X5 ) @ B4 ) ) ) ).
% bij_betwE
thf(fact_88_eq__id__iff,axiom,
! [A: $tType,F2: A > A] :
( ( ! [X: A] :
( ( F2 @ X )
= X ) )
= ( F2
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_89_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B4 @ A4 ) ) ).
% bij_betw_inv
thf(fact_90_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B,G2: A > B,A8: set @ B] :
( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ( F2 @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
= ( bij_betw @ A @ B @ G2 @ A4 @ A8 ) ) ) ).
% bij_betw_cong
thf(fact_91_bij__betw__apply,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,A2: A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( member @ B @ ( F2 @ A2 ) @ B4 ) ) ) ).
% bij_betw_apply
thf(fact_92_bij__betw__id__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B4 )
= ( A4 = B4 ) ) ).
% bij_betw_id_iff
thf(fact_93_bij__betw__iff__bijections,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
? [G: B > A] :
( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ( member @ B @ ( F @ X ) @ B5 )
& ( ( G @ ( F @ X ) )
= X ) ) )
& ! [X: B] :
( ( member @ B @ X @ B5 )
=> ( ( member @ A @ ( G @ X ) @ A5 )
& ( ( F @ ( G @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_94_o__bij,axiom,
! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
( ( ( comp @ B @ A @ A @ G2 @ F2 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ F2 @ G2 )
= ( id @ B ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% o_bij
thf(fact_95_bij__pointE,axiom,
! [B: $tType,A: $tType,F2: A > B,Y: B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X4: A] :
( ( Y
= ( F2 @ X4 ) )
=> ~ ! [X6: A] :
( ( Y
= ( F2 @ X6 ) )
=> ( X6 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_96_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,B4: set @ B,G2: B > C,C3: set @ C] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( ( bij_betw @ B @ C @ G2 @ B4 @ C3 )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 @ C3 ) ) ) ).
% bij_betw_trans
thf(fact_97_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,A4: set @ A,A8: set @ B,F3: B > C,A9: set @ C] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
=> ( ( bij_betw @ B @ C @ F3 @ A8 @ A9 )
= ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 @ A9 ) ) ) ).
% bij_betw_comp_iff
thf(fact_98_bij__betw__imp__inj__on,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ).
% bij_betw_imp_inj_on
thf(fact_99_pointfree__idE,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_100_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_101_arc__imp__path,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_arc @ A @ G2 )
=> ( path_path @ A @ G2 ) ) ) ).
% arc_imp_path
thf(fact_102_inj__on__id,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% inj_on_id
thf(fact_103_simple__path__imp__path,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [G2: real > A] :
( ( path_simple_path @ A @ G2 )
=> ( path_path @ A @ G2 ) ) ) ).
% simple_path_imp_path
thf(fact_104_module__hom__id,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ( real_Vector_linear @ A @ A @ ( id @ A ) ) ) ).
% module_hom_id
thf(fact_105_linear__simps_I3_J,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V55928688vector @ B )
& ( real_V55928688vector @ A ) )
=> ! [F2: A > B] :
( ( real_V1632203528linear @ A @ B @ F2 )
=> ( ( F2 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_simps(3)
thf(fact_106_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V1076094709vector @ B )
& ( real_V1076094709vector @ A ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_107_pathstart__def,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( ( path_pathstart @ A )
= ( ^ [G: real > A] : ( G @ ( zero_zero @ real ) ) ) ) ) ).
% pathstart_def
thf(fact_108_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G2: B > C] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ B @ C @ G2 @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% bij_comp
thf(fact_109_bij__is__inj,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_is_inj
thf(fact_110_right__inverse__linear,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A,G2: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( ( comp @ A @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% right_inverse_linear
thf(fact_111_linear__inverse__left,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A,F3: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( real_Vector_linear @ A @ A @ F3 )
=> ( ( ( comp @ A @ A @ A @ F2 @ F3 )
= ( id @ A ) )
= ( ( comp @ A @ A @ A @ F3 @ F2 )
= ( id @ A ) ) ) ) ) ) ).
% linear_inverse_left
thf(fact_112_left__inverse__linear,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A,G2: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( ( comp @ A @ A @ A @ G2 @ F2 )
= ( id @ A ) )
=> ( real_Vector_linear @ A @ A @ G2 ) ) ) ) ).
% left_inverse_linear
thf(fact_113_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_114_left__right__inverse__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,G2: A > B,H: B > A] :
( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G2 @ H )
= ( id @ B ) )
=> ( F2 = H ) ) ) ).
% left_right_inverse_eq
thf(fact_115_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_116_isomorphism__expand,axiom,
! [A: $tType,B: $tType,F2: B > A,G2: A > B] :
( ( ( ( comp @ B @ A @ A @ F2 @ G2 )
= ( id @ A ) )
& ( ( comp @ A @ B @ B @ G2 @ F2 )
= ( id @ B ) ) )
= ( ! [X: A] :
( ( F2 @ ( G2 @ X ) )
= X )
& ! [X: B] :
( ( G2 @ ( F2 @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_117_bijI_H,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ! [X4: A,Y2: A] :
( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
= ( X4 = Y2 ) )
=> ( ! [Y2: B] :
? [X5: A] :
( Y2
= ( F2 @ X5 ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI'
thf(fact_118_bij__iff,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ! [X: B] :
? [Y3: A] :
( ( ( F2 @ Y3 )
= X )
& ! [Z2: A] :
( ( ( F2 @ Z2 )
= X )
=> ( Z2 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_119_c1__on__open__R2_Ocomplex__of__real__of,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( ( comp @ A @ complex @ complex @ ( poinca1531302022lex_of @ A ) @ ( poinca179318102eal_of @ A ) )
= ( id @ complex ) ) ) ) ).
% c1_on_open_R2.complex_of_real_of
thf(fact_120_c1__on__open__R2_Oreal__of__bij,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( bij_betw @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% c1_on_open_R2.real_of_bij
thf(fact_121_c1__on__open__R2_Oreal__of__bounded__linear,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( real_V1632203528linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% c1_on_open_R2.real_of_bounded_linear
thf(fact_122_c1__on__open__R2_Ocomplex__of__bounded__linear,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( real_V1632203528linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% c1_on_open_R2.complex_of_bounded_linear
thf(fact_123_c1__on__open__R2_Ocomplex__of__linear,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( real_Vector_linear @ A @ complex @ ( poinca1531302022lex_of @ A ) ) ) ) ).
% c1_on_open_R2.complex_of_linear
thf(fact_124_c1__on__open__R2_Oreal__of__linear,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( real_Vector_linear @ complex @ A @ ( poinca179318102eal_of @ A ) ) ) ) ).
% c1_on_open_R2.real_of_linear
thf(fact_125_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X2: C,H: D > A,K: C > D] :
( ( ( F2 @ ( G2 @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X2 )
= ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_126_c1__on__open__R2_Oreal__of__inj,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( inj_on @ complex @ A @ ( poinca179318102eal_of @ A ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% c1_on_open_R2.real_of_inj
thf(fact_127_c1__on__open__R2_Oreal__of__complex__of,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( ( comp @ complex @ A @ A @ ( poinca179318102eal_of @ A ) @ ( poinca1531302022lex_of @ A ) )
= ( id @ A ) ) ) ) ).
% c1_on_open_R2.real_of_complex_of
thf(fact_128_c1__on__open__R2_Ocomplex__of__bij,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ! [F2: A > A,F3: A > ( bounde2145540817linfun @ A @ A ),X7: set @ A] :
( ( poinca1076805489pen_R2 @ A @ F2 @ F3 @ X7 )
=> ( bij_betw @ A @ complex @ ( poinca1531302022lex_of @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ complex ) ) ) ) ) ).
% c1_on_open_R2.complex_of_bij
thf(fact_129_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F2: A > B,Z3: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 @ Z3 )
= ( id @ A @ Z3 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_130_linear__surjective__right__inverse,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V1076094709vector @ B )
& ( real_V1076094709vector @ A ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ? [G4: B > A] :
( ( real_Vector_linear @ B @ A @ G4 )
& ( ( comp @ A @ B @ B @ F2 @ G4 )
= ( id @ B ) ) ) ) ) ) ).
% linear_surjective_right_inverse
thf(fact_131_iso__tuple__update__accessor__eq__assist__idI,axiom,
! [A: $tType,V3: A,F2: A > A,V: A] :
( ( V3
= ( F2 @ V ) )
=> ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F2 @ V3 @ V ) ) ).
% iso_tuple_update_accessor_eq_assist_idI
thf(fact_132_bij__swap__compose__bij,axiom,
! [A: $tType,B: $tType,P: A > B,A2: B,B2: B] :
( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% bij_swap_compose_bij
thf(fact_133_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B,A4: set @ B] :
( ( B2
= ( F2 @ X2 ) )
=> ( ( member @ B @ X2 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_134_swap__nilpotent,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,F2: A > B] :
( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) )
= F2 ) ).
% swap_nilpotent
thf(fact_135_swap__self,axiom,
! [B: $tType,A: $tType,A2: A,F2: A > B] :
( ( swap @ A @ B @ A2 @ A2 @ F2 )
= F2 ) ).
% swap_self
thf(fact_136_swap__apply_I3_J,axiom,
! [A: $tType,B: $tType,C2: B,A2: B,B2: B,F2: B > A] :
( ( C2 != A2 )
=> ( ( C2 != B2 )
=> ( ( swap @ B @ A @ A2 @ B2 @ F2 @ C2 )
= ( F2 @ C2 ) ) ) ) ).
% swap_apply(3)
thf(fact_137_image__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_138_swap__image__eq,axiom,
! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
( ( member @ A @ A2 @ A4 )
=> ( ( member @ A @ B2 @ A4 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
= ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% swap_image_eq
thf(fact_139_inj__on__swap__iff,axiom,
! [B: $tType,A: $tType,A2: A,A4: set @ A,B2: A,F2: A > B] :
( ( member @ A @ A2 @ A4 )
=> ( ( member @ A @ B2 @ A4 )
=> ( ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 )
= ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% inj_on_swap_iff
thf(fact_140_bij__betw__swap__iff,axiom,
! [A: $tType,B: $tType,X2: A,A4: set @ A,Y: A,F2: A > B,B4: set @ B] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( bij_betw @ A @ B @ ( swap @ A @ B @ X2 @ Y @ F2 ) @ A4 @ B4 )
= ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ) ).
% bij_betw_swap_iff
thf(fact_141_swap__comp__involutory,axiom,
! [B: $tType,A: $tType,A2: A,B2: A] :
( ( comp @ ( A > B ) @ ( A > B ) @ ( A > B ) @ ( swap @ A @ B @ A2 @ B2 ) @ ( swap @ A @ B @ A2 @ B2 ) )
= ( id @ ( A > B ) ) ) ).
% swap_comp_involutory
thf(fact_142_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A2: B,B2: B,F2: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_143_bij__swap__iff,axiom,
! [A: $tType,B: $tType,A2: A,B2: A,F2: A > B] :
( ( bij_betw @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ).
% bij_swap_iff
thf(fact_144_swap__id__idempotent,axiom,
! [A: $tType,A2: A,B2: A] :
( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
= ( id @ A ) ) ).
% swap_id_idempotent
thf(fact_145_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_146_swap__id__eq,axiom,
! [A: $tType,X2: A,A2: A,B2: A] :
( ( ( X2 = A2 )
=> ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
= B2 ) )
& ( ( X2 != A2 )
=> ( ( ( X2 = B2 )
=> ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
= A2 ) )
& ( ( X2 != B2 )
=> ( ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) @ X2 )
= X2 ) ) ) ) ) ).
% swap_id_eq
thf(fact_147_comp__swap,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,A2: A,B2: A,G2: A > C] :
( ( comp @ C @ B @ A @ F2 @ ( swap @ A @ C @ A2 @ B2 @ G2 ) )
= ( swap @ A @ B @ A2 @ B2 @ ( comp @ C @ B @ A @ F2 @ G2 ) ) ) ).
% comp_swap
thf(fact_148_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B,G2: C > A,A4: set @ C,X2: B] :
( ( inj_on @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) )
=> ( ( inj_on @ C @ A @ G2 @ A4 )
=> ( ( member @ B @ X2 @ ( image @ A @ B @ F2 @ ( image @ C @ A @ G2 @ A4 ) ) )
=> ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F2 @ G2 ) @ X2 )
= ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G2 ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G2 @ A4 ) @ F2 ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_149_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: B > A,A4: set @ B,G2: C > A,B4: set @ C,H: A > D] :
( ( ( image @ B @ A @ F2 @ A4 )
= ( image @ C @ A @ G2 @ B4 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F2 ) @ A4 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_150_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,R: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G2 @ R ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_151_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( ( image @ A @ B @ F2 @ A4 )
= B4 ) ) ).
% bij_betw_imp_surj_on
thf(fact_152_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: B,B2: B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_153_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_154_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,Y: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ B @ Y @ ( image @ A @ B @ F2 @ A4 ) )
=> ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_155_swap__commute,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A7: A,B3: A] : ( swap @ A @ B @ B3 @ A7 ) ) ) ).
% swap_commute
thf(fact_156_swap__triple,axiom,
! [B: $tType,A: $tType,A2: A,C2: A,B2: A,F2: A > B] :
( ( A2 != C2 )
=> ( ( B2 != C2 )
=> ( ( swap @ A @ B @ A2 @ B2 @ ( swap @ A @ B @ B2 @ C2 @ ( swap @ A @ B @ A2 @ B2 @ F2 ) ) )
= ( swap @ A @ B @ A2 @ C2 @ F2 ) ) ) ) ).
% swap_triple
thf(fact_157_iso__tuple__update__accessor__eq__assist__triv,axiom,
! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
=> ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 ) ) ).
% iso_tuple_update_accessor_eq_assist_triv
thf(fact_158_update__accessor__accessor__eqE,axiom,
! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
=> ( ( Ac @ V )
= X2 ) ) ).
% update_accessor_accessor_eqE
thf(fact_159_update__accessor__updator__eqE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac: B > A,V: B,F2: A > A,V3: B,X2: A] :
( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac @ V @ F2 @ V3 @ X2 )
=> ( ( Upd @ F2 @ V )
= V3 ) ) ).
% update_accessor_updator_eqE
thf(fact_160_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,B2: B,F2: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( ( B2
= ( F2 @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_161_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A4 ) )
=> ( P2 @ X4 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A4 )
=> ( P2 @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_162_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F2: A > B,G2: A > B] :
( ( M = N )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image @ A @ B @ F2 @ M )
= ( image @ A @ B @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_163_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A4 ) )
& ( P2 @ X5 ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P2 @ ( F2 @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_164_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F2: B > A,A4: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F2 @ A4 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A4 )
& ( Z3
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_165_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A4: set @ A,F2: A > B] :
( ( member @ A @ X2 @ A4 )
=> ( member @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% imageI
thf(fact_166_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,G2: A > B,F2: A > A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A4 )
=> ( ( ( G2 @ ( F2 @ X4 ) )
= ( G2 @ ( F2 @ Xa ) ) )
= ( ( G2 @ X4 )
= ( G2 @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F2 @ A4 )
=> ( ( inj_on @ A @ B @ G2 @ ( image @ A @ A @ F2 @ A4 ) )
= ( inj_on @ A @ B @ G2 @ A4 ) ) ) ) ).
% inj_on_image_iff
thf(fact_167_surjD,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X4: B] :
( Y
= ( F2 @ X4 ) ) ) ).
% surjD
thf(fact_168_surjE,axiom,
! [A: $tType,B: $tType,F2: B > A,Y: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X4: B] :
( Y
!= ( F2 @ X4 ) ) ) ).
% surjE
thf(fact_169_surjI,axiom,
! [B: $tType,A: $tType,G2: B > A,F2: A > B] :
( ! [X4: A] :
( ( G2 @ ( F2 @ X4 ) )
= X4 )
=> ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_170_surj__def,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X: B] :
( Y3
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_171_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_172_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X2: B] :
( ( B2
= ( F2 @ X2 ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_173_inj__on__imp__inj__on__swap,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A2: A,B2: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ( member @ A @ B2 @ A4 )
=> ( inj_on @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F2 ) @ A4 ) ) ) ) ).
% inj_on_imp_inj_on_swap
thf(fact_174_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ( F2 @ X2 )
= Y )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( the_inv_into @ A @ B @ A4 @ F2 @ Y )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_175_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( the_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_176_f__the__inv__into__f__bij__betw,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B,X2: B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( ( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( member @ B @ X2 @ B4 ) )
=> ( ( F2 @ ( the_inv_into @ A @ B @ A4 @ F2 @ X2 ) )
= X2 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_177_bij__betw__the__inv__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% bij_betw_the_inv_into
thf(fact_178_swap__general,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
( ( A2 != B2 )
=> ( ( C2 != D2 )
=> ( ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
= ( id @ A ) )
| ? [X4: A,Y2: A,Z4: A] :
( ( X4 != A2 )
& ( Y2 != A2 )
& ( Z4 != A2 )
& ( X4 != Y2 )
& ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
= ( comp @ A @ A @ A @ ( swap @ A @ A @ X4 @ Y2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ Z4 @ ( id @ A ) ) ) ) ) ) ) ) ).
% swap_general
thf(fact_179_swap__id__common,axiom,
! [A: $tType,A2: A,C2: A,B2: A] :
( ( A2 != C2 )
=> ( ( B2 != C2 )
=> ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) )
= ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% swap_id_common
thf(fact_180_swap__id__common_H,axiom,
! [A: $tType,A2: A,B2: A,C2: A] :
( ( A2 != B2 )
=> ( ( A2 != C2 )
=> ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) )
= ( comp @ A @ A @ A @ ( swap @ A @ A @ B2 @ C2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ).
% swap_id_common'
thf(fact_181_swap__id__independent,axiom,
! [A: $tType,A2: A,C2: A,D2: A,B2: A] :
( ( A2 != C2 )
=> ( ( A2 != D2 )
=> ( ( B2 != C2 )
=> ( ( B2 != D2 )
=> ( ( comp @ A @ A @ A @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) )
= ( comp @ A @ A @ A @ ( swap @ A @ A @ C2 @ D2 @ ( id @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ) ) ) ) ) ).
% swap_id_independent
thf(fact_182_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Xa2: D > A,F2: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F2 @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_183_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,F2: A > B,G2: A > B] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z4 )
= ( G2 @ Z4 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ G2 @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_184_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F2: A > B,G2: A > B] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F2 @ Z4 )
= ( G2 @ Z4 ) ) )
=> ( ( comp @ A @ B @ D @ F2 @ X2 )
= ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_185_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F2: A > B,V: D > A] :
( ( image @ D @ B @ ( comp @ A @ B @ D @ F2 @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image @ A @ B @ F2 @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_186_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G2: A > C] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ C @ G2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image @ B @ C @ ( comp @ A @ C @ B @ G2 @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_187_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A4 ) )
= ( member @ A @ A2 @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_188_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F2 @ A4 )
= ( image @ A @ B @ F2 @ B4 ) )
= ( A4 = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_189_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,B2: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X: A] :
( ( B2
= ( F2 @ X ) )
& ! [Y3: A] :
( ( B2
= ( F2 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_190_surj__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_191_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,F3: B > C] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( inj_on @ B @ C @ F3 @ ( image @ A @ B @ F2 @ A4 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F2 ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_192_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G2: C > B,F2: A > C,A4: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G2 @ F2 ) @ A4 )
=> ( inj_on @ C @ B @ G2 @ ( image @ A @ C @ F2 @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_193_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B,A4: set @ A,G2: B > C] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( inj_on @ B @ C @ G2 @ ( image @ A @ B @ F2 @ A4 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G2 @ F2 ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_194_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_betw_imp_surj
thf(fact_195_bij__is__surj,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_is_surj
thf(fact_196_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( bij_betw @ A @ B @ F2 @ A4 @ ( image @ A @ B @ F2 @ A4 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_197_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ( image @ A @ B @ F2 @ A4 )
= B4 )
=> ( bij_betw @ A @ B @ F2 @ A4 @ B4 ) ) ) ).
% bij_betw_imageI
thf(fact_198_bij__betw__def,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
& ( ( image @ A @ B @ F @ A5 )
= B5 ) ) ) ) ).
% bij_betw_def
thf(fact_199_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_200_bij__def,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
& ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ) ).
% bij_def
thf(fact_201_bijI,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI
thf(fact_202_linear__surj__imp__inj,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% linear_surj_imp_inj
thf(fact_203_linear__inj__imp__surj,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% linear_inj_imp_surj
thf(fact_204_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > A,X7: set @ B,G1: A > C,G22: A > C] :
( ( ( image @ B @ A @ F2 @ X7 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ X7 )
=> ( ( comp @ A @ C @ B @ G1 @ F2 @ X4 )
= ( comp @ A @ C @ B @ G22 @ F2 @ X4 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_205_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A4: set @ A] :
( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_206_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A4: set @ A] :
( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_207_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
( ( Inf @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_208_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X7: set @ A,F2: A > B,Y5: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X7 )
=> ( ( ( F2 @ X4 )
= ( F2 @ Y2 ) )
= ( X4 = Y2 ) ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( member @ B @ ( F2 @ X4 ) @ Y5 ) )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ Y5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X7 )
& ( Y2
= ( F2 @ X5 ) ) ) )
=> ( bij_betw @ A @ B @ F2 @ X7 @ Y5 ) ) ) ) ).
% bij_betwI'
thf(fact_209_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G2: B > A,F2: C > B,A4: set @ C] :
( ( Sup @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F2 @ A4 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F2 ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_210_inj__adjoint__iff__surj,axiom,
! [M2: $tType,N2: $tType] :
( ( ( euclid925273238_space @ N2 )
& ( euclid925273238_space @ M2 ) )
=> ! [F2: M2 > N2] :
( ( real_Vector_linear @ M2 @ N2 @ F2 )
=> ( ( inj_on @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
= ( ( image @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) )
= ( top_top @ ( set @ N2 ) ) ) ) ) ) ).
% inj_adjoint_iff_surj
thf(fact_211_surj__adjoint__iff__inj,axiom,
! [N2: $tType,M2: $tType] :
( ( ( euclid925273238_space @ M2 )
& ( euclid925273238_space @ N2 ) )
=> ! [F2: M2 > N2] :
( ( real_Vector_linear @ M2 @ N2 @ F2 )
=> ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
= ( top_top @ ( set @ M2 ) ) )
= ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% surj_adjoint_iff_inj
thf(fact_212_adjoint__adjoint,axiom,
! [M2: $tType,N2: $tType] :
( ( ( euclid925273238_space @ N2 )
& ( euclid925273238_space @ M2 ) )
=> ! [F2: N2 > M2] :
( ( real_Vector_linear @ N2 @ M2 @ F2 )
=> ( ( linear_adjoint @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) )
= F2 ) ) ) ).
% adjoint_adjoint
thf(fact_213_adjoint__linear,axiom,
! [M2: $tType,N2: $tType] :
( ( ( euclid925273238_space @ N2 )
& ( euclid925273238_space @ M2 ) )
=> ! [F2: N2 > M2] :
( ( real_Vector_linear @ N2 @ M2 @ F2 )
=> ( real_Vector_linear @ M2 @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 ) ) ) ) ).
% adjoint_linear
thf(fact_214_linear__surj__adj__imp__inj,axiom,
! [N2: $tType,M2: $tType] :
( ( ( euclid925273238_space @ M2 )
& ( euclid925273238_space @ N2 ) )
=> ! [F2: M2 > N2] :
( ( real_Vector_linear @ M2 @ N2 @ F2 )
=> ( ( ( image @ N2 @ M2 @ ( linear_adjoint @ M2 @ N2 @ F2 ) @ ( top_top @ ( set @ N2 ) ) )
= ( top_top @ ( set @ M2 ) ) )
=> ( inj_on @ M2 @ N2 @ F2 @ ( top_top @ ( set @ M2 ) ) ) ) ) ) ).
% linear_surj_adj_imp_inj
thf(fact_215_coplanar__linear__image__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) )
= ( coplanar @ A @ S ) ) ) ) ) ).
% coplanar_linear_image_eq
thf(fact_216_linear__singular__into__hyperplane,axiom,
! [N2: $tType] :
( ( euclid925273238_space @ N2 )
=> ! [F2: N2 > N2] :
( ( real_Vector_linear @ N2 @ N2 @ F2 )
=> ( ( ~ ( inj_on @ N2 @ N2 @ F2 @ ( top_top @ ( set @ N2 ) ) ) )
= ( ? [A7: N2] :
( ( A7
!= ( zero_zero @ N2 ) )
& ! [X: N2] :
( ( inner_780170721_inner @ N2 @ A7 @ ( F2 @ X ) )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% linear_singular_into_hyperplane
thf(fact_217_adjoint__unique,axiom,
! [A: $tType,B: $tType] :
( ( ( inner_real_inner @ B )
& ( inner_real_inner @ A ) )
=> ! [F2: A > B,G2: B > A] :
( ! [X4: A,Y2: B] :
( ( inner_780170721_inner @ B @ ( F2 @ X4 ) @ Y2 )
= ( inner_780170721_inner @ A @ X4 @ ( G2 @ Y2 ) ) )
=> ( ( linear_adjoint @ A @ B @ F2 )
= G2 ) ) ) ).
% adjoint_unique
thf(fact_218_hyperplane__eq__Ex,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [A2: A,B2: real] :
( ( A2
!= ( zero_zero @ A ) )
=> ~ ! [X4: A] :
( ( inner_780170721_inner @ A @ A2 @ X4 )
!= B2 ) ) ) ).
% hyperplane_eq_Ex
thf(fact_219_adjoint__works,axiom,
! [N2: $tType,M2: $tType] :
( ( ( euclid925273238_space @ M2 )
& ( euclid925273238_space @ N2 ) )
=> ! [F2: N2 > M2,X2: N2,Y: M2] :
( ( real_Vector_linear @ N2 @ M2 @ F2 )
=> ( ( inner_780170721_inner @ N2 @ X2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) )
= ( inner_780170721_inner @ M2 @ ( F2 @ X2 ) @ Y ) ) ) ) ).
% adjoint_works
thf(fact_220_adjoint__clauses_I2_J,axiom,
! [M2: $tType,N2: $tType] :
( ( ( euclid925273238_space @ N2 )
& ( euclid925273238_space @ M2 ) )
=> ! [F2: N2 > M2,Y: M2,X2: N2] :
( ( real_Vector_linear @ N2 @ M2 @ F2 )
=> ( ( inner_780170721_inner @ N2 @ ( linear_adjoint @ N2 @ M2 @ F2 @ Y ) @ X2 )
= ( inner_780170721_inner @ M2 @ Y @ ( F2 @ X2 ) ) ) ) ) ).
% adjoint_clauses(2)
thf(fact_221_vector__eq,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X: A,Y3: A] :
( ( ( inner_780170721_inner @ A @ X @ X )
= ( inner_780170721_inner @ A @ X @ Y3 ) )
& ( ( inner_780170721_inner @ A @ Y3 @ Y3 )
= ( inner_780170721_inner @ A @ X @ X ) ) ) ) ) ) ).
% vector_eq
thf(fact_222_vector__eq__ldot,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [Y: A,Z3: A] :
( ( ! [X: A] :
( ( inner_780170721_inner @ A @ X @ Y )
= ( inner_780170721_inner @ A @ X @ Z3 ) ) )
= ( Y = Z3 ) ) ) ).
% vector_eq_ldot
thf(fact_223_vector__eq__rdot,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X2: A,Y: A] :
( ( ! [Z2: A] :
( ( inner_780170721_inner @ A @ X2 @ Z2 )
= ( inner_780170721_inner @ A @ Y @ Z2 ) ) )
= ( X2 = Y ) ) ) ).
% vector_eq_rdot
thf(fact_224_coplanar__linear__image,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( real_V55928688vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( coplanar @ A @ S )
=> ( ( real_Vector_linear @ A @ B @ F2 )
=> ( coplanar @ B @ ( image @ A @ B @ F2 @ S ) ) ) ) ) ).
% coplanar_linear_image
thf(fact_225_inner__eq__zero__iff,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X2: A] :
( ( ( inner_780170721_inner @ A @ X2 @ X2 )
= ( zero_zero @ real ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% inner_eq_zero_iff
thf(fact_226_inner__zero__right,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X2: A] :
( ( inner_780170721_inner @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% inner_zero_right
thf(fact_227_all__zero__iff,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X2: A] :
( ( ! [U: A] :
( ( inner_780170721_inner @ A @ X2 @ U )
= ( zero_zero @ real ) ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% all_zero_iff
thf(fact_228_inner__zero__left,axiom,
! [A: $tType] :
( ( inner_real_inner @ A )
=> ! [X2: A] :
( ( inner_780170721_inner @ A @ ( zero_zero @ A ) @ X2 )
= ( zero_zero @ real ) ) ) ).
% inner_zero_left
thf(fact_229_aff__dim__injective__linear__image,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid925273238_space @ A )
& ( euclid925273238_space @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( aff_dim @ B @ ( image @ A @ B @ F2 @ S ) )
= ( aff_dim @ A @ S ) ) ) ) ) ).
% aff_dim_injective_linear_image
thf(fact_230_bij__swap__comp,axiom,
! [B: $tType,A: $tType,P: A > B,A2: B,B2: B] :
( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ B @ A @ ( swap @ B @ B @ A2 @ B2 @ ( id @ B ) ) @ P )
= ( swap @ A @ B @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ A2 ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ B2 ) @ P ) ) ) ).
% bij_swap_comp
thf(fact_231_inj__linear__imp__inv__bounded__linear,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A] :
( ( real_V1632203528linear @ A @ A @ F2 )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( real_V1632203528linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% inj_linear_imp_inv_bounded_linear
thf(fact_232_inv__swap__id,axiom,
! [A: $tType,A2: A,B2: A] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) )
= ( swap @ A @ A @ A2 @ B2 @ ( id @ A ) ) ) ).
% inv_swap_id
thf(fact_233_inj__linear__imp__inv__linear,axiom,
! [A: $tType] :
( ( euclid925273238_space @ A )
=> ! [F2: A > A] :
( ( real_Vector_linear @ A @ A @ F2 )
=> ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( real_Vector_linear @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% inj_linear_imp_inv_linear
thf(fact_234_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_235_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B,G2: A > C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ F2 )
= G2 ) ) ).
% o_inv_o_cancel
thf(fact_236_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ) ).
% inv_into_f_f
thf(fact_237_inv__id,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
= ( id @ A ) ) ).
% inv_id
thf(fact_238_inv__equality,axiom,
! [A: $tType,B: $tType,G2: B > A,F2: A > B] :
( ! [X4: A] :
( ( G2 @ ( F2 @ X4 ) )
= X4 )
=> ( ! [Y2: B] :
( ( F2 @ ( G2 @ Y2 ) )
= Y2 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G2 ) ) ) ).
% inv_equality
thf(fact_239_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X2: A,Y: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( ( F2 @ X2 )
= Y )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ Y )
= X2 ) ) ) ) ).
% inv_into_f_eq
thf(fact_240_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B4 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ B4 @ A4 ) ) ).
% bij_betw_inv_into
thf(fact_241_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ( hilbert_inv_into @ B @ A @ A8 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ A2 )
= ( F2 @ A2 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_242_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A8: set @ B,A2: A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
=> ( ( member @ A @ A2 @ A4 )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ A2 ) )
= A2 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_243_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,A8: set @ B,A10: B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A8 )
=> ( ( member @ B @ A10 @ A8 )
=> ( ( F2 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 @ A10 ) )
= A10 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_244_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,Y: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_245_surj__iff__all,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X: A] :
( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_246_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ F2 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ A4 ) )
= A4 ) ) ).
% image_f_inv_f
thf(fact_247_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,G2: A > B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( G2 @ ( F2 @ X4 ) )
= X4 )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= G2 ) ) ) ).
% surj_imp_inv_eq
thf(fact_248_inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ).
% inv_f_f
thf(fact_249_inv__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,X2: A,Y: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X2 )
= Y )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y )
= X2 ) ) ) ).
% inv_f_eq
thf(fact_250_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,G2: B > A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( F2 @ ( G2 @ X4 ) )
= X4 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G2 ) ) ) ).
% inj_imp_inv_eq
thf(fact_251_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_imp_bij_inv
thf(fact_252_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P: A > B,X2: A,Y: B] :
( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( X2
= ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ Y ) )
= ( ( P @ X2 )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_253_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
= F2 ) ) ).
% inv_inv_eq
thf(fact_254_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
% Subclasses (7)
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___Inner__Product_Oreal__inner,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( inner_real_inner @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__vector,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( real_V1076094709vector @ 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___Topological__Spaces_Otopological__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( topolo503727757_space @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Oreal__normed__vector,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( real_V55928688vector @ A ) ) ).
% Type constructors (21)
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A11: $tType,A12: $tType] :
( ( topolo503727757_space @ A12 )
=> ( topolo503727757_space @ ( A11 > A12 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A11: $tType,A12: $tType] :
( ( top @ A12 )
=> ( top @ ( A11 > A12 ) ) ) ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space_1,axiom,
topolo503727757_space @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
! [A11: $tType] : ( top @ ( set @ A11 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_3,axiom,
! [A11: $tType] :
( ( zero @ A11 )
=> ( zero @ ( set @ A11 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_4,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_5,axiom,
top @ $o ).
thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
execut510477386_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_6,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space,axiom,
euclid925273238_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real ).
thf(tcon_Real_Oreal___Inner__Product_Oreal__inner,axiom,
inner_real_inner @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_7,axiom,
zero @ real ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_8,axiom,
real_V55928688vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_9,axiom,
topolo503727757_space @ complex ).
thf(tcon_Complex_Ocomplex___Euclidean__Space_Oeuclidean__space_10,axiom,
euclid925273238_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_11,axiom,
real_V1076094709vector @ complex ).
thf(tcon_Complex_Ocomplex___Inner__Product_Oreal__inner_12,axiom,
inner_real_inner @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_13,axiom,
zero @ complex ).
% Free types (1)
thf(tfree_0,hypothesis,
execut510477386_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( path_pathfinish @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) )
= ( path_pathstart @ complex @ ( comp @ a @ complex @ real @ ( poinca1531302022lex_of @ a ) @ c ) ) ) ).
%------------------------------------------------------------------------------